Question Bank in Math

C O N T E N T S

CHAPTER NO. CHAPTER NAME PAGE NO.

1 Trigonometric Ratios & Identities 1 - 52 Trigonometric Equations & Inverse Circular Functions 6 - 103 Properties and Solution of Triangles 11 - 144 Heights and Distances 15 - 165 Sequence and Progressions 17 - 206 Quadratic Equations and Inequations 21 - 247 Complex Numbers 25 - 298 Permutation and Combination 30 - 329 Binomial Theorem 33 - 3610 Infinite Series 37 - 4011 Straight lines and Pair of Straight lines 41 - 4412 Circles 45 - 4813 Conic Sections - Parabola 49 - 5214 Ellipse and Hyperbola 53 - 5615 Functions, Limit and Continuity 57 - 6216 Differentiability and Differentiation 63 - 6717 Application of Derivatives 68 - 7118 Indefinite Integration 72 - 7719 Definite Integration 78 -8320 Area Under the Curve 84 - 8521 Differential Equations 86 - 8822 Determinants 89 - 9323 Matrices 94 - 9724 Vectors 98 - 10025 Three Dimensional Geometry 101 - 10326 Probability 104 - 10627 Statistics 107 - 11028 Correlation and Regression analysis 111 - 11229 Sets, Relations and Mappings/Functions 113 - 11630 Linear Programming 117 - 11931 Numerical Methods 120 - 121

Quest Tutorials 1Head Office : 44C,Kalusarai New Delhi-16; Ph.(011) 46080363

TRIGONOMETRIC RATIOS & INDENTITIES

1. The value of 1 151 15

2

2

− °+ °

tantan

is

a) 1 b) 3 c) 2/3 d) 2

2. The equation a x b x csin cos ,+ = where | |c a b> +2 2 hasa) a unique solution b) infinite no. of solutions c) no solution d) none of these

3. If sin ,x x+ =cosec 2 then sinn nx x+ cosec is equal toa) 2 b) 2n c) 2n - 1 d) 2n - 2

4. If cos ,A = 34

then 32 sin sinA A2

52

FHGIKJFHGIKJ =

a) 7 b) 8 c) 11 d) none of these

5. The value of sin sin sin sin sin sin sinπ π π π π π π14

314

514

714

914

1114

1314

is

a)1

16 b)1

64 c)1

128 d) none of these.

6. If ( ) ( ) [ ]2/,0,;2/1sin,1sin πβαβαβα ∈=−=+ then tan ( ) tan ( )α β α β+ +2 2 is equal toa) 1 b) -1 c) 0 d) none of these

7. If 1 2 3+ + + + + ∞sin sin sin .... ....x x x is equal to 4 2 3 0+ < < =, ,x xπ then

a)π6 b)

π4 c)

π3 or

π6 d)

π3 or

23π .

8. The value of tan 5θ is

a)5 10

1 10 5

3 5

2 4

tan tan tantan tan

θ θ θθ θ

− +− +

b)5 10

1 10 5

3 5

2 4

tan tan tantan tan

θ θ θθ θ

+ −+ −

c)5 10

1 10 5

5 3

2 4

tan tan tantan tan

θ θ θθ θ

− +− +

d) none of these.

9. For what and only what values of α lying between 0 and π is the inequality αα>αα cossincossin 33 valid ?

a) απ

∈ FHGIKJ0

4,

43,

2ππU b)α

π∈ FHGIKJ0

2, c)α

π π∈ FHG

IKJ4 2

, d) none of these

10. If sin cosA A m+ = and sin cos ,3 3A A n+ = then

a) m m n3 3 0− + = b) n n m3 3 2 0− + =c) m m n3 3 2 0− + = d) m m n3 3 2 0+ + =

11. For x R x x x xn n∈ + + + + FHGIKJ− −, tan tan tan ... tan1

2 212 2

12 22 2 1 1 is equal to


a) 2 cot 2x - 1

2 21 1n nx

− −FHGIKJcot b)

12 2

2 21 1n nx x− −FHGIKJ −cot cot

c) cot cotx xn221−

FHGIKJ − d) none of these

12. If y = −+

sec tansec tan '

2

2

θ θθ θ

then

a) 331

≤≤ y b) y ∉LNMOQP

13

3, c) − < < −3 13

y d) none of these.

13. If δγβα ,,, are the smallest positive angles in ascending order of magnitude which have their sines equal to the positive quantity k, then the

value of 4 sin α β γ δ2

32

22 2

+ + +sin sin sin is equal to

a) 2 1− k b) 2 1+ k c)12+ k

d) none of these

14. The expression cosec2 2 2 2A A A Acot sec tan− − − −(cot tan ) (sec )2 2 2 2 1A A A Acosec is equal to

a) 1 b) -1 c) 0 d) none of these.

15. The values of θ θ( )0 360< < ° satisfying cosec θ + 2 = 0 are

a) 210°, 300° b) 240°, 300° c) 210°, 240° d) 210°, 330°

16. If sin sin ,x x+ =2 1 then the value of cos cos cos cos cos cos ,12 10 8 6 4 23 3 2 2x x x x x x+ + + + + − is equal to

a) 0 b) 1 c) 2 d) sin2 x.

17. If A, B, C are acute positive angles such that A + B + C = π and cot A cot B cot C = K, then

a) K ≤1

3 3b) K ≥

13 3

c) K <19

d) K >13

18. If sinβ is the GM between sinα and cosα , then cos 2β =

a) 24

2sin πα−FHGIKJ b) 2

42cos π

α−FHGIKJ c) 2

42sin π

α+FHGIKJ d) none of these

19. sin2 A + sin2 ( A - B ) + 2 sin A cos B sin ( B - A ) is equal toa) sin2 A b) sin2 B c) cos2 A d) cos2 B

20. The value of the determinant

sin sin tansin tan sintan sin sin

2 2

2 2

2 2

13 77 13577 135 13

135 13 77

° ° °° ° °° ° °

is equal to

a) -1 b) 0 c) 1 d) 2


21. If tan secA A+ = 2 and A is less than 90°, then sin cosA A+ is equal to

a) 2 / 5 b) 3 / 5 c) 4 / 5 d) 7 / 5

22. Let n be a fixed positive integer such that sin ( / ) cos( / ) / ,π π2 2 2n n n+ = then

a) n = 4 b) n = 5 c) n = 6 d) none of these

23. Number of solutions of the equations tan x + sec x = 2 cos x lying in the interval [ 0, 2 π ] is

a) 0 b) 1 c) 2 d) 3

24. If sec tan ,A A− =14

then

a) sin /2 8 17A = b) cos /A = 15 17c) sin cos /A A+ = 23 17 d) cos sin /A A− = 7 17

25. If 24

3 2 02sin ( ) cos ,x x+ + >π

then

a) cos( )26

12

x − > −π

b) sin( )26

12

x − < −π

c) sin( )26

12

x − > −π

d) cos( )26

12

x − < −π

26. sec( )

22

4θ =

+xy

x y, where x R y R∈ ∈, , is true if and only if

a) x y+ ≠ 0 b) x y x= ≠, 0 c) x = y d) x y≠ ≠0 0,

27. If tan ,π9x and tan 5

18π

are in AP and tan ,π9y and tan 7

18π

are also in AP then

a) 2x y= b) x y> c) x y= d) none of these

28. The minimum value of cos cos2θ θ+ for real values of θ is

a) −98 b) 0 c) - 2 (d)none of these

29. If 0 180°< < °θ then 2 2 2 2 1+ + + + +... ( cos )θ , there being n number of 2's, is equal to

a) 22

cos θn b) 2

2 1cos θn− c) 2

2 1cos θn+ d) none of these


30. If tan tanα β2 2

and are the roots of the equation 8 26 15 02x x− + = then cos( )α β+ is equal to

a) −627725 b)

627725 c) - 1 d) none of these

31. If 21

sinsin cos

αα α

λ+ +

= then 11

+ −+

sin cossinα α

α is equal to

a)1λ

b) λ c) 1− λ d) 1+ λ

32. If | tan | ,A < 1 and | A | is acute then 1 2 1 21 2 1 2+ + −+ − −

sin sinsin sin

A AA A

is equal to

a) tan A b) - tan A c) cot A d) - cot A

33. The set of all possible values of α in [ , ]−π π such that 11−+

sinsin

αα

is equal to secα - tanα is

a) 02

, πLNMIKJ b) 0

2 2, ,π π

πLNMIKJ∪FHGIKJ c) [ , ]−π 0 d) −FHG

IKJ

π π2 2

,

34. If tan tan tan tanθ θπ

θπ

θ+ +FHGIKJ + −FHG

IKJ =3 3

3k then k is equal to

a) 1 b) 3 c) 1/3 d) none of these

35. Let n be an odd integer. If sin n brr

r

n

θ θ==∑ sin

0

for all real θ then

a) b b0 11 3= =, b) b b n0 10= =,

c) b b n0 11= − =, d) b b n n0 120 3 3= = − −,

36. The value of tan 63° - cot 63° is equal to

a) 25 1

10 2 5+

−. b)2

5 110 2 5

++. c)

5 14

10 2 5−−. d) none of these

37. The sum of the real roots of cos sin6 4 1x x+ = in the interval − ≤ ≤π πx is equal toa) 0 b) π c) - π d) none of these

38. If ABCD is a convex quadrilateral such that 4 sec A + 5 = 0 then the quadratic equation whose roots are tan A and cosec A is

a) 12 29 15 02x x− + = b) 12 11 15 02x x− − = c) 12 11 15 02x x+ − = d) none of these


ANSWERS

1. c 2. c 3. a 4. c 5. b 6. a 7. d8. a 9. a 10. c 11. b 12. a 13. b 14. c15. d 16. d 17. a 18. a 19. c 20. b 21. d22. c 23. d 24. c 25. a 26. b 27. a 28. a29. a 30. a 31. b 32. c 33. d 34. b 35. b36. a 37. a 38. b 39. b 40. c 41. d 42. a43. d 44. b 45. b 46. c 47. c 48. d

39. =°++°+° 89tanlog....2tanlog1tanloga) 1 b) 0 c) 4/π d) none of these

40. °−° 20sec20eccos3 is equal to

a) 2 b) °° 40sin/20sin2 c) 4 d) °° 40sin/20sin4

41. If 1

tan+

=aaA and

121tan+

=a

B , then the value of BA + is

a) 0 b) 2/π c) 3/π d) 4/π

42. =−− x2tanx3tanx5tana) x2tanx3tanx5tan b) x2cosx3cosx5cosc) x2sinx3sinx5sin d) xtanx2tanx8tan

43. The value of °+°−°−° 9tan27tan63tan81tana) 1 b) 2 c) 3 d) 4

44. If θ+θ= 148 cossinA , then for all values of θ ,a) 1A ≥ b) 1A0 ≤< c) 3A21 ≤< d) none of these

45. If a)sin( =α+θ and b)sin( =β+θ , then )cos(ab4)(2cos β−α−β−α is equal to

a) 22 ba1 −− b) 22 b2a21 −− c) 22 ba2 ++ d) 22 ba2 −−

46. If 1)xtan1(log)xtan1(logxcoslogxsinlog 3333 −=+−−−− , then =x2tana) - 2 b) 3/2 c) 3/2 d) 6

47. If °°°= 40cos20cos10cosx , then the value of x is

a) °10tan)8/1( b) °10eccos)8/1( c) °10cot)8/1( d) °10sec)8/1(

48. If ∑=

=n

0m

mm

3 xcosCx3sinxsin is an identity in x , where n10 C.......,,C,C are constants and 0Cn ≠ , then the value of n is

a) 17 b) 27 c) 16 d) 6


Trigonometric Equations & Inverse Circular Functions

1. The general solution of sin sin sin cos cos cosx x x x x x− + = − +3 2 3 3 2 3 is

a) nπ π+

8b)nπ π2 8+ c) ( )− +FHG

IKJ1

2 8n nπ π d) 2 3

21nπ + FHGIKJ

−cos

2. The general value of θ satisfying the equation 2 3 2 02sin sinθ θ− − = is

a) n nππ

+ −( )16

b) n nππ

+ −( )12

c) n nππ

+ −( )1 56

d) n nππ

+ −( )1 76

3. The set of values of x for which 12tan3tan1

2tan3tan=

+−

xxxx is

a) φ b)π4RSTUVW

c) n nππ

+ =RSTUVW4

1 2 3, , , ,.... d) 24

1 2 3n nππ

+ =RSTUVW, , , ,....

4. If βα, are different values of x satisfying a x b x ccos sin ,+ = then tan α β+FHGIKJ =2

a) a + b b) a - b c) b / a d) a / b.

5. If α is a root of 25 5 12 02

2cos cos , ,θ θπ

α π+ − = < < then α2sin is equal to

a) 2425

b) −2425

c) 1318

d) −1318

6. The number of pairs ( x, y ) satisfying the equations sin x + sin y = sin ( x + y ) and | x | + | y | = 1 isa) 2 b) 4 c) 6 d) infinite.

7. The expression ( 1 + tan x + tan2 x ) ( 1 - cot x + cot2 x ) has the positive values for x, given by

a) 02

≤ ≤x πb) 0 ≤ ≤x π c) for all x R∈ d) x ≥ 0.

8. If the complex numbers (sin x + i cos 2x) and (cos x - i sin 2x) are conjugate to each other, then x is equal to

a) nπ b) n +FHGIKJ

12

π c) 0 d) none of these

9. The values of x & y satisfying the system of equations 2 1 16 42 2sin cos sin cos,x y x y+ += = are given by

a) x n y nn= + − = ±ππ

ππ( )1

62

3and b)

6)1( 1 π−+π= +nnx and

62 π

±π= ny

c) x n y nn= + − = ±ππ

ππ( )1

62 2

3and d) x n y nn= + − = ±+π

ππ

π( )16

2 23

1 and


10. The number of all possible ordered pairs ( x, y ), x y R, ∈ satisfying the system of equations

x y x y+ = + =23

32

π , cos cos is

a) 0 b) 1 c) infinite d) none of these.

11. The value of x between 0 and 2 π which satisfy the equation sin x 8 12cos x = are in AP with common difference

a)π4 b)

π8 c)

38π

d)58π

12. If 16

sin ,cos , tanx x x are GP, then x is equal to

a) n n Zππ

± ∈3

, b) 23

n n Zππ

± ∈, c) n n Znππ

+ − ∈( ) ,13 d) none of these

13. From the identity sin sin sin3 3 4 3x x x= − it follows that if x is real and | x | < 1, then

a) 1|43| 3 >− xx b) 1|43| 3 ≤− xx c) 1|43| 3 <− xx d) Nothing can be said about

3 4 3x x− .

14. 4 15

1239

1 1tan tan− −− is equal to

a) π b) π / 2 c) π / 3 d) π / 4

15. The value of sin ( 2 sin-1 (0.8)) is equal toa) sin 1.2° b) sin 1.6° c) 0.48 d) 0.96

16. If sin ,− −+ FHGIKJ =

1 1

554 2

x cosec π then x =

a) 4 b) 5 c) 1 d) 3

17. The value of sin (cot (cot (tan )))− −1 1 x is

a)xx

2

2

21++

b)xx

2

2

12++

c)12 +x

xd)

122x +

18. The greatest and least values of (sin ) (cos )− −+1 3 1 3x x are

a)−π π2 2

, b) −π π3 3

8 8, c)

π π3 3

327

8, d) none of these


19. If sin sin sin ,− − −+ + =1 1 1 32

x y z π the value of x y z

x y z100 100 100

101 101 101

9+ + −

+ + is

a) 0 b) 1 c) 2 d) 3

20. The number of real solutions of tan ( ) sin /− −+ + + + =1 1 21 1 2x x x x π is

a) zero b) one c) two d) infinite

21. If sinα sin ( 60° -α ) sin ( 60° + α ) = 1 / 8, and n ,∈I then

a) α π π= + −n n( ) /1 6 b) α π π= + −( / ) ( ) /n n3 1 18

c) α π π= = −n n( ) /1 3 d) α π π= + −( / ) ( ) /n n3 1 9

22. The value of sin ( tan-1 x ) is equal to

a)xx1 2−

b)1

1 2+ xc)

xx1 2+

d)1

1 2− x

23. 2 4 32 2cos cos sin .x x x+ = If

a) cos x = − −2 195

b) cos x = − +2 195

c) sin x = − −2 195

d) sin x = − +2 195

24. The number of all possible triplets ( a1 , a2 , a3 ) such that a1 + a2 cos 2x + a3 sin2 x = 0 for all x isa) 0 b) 1 c) 3 d) infinite

25. In a triangle ABC, the angle A is greater than the angle B. If the values of the angles A and B satisfy the equation 3 sin x - 4 sin3 x - k = 0,0 < k < 1, then the measure of angle C isa) π / 3 b) π / 2 c) 2π / 3 d) 5π / 6.

26. The principal value of sin cos cos− −−FHGIKJ +

FHGIKJ

1 132

76π

is

a) 5 π / 6 b) π / 2 c) 3π / 2 d) none of these

27. If sin cos ,θ θ+ =7 5 then tan( θ / 2 ) is a root of the equation

a) x x2 6 1 0− + = b) 6 1 02x x− − = c) 6 1 02x x+ + = d) x x2 6 0− + =

28.sin

cos3

2 2 112

θθ +

= if

a) θ ππ

= +n6

b) θ ππ

= −26

n c) θ ππ

= + −n n( )16

d) θ ππ

= −n6

29. A solution of the equation log sin log cos log ( tan ) log ( tan )2 2 2 21 1 1 0x x x x− − − − + + = is given by

a) tan x = -1 b) tan x = 1 c) tan 2 x = -1 d) tan 2 x = 1


30. 6 2 22 2tan cos cosx x x− = ifa) cos 2 x = -1 b) cos 2 x = 1 c) cos 2 x = -1/2 d) cos 2 x = 1/2

31. The equation cos ( ) cos ( )4 22 3 0x a x a− + − + = possess a solution if

a) a > - 3 b) a < - 2 c) 23 −≤≤− a d) a is any positive integer

32. If A = tan / , tan / ,− −=1 11 7 1 3B thena) cos 2 A = sin 2 A b) cos 2 A = sin 2 B c) cos 2 A = cos 2 B d) cos 2 A = sin 4 B

33. If u = −− −cot tan tan tan ,1 1α α then tan π4 2−FHGIKJu

is equal to

a) tanα b) cotα c) tanα d) cotα .

34. If cosec− − −= +1 1 12 7 3 5x cot cos ( / ) then the value of x isa) 44 / 117 b) 125 / 117 c) 24 / 7 d) 5 / 3

35. 2 3 31 1 1tan ( tan tan cot )− − −−cosec is equal toa) π / 16 b) π / 6 c) π / 3 d) π / 2

36. If 1cossin =+ AA , then A2sin is equal toa) 1 b) 2 c) 0 d) 1/2

37. The value of 8

7sin8

5sin8

3sin8

sin 2222 ππππ+++ is

a) 1 b) 2 c)811 d)

812

38. The value of cos1° cos 2° cos 3° ..... cos 179° is equal to

a)21

b) 0 c) 1 d) 2

39. The number of solutions of the equation 8)cos(sin2)cos(sin3 33 =+−+ xxxx area) 0 b) 1 c) 2 d) 3

40. If 0qtanptan =θ−θ , then the values of θ form a series ina) A.P b) G.P c) H.P d) none of these

41. If α and β are the solutions of csecbtana =θ+θ , then =β+α )(tan

a) 22 caac2−

b) 22 acac2−

c) 22 caac2+

d) none of these

42. The number of roots of the equation 2/xtan2x π=+ in the interval ]2,0[ π isa) 1 b) 2 c) 3 d) infinite


43. )]13/1(tan)7/1([tantan 11 −− + is equal toa) 2/9 b) 9/2 c) 7/9 d) 9/7

44. If 1)xcos5/1(sinsin 11 =+ −− , then x =a) 1 b) 0 c) 4/5 d) 1/5

45. )x21(cosxsin2 211 −= −− is true for

a) 1x1 ≤≤− b) 2/12/1 ≤≤− x c) 1x0 ≤≤ d) none of these

46. If 2)6/52sin(3cos −=π++ xx , then =xa) )16()3/( −π k b) )16()3/( +π k c) )14()3/( +π k d) )12()3/( +π k

ANSWERS

1. b 2. d 3. c 4. c 5. b 6. c 7. c8. d 9. c 10. a 11. a 12. b 13. b 14. d15. d 16. d 17. c 18. c 19. a 20. c 21. b22. c 23. b 24. d 25. c 26. b 27. b 28. c29. d 30. d 31. c 32. d 33. a 34. b 35. c36. c 37. b 38. b 39. a 40. a 41. a 42. c43. a 44. d 45. c 46. b


PROPERTIES AND SOLUTION OF TRIANGEL

1. The perimeter of a ∆ ABC is 6 times the arithmetic mean of the sines of its angles. If the side a is 1, then the angle A is

a)π6 b)

π3 c)

π2 d) π

2. The area of the circle and the area of a regular polygon of n sides and of perimeter equal to that of the circle are in the ratio of

a) tan :π πn nFHGIKJ b)cos :π π

n nFHGIKJ c) sin :π π

n nd)cot :π π

n nFHGIKJ

3. In a triangle ABC, the line joining the circumcentre to the incentre is parallel to BC, then cos B + cos C =a) 3 / 2 b)1 c) 3 / 4 d)1 / 2.

4. The ex - radii of a triangle r1, r2, r3 are in harmonic progression, then the sides a, b, c area) in H.P. b) in A.P. c) in G.P. d) none of these

5. If a cos A = b cos B, then the triangle isa) equilateral b) right angledc) isosceles d) isosceles or right angled

6. Two straight roads intersect at an angle of 60°. A bus on one road is 2 km. away from the intersection and a car on the other road is 3 km. awayfrom the intersection. Then the direct distance between the two vehicle is

a) 1 km b) 2 km c) 4 km d) 7 km

7. In any triangle ABC, sin2A A

A+ +

∑sin

sin1

is always greater than


8. In any ∆ ABC if 2 cos caB /= then the triangle isa) right angled b) equilateral c) isosceles d) none of these.

9. In an equilateral triangle, the in - radius, circum - radius and one of the ex - radii are in the ratio

a) 2 : 3 : 5 b) 1 : 2 : 3 c) 1 : 3 : 7 d) 3 : 7 : 9.

10. If in a ∆ ABC, ∆ = a2 - ( b - c )2 , then tan A =a) 15 / 16 b) 8 / 15 c) 8 / 17 d) 1 / 2.

11. In a triangle the length of the two larger sides are 24 and 22, respectively. If the angles are in AP, then the third side is

a) 13212 + b) 12 - 2 3 c) 2 3 +2 d) 2 3 - 2

12. If in a triangle ABC, BA 2sinsin = and BA 22 cos3cos2 = , then the ∆ ABC is

a) right angled b) obtuse angled c) isosceles d) equilateral


13. Points D, E are taken on the side BC of a triangle ABC, such that BD = DE = EC. If ∠ = ∠ = ∠ =BAD x DAE y EAC z, , , then the value

of sin ( ) sin ( )sin sinx y y z

x z+ + is equal to


14. If A + B + C = π , n Z∈ , then the tan tan tannA nB nC+ + is equal to

a) 0 b) 1 c) tan tan tannA nB nC d) none of these

15. If p p p1 2 3, , are altitudes of a triangle ABC from the vertices A, B, C and ∆ , the area of the triangle, then 23

22

21

−−− ++ ppp is equal to

a)a b c+ +

∆b)a b c2 2 2

24+ +∆

c)a b c2 2 2

2

+ +∆

d) none of these

16. If in a triangle ABC, sinsin

sin ( )sin ( )

,AC

A BB C

=−−

then

a) a, b, c are in AP b)a b c2 2 2, , are in AP c) a, b, c are in HP d) a b c2 2 2, , are in HP

17. If R is the radius of circumscribing circle of a regular polygon of n-sides having length of side as a , then R =

a)a

n2sin πFHGIKJ b)

an2

cos πFHGIKJ c)

an2

cosec πFHGIKJ d)

an2 2

cosec πFHGIKJ

18. If p p p1 2 3, , are respectively the perpendiculars from the vertices of a triangle to the opposite sides, then cos cos cosAp

Bp

Cp1 2 3

+ + is

equal to

a)1r b)

1R c)

1∆

d) none of these

19. If rr

rr1

2

3

= , then

a) A = 90° b) B = 90° c) oC 90= d) none of these

20. If the angles of a triangle are 30° and 45° and the included side is ( )3 1+ , then the area of the triangle is

a)13 1− b) 3 1+ c)

13 1+ d) none of these

21. In a triangle ABC, ∠ =B π3

and ∠ =C π4. Let D divide BC internally in the ratio 1 : 3. Then

sinsin

∠∠BADCAD

equals

a)16 b)

13 c)

13 d)

23

22. The two adjacent sides of a cyclic quadrilateral are 2 and 5 and the angle between them is 60°. If the third side is 3, the remaining fourth side is


a) 2 b) 3 c) 4 d) 5

23. If the data given to construct a triangle ABC are a = 5, b = 7, sin A = 3 / 4, then it is possible to constructa) only one triangle b) two triangles c) infinitely many triangles d) no trianlges

24. In triangle ABC, A = 30°, b = 8, a = 6, then B = sin-1 x, where x =a) 1 / 2 b) 1 / 3 c) 2 / 3 d) 1

25. If A = 30°, a = 7, b = 8 in ∆ ABC, then B hasa) one solution b) two solutions c) no solution d) none of these.

26. In a triangle the angles are in A.P. and the lengths of the two larger sides are 10 and 9 respectively, then the length of the third side can be

a) 65± b) 0.7 c) 5 6− d) none of these

27.The smallest angle of the triangle whose sides are 6 12 48 24+ , , is

a)π3 b)

π4 c)

π6 d) none of these

28. In a ∆ ABC, a, b, A are given and c1 , c2 are two values of the third side c. The sum of the areas of two triangles with sides a, b, c1 and a,b, c2 isa) (1/2) b2 sin 2A b) (1/2) a2 sin 2A c) b2 sin 2 A d) none of these

29. If in a triangle ABC, ( a + b + c ) ( b + c - a ) = bc, then the triangle isa) equilateral b) right anlged c) obtuse angled d) none of these

30. If in a triangle ABC, a2 , b2 , c2 , the squares of the lengths of the sides of the triangles are in arithmetical progression, then b3 is equal toa) 2abc cos A b) 2abc cos B c) 2abc cos C d) none of these

31. If the angles of a triangle are in the ratio 1 : 3 : 5 andθ denotes the smallest angle, then the ratio of the largest side to the smallest side ofthe triangle is

a)32sin cossinθ θθ+

b)32cos sinsinθ θθ−

c)cos sin

sinθ θ

θ+ 32

d)32cos sinsinθ θθ+

32.The vertical angle of a triangle is divided into two parts, such that the tangent of one part is 3 times the tangent of the other and the differenceof these parts is 30°, then the triangle isa) isosceles b) right angled c) obtuse angled d) none of these

33. In a cyclic quadrilateral ABCD; a, b, c, d denote the length of the sides AB, BC, CD and DA respectively, then cos A is equal to

a)a b c d

ab cd

2 2 2 2

2+ − −

+( ) b)b c d a

bc da

2 2 2 2

2+ − −

+( ) c)c d a b

cd ab

2 2 2 2

2+ − −

+( ) d)d a b c

da bc

2 2 2 2

2+ − −

+( )34. In a triangle ABC, sin A + sin B + sin C is maximum when the triangle is

a) right angled b) isosceles c) equilateral d) obtuse angle

35. If the angles A and B of the triangle ABC satisfy the equation sin A + sin B = 3 (cos B-cos A ) then they differ bya) π / 6 b) π / 3 c) π / 4 d) π / 2


36.The sides of a triangle are 17, 25, 28. The length of the largest altitude isa) 15 b) 84 / 5 c) 420 / 17 d) 210 / 17

37. In a triangle ABC if cos cos cos ,Aa

Bb

Cc

abc

+ + = then A is

a) an acute angle b) an obtuse angle c) a right angle d) equal to B-C

38. If in a triangle ABC, the line joining the circumcentre 0 and the incentre I is parallel to BC, thena) r = R cos A b) r = R sin A c) R = r cos A d) R = r sin A

39.Suppose that xx 3sin.sin3 ∑=

=n

mm mxc

0,cos is an identity in x, where 210 c,c,c are constants and 0cn ≠ , then the value of n is

a) 4 b) 5 c) 9 d) 6

40. In any ( )CBa −∑ sinABC,∆ 3 is equal to

a) 0 b) ( )cba ++3 c) 3 abc d) cabcab ++

ANSWERS

1. a 2. a 3. b 4. b 5. d 6. d 7. a8. c 9. b 10. b 11. a 12. b 13. c 14. c15. b 16. b 17. c 18. b 19. c 20. a 21. a22. a 23. d 24. c 25. b 26. a 27. c 28. a29. c 30. b 31. d 32. b 33. d 34. c 35. b36. c 37. c 38. a 39. d 40. a


HEIGHTS & DISTANCES

1. A flag staff of 5 mt high stands on a building of 25 mt high. At an observer at a height of 30 mt, the flag staff and the building subtend equalangles. The distance of the observer from the top of the flag staff is

a)5 32

b) 532

c) 5 23

d) none of these

2. If a flag-staff of 6 metres high placed on the top of a tower throws a shadow of 2 3 metres along the ground then the angle (in degrees)that the sun makes with the ground isa) 60° b) 30° c) 45° d) none of these

3. The top of a hill observed from the top and bottom of a building of height h is at angles of elevation p and q respectively. The height of the hillis

a)h qq pcot

cot cot− b)h pp qcot

cot cot− c)h pp qtan

tan tan− d) none of these

4. On the level ground the angle of elevation of the top of a tower is 30°. On moving 20 mt. nearer the tower, the angle of elevation is found tobe 60°. The height of the tower is

a) 10 mt b) 20 mt c) 10 3 mt d) none of these

5. A person standing on the bank of a river observes that the angle subtended by a tree on the opposite bank is 60°, when he retires 40 metresfrom the bank he finds the angle to be 30°. Then the breadth of the river isa) 40 m b) 60 m c) 20 m d) 30 m.

6. A tree is broken by wind, its upper part touches the ground at a point 10 metres from the foot of the tree and makes an angle of 45° with theground. The entire length of the tree is

a) 15 metres b) 20 metres c) 10 1 2( )+ metres d) 10 1 32

+FHGIKJ metres.

7. A vertical pole subtends an angle tan /−11 2 at a point P on the ground. The angle subtended by the upper half of the pole at the point Pis

a) tan ( / )−1 1 4 b) tan ( / )−1 2 9 c) tan ( / )−1 1 8 d) tan ( / )−1 2 3

8. An aeroplane flying at a height of 300 m above the ground passes vertically above another plane at an instant when the angles of elevationof the two planes from the same point on the ground are 60° and 45° respectively. The height of the lower plane from the ground is

a) 100 3 m b) 100 3/ m c) 50 m d) 150 3 1( )+ m

9. A pole 50 m high stands on a building 250 m high. To an observer at a height of 300 m, the building and the pole subtends equal angles. Thehorizontal distance of the observer from the pole is

a) 25 m b) 50 m c) 25 6 m d) 25 3 m

10.A man in a boat rowing away from a cliff 150 metres high observes that it takes 2 minutes to change the anlge of elevation of the top of thecliff from 60° to 45°. The speed of the boat is

a) ( / ) ( )1 2 9 3 3− km / h b) ( / ) ( )1 2 9 3 3+ km / h

c) ( / ) ( )1 2 9 3 km / h d) none of these


11. When the length of the shadow of a pole is equal to the height of the pole then the elevation of source of light isa) 30° b) 45° c) 60° d) 75°

12.From the top of a light house 60 m high with its base at sea level the angle of depression of a boat is 15°. The distance of the boat from thelight house is

a) m

+−131360 b) m

−+131360 c) m

+−131330 d) m

−+131330

13.On the level ground, the angle of the top of the tower is 30°. On moving 20 metres nearer, the angle of elevation is 60°. Then the height ofthe tower is

a) 320 metres b) 310 metres c) ( )1310 − metres d) none of these

14.The angle of elevation of the top of a tower from a point 20 metres away from its base is 45°. The height of the tower is

a) 10 m b) 20m c) 40 m d) m320

15.A tree is broken by wind and its upper part touches the ground at a point 10 metres from the foot of the tree and makes an angle of 45° withthe ground. The entire length of the tree is

a) 15m b) 20m c) ( )m2110 + d) m

+23110

16.Angle of depression from the top of a light house of two boats are 45° and 30° due east which are 60m apart. The height of the light houseis

a) 360 b) ( )1330 − c) ( )1330 + d) none of these

17.The angle of elevation of the top of a tower at a point G on the ground is 30°. On walking 20m towards the tower the angle of elevationbecomes 60°. The height of the tower is equal to

a) 310

b) 320 c) 320

d) 310

18.Two posts are 25 metres and 15 metres high and the line joining their tops makes an angle of 45° with the horizontal. The distance betweenposts is

a) 5m b) m210

c) 10 m d) m210

19.A portion of a 30m long tree is broken by tornado and the top struck up the ground making an angle of 30° with ground level. The height ofthe point where the tree is broken is equal to

a) m330

b) 10m c) m330 d) 60 m

ANSWERS

1. b 2. a 3. b 4. c 5. c 6. c 7. b8. a 9. c 10. a 11. b 12. b 13. b 14. b15. c 16. c 17. d 18. c 19. b


SEQUENCE AND PROGRESSIONS

1. The value of .,....................16.8.4.2 32/116/18/14/1 is

a) 1 b) 2 c) 3/2 d) 5/2

2. The third term of a G.P. is 4. The product of first five terms is

a) 43 b) 45 c) 44 d) none of these3. Sum of three numbers in GP be 14. If one is added to first and second and 1 is subtracted from the third, the new numbers are in AP. The smallest

of them is

a) 2 b) 4 c) 6 d) 84. If a, b, c, d are in HP, then

a) a b c d+ > + b) a c b d+ > + c) a d b c+ > + d) none of these.5. If a, b, c, d are in HP, then

a) ab > cd b) ac > bd c) ad > bc d) none of these

6. If x, y, z are in G.P. & zyx cba == then

a) log logb aa c= b) log logc ab c= c) log logb ca b= d) none of these

7. If a,b, c are in HP then ab c

bc a

ca b+ + +

, , will be in

a) AP b) GP c) HP d) none of these

8. The sum of first 10 terms of the series xx

xx

xx

+FHGIKJ + +FHG

IKJ + +FHG

IKJ +

1 1 122

2

23

3

2

....... is

a)xx

xx

20

2

22

20

11

1 20−−

FHGIKJ

+FHGIKJ + b)

xx

xx

18

2

11

911

1 20−−

FHGIKJ

+FHGIKJ +

c)xx

xx

18

2

11

911

1 20−−

FHGIKJ

−FHGIKJ + d) none of these

9. If n Arithmetic means are inserted between two quantities a and b, then their sum is equal to

a) n a b+b g b) n a b2

+b g c) 2n a b+b g d)n a b2

−b g10. If log 2, log 2 1x −c h and log 2 3x +c h are in AP, then the value of x is given by

a) 5/2 b) log2 5 c) log3 5 d) log5 3

11. If the AM of the roots of a quadratic in x is A and GM is G, then the quadratic is

a) x Ax G2 2 0− + = b) x Ax G2 2 0− + = c) x Ax G2 22 0− + = d) x Ax G2 0− + =

12. The sum of n term of the series 1 2 3 4 5 62 2 2 2 2 2− + − + − +...... is ( n is even no.)

a) -n n +12b g

b) n n +12b g

c) − +n n 1b g (d) none of these


13. If a, b, c are in GP, then log ,log ,loga b cx x x are in

a) AP b) GP c) HP d) none of these14. If the sum of n terms of a series is an2 + bn, then the series is ( a, b are constants )

a) an AP b) a GP c) AP d) none of these

15. Let sn denote the sum of first n terms of an A.P. If s sn n2 3= then the ratio s sn n3 / is equal to

a) 4 b) 6 c) 8 d) 10

16. If the roots of the equation x x x3 212 39 28 0− + − = are in A.P., then their common difference will be

a) ±1 b) ±2 c) ±3 d) ±417. log32, log62 log122 are in

a) A.P. b) G.P. c) H.P. d) none of these

18. If a, b, c, d, e, f are in A.P., then e - c is equal toa) 2 ( c - a ) b) 2 ( d - c ) c) 2 ( f - d ) d) d - c.

19. The sum of n terms of the series 12

34

781516

+ + + +...... is

a) 2n - n - 1 b) 1 - 2-n c) n + 2-n - 1 d) 2n - 1.20. The maximum value of the sum of the A.P. 30, 27, 24, 21, .... is

a) 165 b) 168 c) 171 d) 180.

21. The least value of n for which the sum of the series 3 + 6 + 9 + .... to n terms exceeds 1000 is

a) 25 b) 26 c) 27 d) 28.

22. If (A1, A2) ( G1, G2 ) ( H1, H2 ) are two (arithmetic), ( geometric ), ( harmonic ) means between two positive real numbers a and b, then value

of G GH H

H HA A

1 2

1 2

1 2

1 2

. ++

is

a) ab b) a b2 2+ c) b a a b+ d) none of these.

23. Sum of the series 1 11 2

11 2 3

11 2

++

++ +

+ ++ + +

........ n

is

a)21n

n + b) nn +1 c)

nn2 1( )+ d)

nn

2

2 1( ).

−

24. Sum to n terms of the series 1 + ( 1 + 2 ) + ( 1 + 2 + 3 ) + ... is

a)12

1 2n n n( ) ( )+ + b) 13

1 2n n n( ) ( )+ + c)14

1 2 1n n n( ) ( )+ + d) none of these

25. The sum to n terms of the series 23

89

2627

8081

+ + + +.... is equal to

a) 1 1 3− ( / )n b) 2 122 3− ( / )n c) n n n− ( / )1 3 d) none of these


26. If x, y, z are the pth, qth and rth terms respectively of an A.P. as well as that of a G.P., then x y zy z z x x y− − − is equal to

a) x y z b) 0 c) 1 d) none of these27. If log ( a + c ), log ( c - a ), log ( a - 2b + c ) are in A.P., then

a) a, b, c are in A.P. b) a2, b2, c2 are in A.P. c) a, b, c are in G.P. d) a, b, c are in H.P.28. The coefficient of x99 in the expansion of ( x - 1 ) ( x - 2 ) .... ( x - 100 ) is

a) 5050 b) -5050 c) 3300 d) -3310

29. If x = 1 + y + y2 + .......to ∞ , then y is

a)xx −1 b)

xx1− c)

xx−1

d)1− xx

30. The first term of an A.P of consecutive integers is 12 +p . The sum of ( )12 +p terms of this series can be expressed as

a) ( )21+p b) ( ) ( )2112 ++ pp c) ( )31+p d) ( )33 1++ pp

31. If the sum of first n terms of an A.P is 2QnPn + where P and Q are constants, then common difference of A.P will be

a) QP + b) QP − c) 2P d) 2Q

32. If a,b,c are in A.P, then 0,10,10,10 101010 ≠+++ xcxbxax are ina) A.P b) G.P only when x >0 c) G.P for all x d) G.P only when x < 0

33. Let the sequence ,,, 321 naaaa ⋅⋅⋅⋅⋅ form an A.P, then2

2212

24

23

22

21 .. nn aaaaaa −++−+− − is equal to

a) ( )22

2112 naa

nn

−−

b) ( )21221

2 aann

n −−

c) ( )22

211 naa

nn

++

d) none of these

34. The sum of the series ∞+×

+×

+×

o.........t15111

1171

731 is

a)31

b)61

c)91

d)121

35. If the first, second and last terms of an arithmetic series are a, b and c respectively. Then the number of terms are

a) )cba(21

++ b) )a2cb(21

−+ c) )ab/()a2cb( −−+ d) none of these

36. If the sum of n terms of two arithmetic series are in the ratio 27n4:1n7 ++ , then their 11th terms are in the ratioa) 3 : 4 b) 4 : 3 c) 78 : 61 d) 152 : 119

37. If ba,ac,cb +++ are in H.P., then 222 c,b,a are ina) A.P b) G.P c) H.P d) none of these


38. =+++ terms....888888 n

a)9n8)110(

8180 n −− b) )110(

8110 n − c)

9n8)110(

8180 n +− d) none of these

39. If nn

1n1n

baba

++ ++

is the H.M of a and b, then the value of n is

a) 1 b) - 1/2 c) - 1 d) 0

ANSWERS

1. b 2. b 3. a 4. c 5. c 6. c 7. c8. a 9. b 10. b 11. c 12. a 13. c 14. a15. b 16. c 17. c 18. b 19. c 20. a 21. b22. d 23. a 24. d 25. d 26. c 27. d 28. b29. c 30. d 31. d 32. c 33. a 34. d 35. c36. b 37. a 38. a 39. c


QUADRATIC EQUATIONS AND INEQUATIONS

1. The roots of the equation x x x2 6 2− − = + are

a) -2, 1, 4 b) 0, 2, 4 c) 0, 1, 4 d) -2, 2, 4

2. If f x ax bx c g x ax bx c( ) , ( )= + + = − + +2 2 where ac f x g x≠ =0 0, ( ) ( )then hasa) at least three real roots b) no real roots c) at least two real roots d) two real roots and two imaginary roots.

3. The equation 22

1 02

2 2 22cos sin ,x x xx

xFHGIKJ = + ≤ ≤

π has

a) no real solution b) one real solution c) more than one real solution d) none of these.

4. The number of real roots of the equation ( ) ( ) ( )x x x− + − + − =1 2 3 02 2 2 isa) 1 b) 2 c) 3 d) none of these.

5. The roots of the equation log ( ) ( )22 4 5 2x x x− + = − are

a) 4, 5 b) 2, -3 c) 2, 3 d) 3, 5

6. The value of k for which the equation 3 2 1 3 2 02 2 2x x k k k+ + + − + =( ) has roots of opposite sign, lies in the interval

a) ( , )−∞ 0 b) ( , )−∞ −1 c) (1, 2) d) ( 3/2, 2 )

7. The quadratic equation whose roots are reciprocal of the roots of the equation ax bx c2 0+ + = is

a) cx bx a2 0+ + = b) bx cx a2 0+ + = c) cx ax b2 0+ + = d) bx ax c2 0+ + =

8. If one root of the equation 5 13 02x x k+ + = is reciprocal of other, then the value of k isa) 0 b) 5 c) 1 / 6 d) 6

9. If the root of the equation x px q2 0− + = differ by unity, then

a) p q2 4= b) p q2 4 1= + c) p q2 4 1= − d) none of these

10.The number of real roots of the equation x x2 3 2 0− + = isa) 4 b) 3 c) 2 d) 1

11. If x = 1+ i is a root of the equation x ix i3 1 0− + − = , then the other real root isa) 1 b) -1 c) 0 d) none of these

12.The number of solutions of the equation 2 5 5sin ex x xc h = + − isa) 0 b) 1 c) 2 d) Infinitely many

13. If c and d are roots of the equation x a x b k− − − =b gb g 0 , then a , b are roots of the equation

a) x c x d k− − − =b gb g 0 b) x c x d k− − + =b gb g 0

c) x a x c k− − + =b gb g 0 d) x b x d k− − + =b gb g 0


14. If x x2 3 2− + is a factor of x px q4 2− + , then the values of p and q area) 5, -4 b) 5, 4 c) -5 , 4 d) -5, -4,

15. If the expression x x a2 11− + and x x a2 14 2− + have a common factor, then the values of 'a' area) 0,24 b) 0, -24 c) 1, -1 d) -2, 1

16. If the sum of the roots of the equation a x a x a+ + + + + =1 2 3 3 4 02b g b g b g is -1, then the product of the roots is

a) 0 b) 1 c) 2 d) 3

17.The values of x satisfying x = + + + ∞6 6 6 ... are

a) 3, -2 b) -2 c) 3 d) none of these

18. If 2 3+ i is a root of x px q2 0+ + = where p q R, ,∈ then

a) p q= − =4 7, b) p q= =4 7, c) p q= = −4 7, d) p q= − = −4 7,

19. Let f x x xb g = + +2 4 1 . Then

a) f xb g > 0 for all x b) f xb g > 1when x ≥ 0 c) f xb g ≥ 1when x ≤ −4 d) f x f xb g b g= − for all x

20.The adjoining figure shows the graph of y ax bx c= + +2 . Then

a) a < 0 b) b ac2 4< c) c > 0 d) none of these

21.The diagram shows the graph of y ax bx c= + +2 . Then,

a) a > 0 b) b < 0 c) c > 0 d) b ac2 4 0− =

22. If x x2 1 0− + = , Then the value of x n3 is

a) -1, 1 b) 1 c) -1 d) 0

23.The number of positive real roots of x x4 4 1 0− − = is

a) 3 b) 2 c) 1 d) 0


24. The number of negative real roots of x x4 4 1 0− − = isa) 3 b) 2 c) 1 d) 0

25. The number of complex roots of the equation x x4 4 1 0− − = isa) 3 b) 2 c) 1 d) 0

26. The number of values of k for which the equation x x k3 3 0− + = has two distinct roots lying in the interval (0, 1) area) 3 b) 2 c) infinite d) none of these

27. If βα, are roots of ax bx c2 0+ + = , then the equation ax bx x c x2 21 1 0− − + − =b g b g has roots

a)αα

ββ1 1− −

, b)1 1− −αα

ββ

, c)α

αβ

β+ +1 1, d)

αα

ββ

+ +1 1,

28. If one root of the equation

7 62 2

3 70

xx

x= is -9, the other roots are

a) 2, 6 b) 3, 6 c) 2, 7 d) 3, 7

29.Given that ax bx c2 0+ + = has no real roots and a b c+ + < 0, then

a) c = 0 b) c > 0 c) c < 0 d)none of these

30. Let α and β be the roots of the equation x x2 1 0+ + = . The equation whose roots are α β19 7, is

a) x x2 1 0− − = b) x x2 1 0− + = c) x x2 1 0+ − = d) x x2 1 0+ + =

31.The value of 'k' for which one of the roots of x x k2 3 0− + = , is double of one of the roots of x x k2 0− + = is

a) 1 b) -2 c) 2 d) none of these

32. If a, b, c are all positive and in H.P., then the roots of ax bx c2 2 0+ + = area) Real b) Imaginary c) Rational d) Equal

33. If 7 172 4 5log ( ) ,x x x x− + = − may have values

a) 2,3 b) 7 c) -2, -3 d) 2, -3

34.The equation x x x+ − − = −1 1 4 1 hasa) no solution b) one solution c) two solution d) more than two solutions

35.The equation log log3 323 1 0+ + + =x xd i c h has

a) no solution b) one solution c) two solutions d) more than two solutions


ANSWERS1. d 2. c 3. a 4. d 5. c 6. c 7. a8. b 9. b 10. a 11. b 12. a 13. b 14. b15. a 16. c 17. a 18. a 19. c 20. a 21. c22. a 23. c 24. c 25. b 26. d 27. c 28. c29. c 30. d 31. b 32. b 33. a 34. a 35. a36. a 37. b 38. a 39. a 40. b

36. The number of roots of the equation x x x x3 2 2 0+ + + =sin in [ , ]−2 2π π isa) 1 b) 2 c) 3 d) none of these

37. The line y + =14 0 cuts the curve whose equation is x x x y2 1 0+ + + =c h at

a) three real points b) One real point c) at least one real point d) no real point

38. If sin ,cosα α are the roots of the equajtion ax bx c2 0+ + = , then

a) a b ac2 2 2 0− + = b) a c b c+ = −b g2 2 2

c) a b ac2 2 2 0+ − = c) a c b c− = +b g2 2 2

39. If the product of the roots of the equation ( ) 0123 ln23 =−−+ kekxx is 7, then the roots are real only if

a) k = 2 b) k = −2 c) k = 4 d) k = −4

40. If βα, are the roots of the equation 8 3 27 02x x− + = then the value of α β β α2 1 3 2 1 3/ /

/ /c h c h+ is:

a) 1/3 b) 1/4 c) 1/5 d) 1/6


Complex Number

1. The product of cube roots of -1 is equal toa) 0 b) 1 c) -1 d) none of these.

2. Which of the following is correct ?a) 1 + i > 2 - i b) 2 + i > 1 + i c) 2 - i > 1 + i d) none of these

3. Let z be a purely imaginary number such that Im (z) > 0. Then arg (z) is equal toa) π b) π / 2 c) 0 d) -π / 2

4. Let z be any non-zero complex number. Then, arg (z) + arg ( z ) is equal toa) π b) -π c) 0 d) π / 2

5. The smallest positive integer n for which 11

1+−FHGIKJ =ii

n

a) 3 b) 2 c) 4 d) none of these.

6. Let z be a complex number. Then the angle between vectors z and iz isa) π b) 0 c) π / 2 d) none of these.

7. The locus of the points z satisfying the condition arg zz−+FHGIKJ =11 3

π is

a) parabola b) circle c) pair of straight lines d) none of these.

8. If z = x + iy and w izz i

=−−

1 , then | w | = 1 implies that in the complex plane

a) z lies on imaginary axis b) z lies on real axis c) z lies on unit circle d) none of these

9. Let z be a complex number such that | z | = 4 and arg (z) = 56π , then z =

a) − +2 3 2 i b) 2 3 2+ i c) 2 3 2− i d) − +3 i10. Let 3 - i and 2 + i be affixes of two points A and B in the argand plane and P represents the complex number z = x + i y. Then the locus of P

if |2||3| iziz −−=+− isa) circle on AB as diameter b) the line ABc) the perpendicular bisector of AB d) none of these

11. If the complex numbers sin x + i cos 2 x and cos x - i sin 2 x are conjugate to each other, then x is equal toa) n π b) ( n + 1/2 ) π c) 0 d) none of these.

12. Common roots of the equations z z z3 22 2 1 0+ + + = and z z1985 100 1 0+ + = are

a) w , w2 b) 1, w, w2 c) -1, w, w2 d) - w, - w2

13. The inequality | z - 2 | < | z - 4 | represents the half plane

a) Re ( z) ≥ 3 b) Re ( z ) ≤ 3 c) Re ( z ) = 3 d) none of these

14. The cube roots of unity

a) are collinear b) lie on a circle of radius 3 c) from an equilateral triangle d) none of these.


15. If z is a complex number such that z iz i−+

=55

1, then the locus of z is

a) x-axis b) straight line y = 5

c) a circle passing through the origin (d)none of these.16. For any complex number z, the minimum value of | z | + | z - 1| is

a) 1 b) 0 c) 1 / 2 d) 3 / 2.

17. If 0|| 22 =+ zz , then the locus of z is

a) a circle b) a straight line c) a pair of straight line d) none of these.

18. If log | | | || |

,3

2 12

2z zz

− ++

FHG

IKJ > then the locus of z is

a) | z | = 5 b) | z | < 5 c) | z | > 5 d) none of these.

19. Let z ( ≠ 2 ) be a complex number such that log1/2 | z - 2 | > log1/2 | z | , then

a) Re (z) > 1 b) Im (z) > 1 c) Re (z) = 1 d) Im (z) = 1.

20. The product of n th roots of unity is

a) 1 b) -1 c) ( -1 )n - 1 d) ( -1 )n.

21. If p is a multiple of n, then the sum of pth powers of nth roots of unity is

a) p b) n c) 0 d) none of these.

22. If xn = cos π π2 2n ni n NFHGIKJ +FHGIKJ ∈sin , , then ∞xxxx .......... 321 is equal to


23. The region of the Argand diagram defined by | z - 1 | + | z + 1 | ≤ 4 is

a) interior of an ellipse b) exterior of a circle

c) interior and boundary of an ellipse d) none of these.

24. If θ= irez , then | ei z | is equal to

a) θ− sinre b) θsinrer − c) θcosre− d) θcosrer −

25. If | z - 25 i | ≤ 15, then | max arg(z) - min arg (z) | =

a) cos− FHGIKJ

1 35

b) π−−FHGIKJ

−2 35

1cos c)π2

35

1+ FHGIKJ

−cos d) sin cos− −FHGIKJ −

FHGIKJ

1 135

35


26. The locus of the points representing the complex numbers z for which

| z | - 2 = | z - i | - | z + 5 i | = 0 is

a) a circle with centre at the origin b) a straight line passing through the origin

c) the single point ( 0, -2 ) d) none of these.

27. If m, n, p, q are consecutive integers then the value of i i i im n p q+ + + is


28. The value of z satisfying the equation log log ... logz z zn+ + + =2 0 is

a) cos( )

sin( )

, , ,....41

41

1 2mn n

i mn n

mπ π+

++

= b) cos( )

sin( )

, , ,....41

41

1 2mn n

i mn n

mπ π+

−+

=

c) sin cos , , ,....4 4 1 2mn

i mn

mπ π+ = d) 0

29. If z z zn1 2 1= = = =.... , then the value of z z zn1 2+ + +.... is

a) n b)1 1 1

1 2z z zn+ + +.... c) 0 d) none of these.

30. The general value of θ which satisfies the equation

(cos sin ) (cos sin ) (cos sin ) ...(cos( ) sin ( ) )θ θ θ θ θ θ θ θ+ + + − + − =i i i n i n3 3 5 5 2 1 2 1 1 is

a)rnπ2 b)

( )rn−12

πc)( )2 1

3

rn+ π

d)2

2

rnπ .

31. If the area of the∆ on the complex plane formed by the complex numbers z, iz and z + iz is a | z |2, then the constant 'a' is equal to

a) 1 / 2 b) 2 / 3 c) 3 / 4 d) none of these.

32. The value of ii is

a) ω b) 2ω− c) π / 2 d) none of these

33. The value of the expression

1 1 1 1 2 1 2 1 3 1 3 1 1 12 2 2 2+FHG

IKJ +FHG

IKJ + +FHG

IKJ +FHG

IKJ + +FHG

IKJ +FHG

IKJ + + +FHG

IKJ +FHG

IKJω ω ω ω ω ω ω ω

... ,n n where w is an imaginary cube

root of unity is

a)n n( )2 2

3+

b)n n( )2 2

3−

c)n n( )2 13+

d) none of these.

34. The number of solutions of z z2 2 0+ = is

a) 2 b) 3 c) 4 d) 5


35. i i i2 4 6+ + + +... up to ( 2n + 1 ) terms =

a) i b) -i c) - 1 d) 1

36. If )1(≠ω is a cube root of unity, then the value of }4/){( 2310 π−πω+ω is

a) − 3 2/ b) −1 2/ c)45π−

d) 3 2/

37. Let z andω be two complex numbers such that | z | = |ω | and Arg z + Arg ω = π . Then z equals

a) ω b) −ω c) ω d) −ω

38. If )1(≠ω is a cube root of unity, then ∆ =+ +

− − −− − + − −

1 11 1 1

1 1

2 2

2

iii i

ω ωω

ω equals

a) 0 b) 1 c) i d) ω

39. Let zk ( k = 0, 1, 2, ..., 6 ) be the roots of the equation ( ) ,z z+ + =1 07 7 then Re( )zkk=∑0

6

is equal to

a) 3 - 2i b) 0 c) - 7/2 d) 3 + 2i

40. If m and x are two real numbers, then m

xmi

xixie

−+−

111cot2

is equal to

a) cos x + i sin x b) m / 2 c) 1 d) ( m + 1 ) / 2.

41. The value of

−

∑

= 112cos

112sin

10

1

ππ kik

k is

a) i b) - i c) 1 d) -1

42. If 1, α α α1 2 1, ,..., n− are the n, nth roots of unity, then the value of )1)......(1()1( 121 −α+α+α+ n is

a) n b) - n c) - n / 2 + 1 d) none of these.

43. If z z z z1 2 1 2+ = − , then the difference of the arguments of z1 and z2 is

a) 0 b) π / 2 c) π d) 2π

44. If z lies on the circle z i− =2 2 2 then arg zz−+FHGIKJ22

is equal to

a) π / 3 b) π / 4 c) π / 6 d) π / 8

45. If | z | = 2, then area of the triangle whose sides are magnitudes of the complex numbers z , w z and z+ wz is

a) 3 b) 2 3 c) 4 3 d) 16 3


46. The curve represented by | z | = Re ( z ) + 2 is

a) a straight line b) a circle c) an ellipse d) none of these

47. If )1(≠α is a root of x5 = 1, then the value of ( ) ( ) ( ) ( )α α α α− − − −1 1 1 12 3 4 is

a) 5 b) 10 c) - 5 d) - 10

48. If α is a root of x7 = 1, with 1≠α , then value of α α α101 102 205+ + +... is

a) 1 b) 0 c) - 104 d) 104

49. The value of ( ) / ( )i i i i i i5 6 7 8 9 1+ + + + + is

a)121( )+ i b)

121( )− i c)

12

1( )− − i d)12

1( )− + i

50. The system of equations | |z i+ − =1 2 and | z | = 3 has

a) no solution b) one solution c) two solutions d) infinite number of solutions.

ANSWERS

1. c 2. d 3. b 4. c 5. c 6. c 7. b8. b 9. a 10. c 11. d 12. a 13. d 14. c15. a 16. a 17. c 18. c 19. a 20. c 21. b22. b 23. c 24. a 25. b 26. c 27. c 28. a29. b 30. d 31. a 32. d 33. a 34. c 35. c36. c 37. d 38. a 39. c 40. c 41. a 42. d43. b 44. b 45. a 46. d 47. a 48. b 49. a50. a


Permutation and Combination

1. A box contains two white balls, three black balls and four red balls. In how many ways can three balls be drawn from the box if at least oneblack ball is to be included in the draw ? All balls are dissimilara) 129 b) 84 c) 64 d) none of these.

2. The exponent of 3 in 100! isa) 33 b) 44 c) 48 d) 52

3. A person wishes to make up as many different parties as he can out of his 20 friends such that each party consists of the same number ofpersons. The number of friends he should invite at a time isa) 5 b) 10 c) 8 d) none of these

4. If a denotes the number of permutations of x + 2 things taken all at a time, b the number of permutations of x things taken 11 at a time andc the number of permutations of x - 11 things taken all at a time such that a = 182 bc, then the value of x isa) 15 b) 12 c) 10 d) 18

5. The number of natural numbers smaller than 104, in the decimal notation of which all the digits are different isa) 5274 b) 5265 c) 4676 d) none of these

6. The sum of all the numbers that can be formed with the digits 2, 3, 4, 5 taken all at a time isa) 93324 b) 66666 c) 84844 d) none of these

7. All possible two-factor products are formed from the numbers 1,2,...., 100. The number of factors out of the total obtained which are multipleof 3 is.a) 2211 b) 4950 c) 2739 d) none of these

8. The number of ordered triplets of positive integers which are solutions of the equation x + y + z = 100 isa) 6005 b) 4851 c) 5081 d) none of these

9. The number of ways in which 52 cards can be divided into 4 sets, three of them having 17 cards each and the fourth one having just one card

a)5217 3!!b g b)

( ) !3!17

!523 c) 51

17 3!!b g d)

5117 33!! !b g

10. The total number of all proper factors of 75600 isa) 120 b) 119 c) 118 d) none of these

11. The number of ways in which 12 different balls can be divided between two friends, one receiving 8 and the other 4, is

a)128 4!! ! b)

12 28 4! !! ! c)

128 4 2

!! ! ! d) none of these

12. The least positive integer n for which is n n nC C C− −+ <15

16 7 is

a) 12 b) 13 c) 14 d) 15

13. The number of five digit numbers (without repetition) which are divisible by 9 and which can be formed by using the digits 0, 1, 2, 3, 4, 7, and8 isa) 212 b) 214 c) 412 d) none of these

14. A binary sequence is a sequence of 0's and 1's. The number of n - digit binary sequence which contain odd number of 0's is

a) 12 −n b) 2 2n − c) 2 21n− + d) none of these


15. The tens digit of 1! + 2! + 3! +........+ 49! isa) 1 b) 2 c) 3 d) 4

16. If n nC C4 5, and n C6 , are in A P, the value of n can bea) 14 b) 7 c) 7 and 14 d) none of these

17. A parallelogram is cut by two sets of m lines parallel to its sides. The number of parallelograms thus formed is

a) ( )mC22 b) ( )m C+1

22 c) ( )m C+2

22 d) none of these

18. A code word consists of three letters of the English alphabet followed by two digits of the decimal system. If neither letter nor digit isrepeated in any code word, the the total number of code words isa) 1404000 b) 16848000 c) 2808000 d) none of these

19. There are four balls of different colours and four boxes of colours, same as those of the balls. The number of ways in which the balls, oneeach in a box, could be placed such that a ball does not go to a box of its own colour isa) 9 b) 24 c) 12 d) none of these

20. The number of ways in which four persons be seated at a round table, so that all shall not have the same neighbours in any two arrangementsisa) 24 b) 6 c) 3 d) 4

21. A library has a copies of one book, b copies of each of two books, c copies of each of three books, and single copy of d books. The totalnumber of ways in which these books can be distributed is

a) a b c da b c

+ + +b g!! ! !

b) a b c da b c+ + +2 3

2 3b g!

!( !) ( !)c)a b c d

a b c+ + +2 3b g!!( ! !

d) none of these

22. The number of parallelograms that can be formed from a set of four parallel lines intersecting another set of three parallel lines isa) 6 b) 18 c) 12 d) 9

23. The number of 5 - digit even numbers that can be made with the digits 0, 1, 2 and 3 isa) 384 b) 192 c) 768 d) none of these

24. There are 10 points in a plane of wlhich no three points are collinear and 4 points are concyclic. The number of different circles that can bedrawn through at least 3 points of these points isa) 116 b) 120 c) 117 d) none of these

25. In a polygon no three diagonals are concurrent. If the total number of points of intersection of diagonals interior to the polygon be 70 thenthe number of diagonals of the polygon isa) 20 b) 28 c) 9 d) none of these

26. The number of triangles that can be formed with 10 points as vertices, n of them being collinear, is 110. The n isa) 3 b) 4 c) 5 d) 6

27. The number of possible outcomes in a throw of n ordinary dice in which at least one of the dice shows an odd number isa) 6 1n − b) 3 1n − c) 6 3n n− d) none of these

28. From 4 gentelmen and 6 ladies a committee of five is to be selected. The number of ways in which the committee can be formed so thatgentlemen are in majority isa) 66 b) 156 c) 60 d) none of these

29. The number of ways in which the letters of the word ARTICLE can be rearranged so that the even places are always occupied byconsonants isa) 576 b) 4

3 4C × !b g c) 2 4!b g d) none of these


30. The number of ways in which a couple can sit around a table with 6 guests if the couple take consecutive seats isa) 1440 b) 720 c) 5040 d) none of these

31. There are 4 mangoes, 3 apples, 2 oranges and 1 each of 3 other varieties of fruits. The number of ways of selecting at least one fruit of eachkind isa) 10 ! b) 9 ! c) 4 ! d) none of these

32. The number of proper divisors of rqp 15.6.2 is

a) p q q r r+ + + + +1 1 1b gb gb g b) p q q r r+ + + + + −1 1 1 2b gb gb gc) p q q r r+ + −b gb g 2 d) none of these

33. The number of ways to give 16 different things to three persons A, B, C so that B gets 1 more than A and C gets 2 more than B, is

a)164 7

!!5! ! b) 4 7!5! ! c)

163 5 8!! ! d) none of these

34. If 4

)a2a(2

)a2a( CC −− = , then =aa) 2 b) 3 c) 4 d) none of these

35. The number of diagonals for n sided polygon isa) 2/)1n(n − b) 6/)2n()1n(n −− c) )1n(n − d) 2/)3n(n −

36. A bag contains unlimited number of black, blue, orange and red balls. The number of ways to select 10 balls so that the selection includesat least one ball of each colour isa) 1001 b) 286 c) 270 d) none of these

37. Six Xs have to be placed in the squares of the figure given below such that each row contains at least one X. The number of ways in whichthis can be done is

1R

2R

3R


38. The number of ways to select 3 numbers in A.P from the first 1n2 + natural numbers is given by

a) 2)1n(41

− b)5n2

c) 2n d) none of these

ANSWERS

1. c 2. c 3. b 4. b 5. a 6. a 7. c8. b 9. b 10. c 11. a 12. c 13. d 14. a15. a 16. a 17. c 18. a 19. a 20. c 21. b22. b 23. a 24. c 25. a 26. c 27. c 28. a29. a 30. a 31. c 32. b 33. a 34. b 35. d36. d 37. a 38. c


BINOMIAL THEOREM

1. The number of terms in the expansion of 2 3 4x y z n+ −b g is

a) n +1 b) n + 3 c)n n+ +1 2

2b gb g

d) none of these

2. Given positive integers r n> >1 2, and the coefficients of (3r)th and (r + 2)th terms in the binomial expansion of 1 2+ x nb g are equal.Thena) n = 2r b) n = 3 r c) n = 2r + 1 d) none of these

3. The sum of the series 20

0

10

rrC

=∑ is

a) 220 b) 219 c) 2 12

19 2010+ C d) 2 1

219 20

10− C

4. If | x | < 1, then the coefficient of xn in the expansion of 1 2 3 2+ + +x x x .....c h is

a) n b) n - 1 c) n + 2 d) n +1

5. If A and B are coefficients of xn in the expansions of 1 2+ x nb g and 1 2 1+ −x nb g respectively, thena) A = B b) 2 A + B c) A = 2 B d) none of these

6. If the ( r + 1)th term in the expansion of18

3

3

+ab

ba contains a and b having the same power, then the value of r is

a) 9 b) 10 c) 8 d) 6

7. The coefficient of x4 in the expansion of 1 2 3 11+ + +x x xc h is

a) 900 b) 909 c) 990 d) 999

8. If the coefficient of the middle term in the expansion of 1 2 2+ +x nb g is p and the coeficients of middle terms in the expansion of ( ) 121 ++ nxare q and r , thena) p + q = r b) p + r = q c) p = q + r d) p + q + r = 0

9. The coefficient of x5 in the expansion of 1 12 5 4+ +x xc h b g is


10. The 14th term from the end in the expansion of x y−d i17 is

a) 175

6 5C x y−d i b) 17 6

11 3C x yd i c) 174

13 2 2C x y/ d) none of these

11. If the sum of the coefficients in the expansion of α α2 2 512 1x x− +c h vanishes, then the value of α is

a) 2 b) -1 c) 1 d) -2


12. If n is even, then the greatest coefficient in the expansion of x a n+b g is

a) nCn21+ b) nCn

21− c)

n C n2

d) none of these

13. If there is a term containing x r2 in xx

n

+FHGIKJ

−12

3

, then

a) n - 2r is an integral multiple of 3 b) n - 2r is evenc) n - 2r is odd d) none of these

14. The value of the sum of the series 3 8 13 180 1 2 3. ...n n n nC C C C− + − + , when n is an even natural no.

a) 0 b) 3n c) 5n d) none of these

15. Let 10

+ ==∑x C xn

rr

nrb g and C

CCC

CC

n CC k

n nn

n

1

0

2

1

3

2 1

2 3 1 1+ + + + = +−

... b g , then the value of k is

a) 1/2 b) 2 c) 1/3 d) 3

16. The value of 2 C C C C C0

2

1

3

2

4

3

11

1022

23

24

211

+ + + +... is

a)3 111

11 −b)2 111

11 −c)

111113 − d)

11 111

2 −

17. If x + y = 1, then r C x ynr

r n r

r

n2

0

−

=∑ equals

a) n x y b) n x x y n( )+ c) n x n x y( )+ d) none of these.

18. The coefficient of xn in the expansion of 1

1 3( ) ( )− −x x is

a)3 12 3

1

1

n

n

+

+

−.

b) 3 13

1

1

n

n

+

+

− c) 2 3 13

1

1

n

n

+

+

−FHG

IKJ d) none of these

19. If ( r + 1 )th term is the first negative term in the expansion of ( 1 + x )7/2 , then the value of r isa) 5 b) 6 c) 4 d) 7

20. The approximate value of ( . ) /7 995 1 3 correct to four decimal places isa) 1.9995 b) 1.9996 c) 1.9990 d) 1.9991

21. The term independent of x in the expansion of ( ) ( / )1 1 1+ +x xn n is

a) C C C n Cn02

12

22 22 3 1+ + + + +. ..... ( ) b) ( .... )C C Cn0 1

2+ + +

c) C C Cn02

12 2+ + +.... d) none of these.

22. The largest term in the expansion of ( 3 + 2 x )50 where x = 1/5 isa) 5th b) 51st c) 7th d) none of these


23. If the coefficients of x7 and x8 in ( 2 + x / 3)n are equal, then n is equal toa) 56 b) 55 c) 45 d) 15

24. The ratio of the coefficient of x15 to the term independent of x in xx

2152

+FHGIKJ is

a) 1 / 4 b) 1 / 16 c) 1 / 32 d) 1 / 64.

25. For each n x yn n∈ +− −N, 2 1 2 1 is divisible bya) x + y b) ( x + y )2 c) x3 + y3 d) none of these

26. The middle term of ( 1 + x )2n is

a) 1 3 5 2 1 2. . ........( )!n

nxn n− b) 2n xn!

c) n xn! d) none of these

27. Coefficient of x3 in the expansion of ( ) /1 2 1 2− −x isa) - 5/2 b) 5/2 c) 3/2 d) - 3/2

28. The greatest integer less than or equal to ( )2 1 6+ isa) 196 b) 197 c) 198 d) 199

29. The number of distinct terms in the expansion of ( ..... )x x xn1 23+ + + is

a) n C+13 b) n C+2

3 c) n C+33 d) none of these

30. If ( ) .... ,1 20 1 2

22

2+ + = + + + +x x a a x a x a xnn

n then value of a a a a a n02

12

22

32

22− + − + +.... is

a) n! b) 2nnC c) n! / 2 d) none of these

31. The remainder when 22000 is divided by 17 isa) 1 b) 2 c) 8 d) none of these

32. If n is a natural number which is not a multiple of 3 and ( 1 + x + x2 )n = a xrr

r

n

,=∑0

2

then value of ( ) ( ) ( )−=∑ 10

r

r

n

rn

ra C is

a) - 1 b) 2 c) 0 d) ( - 1)n

33. The coefficient of x3 in the expansion of ( )1 2 5− +x x isa) 10 b) - 20 c) - 50 d) - 30

34. The coefficient of x3 in the expansion of ( ) ( )1 15 2 3 4− + + +x x x x isa) 4 b) - 4 c) 0 d) none of these

35. 1 2 31 2 3. . . .... .n n n nnC C C n C+ + + + is equal to

a) n n n( ) .+14

2 b) 2 n + 1 - 3 c) n . 2 n - 1 d) none of these


36. The value of r CC

nr

nrr

.−=

∑11

10

is equal to

a) 5 2 9( )n − b) 10 n c) 9 ( n - 4 ) d) none of these

37. The sum 20 020

120

220

10C C C C+ + + +..... is equal to

a) 2 20!10

202+

( !)b) 2

12

20!10

192− .

( !)c) 219 20

10+ C d) none of these

38. The sum of the last ten coefficients in the expansion of (1 + x)19 when expanded in ascending powers of x isa) 218 b) 219 c) 2 18 - 19C10 d) none of these

ANSWERS

1. c 2. a 3. c 4. d 5. c 6. a 7. c8. c 9. b 10. c 11. c 12. c 13. a 14. a15. b 16. a 17. c 18. a 19. a 20. b 21. c22. c 23. b 24. c 25. a 26. a 27. b 28. b29. b 30. a 31. a 32. c 33. d 34. c 35. c36. a 37. d 38. a


Infinite Series

1. The product of the following series 11

12

13

12

13

14

15! ! !

....! ! ! !

....+ + +FHG

IKJ − + − +FHG

IKJ is

a) e -1 b) 1 - e-1 c) e - 1 d) none of these

2. The sum of the series 1 12

14

16!

+ + + +! !

..... is

a) e e− −1 b)12

1( )e e+ − c)12

1( )e e− − d) none of these


13

14

2 2 3

++

++ +

++ + +

+a a a a a a! ! !

.... is

a)e ea

a −−1

b)ea

a

−1c)e aa

a −−1

d)e ea

a a+−

−

1


45

67

89! ! ! !

...+ + + + is

a) e b) e-1 c)12

1( )e e+ − d) none of these


52

73

+ + + +! ! !

.... is

a) 2e b) e - 1 c) 3e d) 5e - 7


33

44

2 2 2

+ + + +! ! !

.... is

a) e/2 b) e c) 3e / 2 d) 2e


23

24

2

+ + + +! ! !

..... is

a) e 2 b) e 2 + 1 c) e 2 - 1 d) none of these

8. The sum of the series 1 40!

2 51

3 62

4 73

5 84

. .!

.!

.!

.!

....+ + + + + is

a) 11 e b) 10 e c) 9 e d) 8 e

9. The sum of the series 1 . + 3 + 2 41 2

3 51 2 3

4 61 2 3 4

..

.. .

.. . .

....+ + + is

a) e b) 2e c) 3e d)1+ 4e

10. The sum of the series a ba

a ba

a ba

−+

−FHGIKJ +

−FHGIKJ +

12

13

2 3

.... is

a) 2 log ( a / b ) b) log 2 - 1 c) log ( b / a ) d) none of these


11. The sum of the series log4 2 - log8 2 + log16 2 - ..... is

a) e2 b) 1 + log2 c) log3 - 2 d) 1 - log2


12 2

13 2

14 22 3 4− + − +

. . .... is

a) log 2 b) log ( 1/2 ) c) log ( 3/2 ) d) none of these

13. The sum of the series .......31

21

61

31

21

41

31

21

21

3322 +

++

−−

+ is

a) log 2 b) 2/3log c) log 3 d) log 20

14. ( 1 - x )3/2 can be expanded in ascending powers of x ifa) − < <1 1x b) x < −1 c) x > 1 d) none of these

15. In the expansion of ( 1 + x ) -2, | x | < 1, the 5th term is

a) - 6x5 b) 54

4C x c) −5 4x d) none of these

16. The coefficient of x5 in the expansion of 11

12+

+<

xx

x,| | , is

a) -1 b) 2 c) 0 d) - 2

17. The coefficient of xn in the expansion of e x2 3+ is

a)2n

n!b)en

n3 2.!

c)en

n2 3.!

d) none of these

18. In the expansion of e xx

x − −12 in ascending powers of x, the fourth term is

a)15

3

!x b)

14

4

!x c)

13

3

!x d) none of these

19. If | x | < 1, the coefficient of x3 in the expansion of 11e xx . ( )+

is

a)176 b) −

176 c) −

116 d) none of these

20. The constant term in the expansion of x xxe+ −log ( )13

a) −13 b) 0 c) −

12 d)

13

21. 10!

21

32

43

+ + + + ∞! ! !

... . to is equal to

a) 4e b) 3e c) 2e d) none of these


22.1

2 11

2

nx

n

n

−=

∞

∑ . is equal to

a)

−+xxx

11log

2 b)12

11

2

2log +−FHGIKJ

xx

c)1 12

− loge ed) none of these

23. If x x x x y− + − + ∞ =2 3 4

2 3 4.... to then y y y

+ + + ∞2 3

2 3! !.... to is equal to

a) - x b) x c) x + 1 d) none of these

24. If e ee

x x

x

5

3

+ is expanded in a series of ascending powers of x and n is an odd natural number, then the coefficient of xn is

a) 2 n

n !b)

22

1n

n

+

( )!c)

22

2n

n( )! d) none of these

25.nnn

2

1 !=

∞

∑ is equal to

a) 2 e b) 3 e c)12e d) none of these

26. If ex

B B x B x B xx

nn

1 0 1 22

−= + + + + +.... ....., then B Bn n− −1 equals

a)1n! b)

11( )!n − c)

1 11n n! ( )!

−− d) 1

27. ex x x x− − − + − − − +1 1

21 1

31 1

412 3 4( ) ( ) ( ) .....

is equal toa) log ( x - 1 ) b) log x c) x d) none of these

28. The sum of the series log log log .... ,4 8 162 2 2− + − is

a) e2 b) loge 2 1+ c) loge 3 2− d) 1 2− log .e

29. The coefficient of xn in the expansion of log ( ),1 2+ +x x when n is not a multiple of 3, is

a) −2n b)

1n c)

2n d) none of these.

30. The coefficient of xn in the expansion of log ( )a x1+ is

a)( )− −1 1n

nb)

( ) log− −1 1n

ane c) ( ) log− −1 1n

ena d)

( ) log .−1 n

ane


ANSWERS

1. b 2. b 3. a 4. b 5. c 6. d 7. d8. a 9. d 10. d 11. d 12. c 13. b 14. a15. b 16. d 17. b 18. a 19. d 20. a 21. c22. a 23. b 24. d 25. a 26. a 27. c 28. d29. b 30. b


STRAIGHT LINES AND PAIR OF STRAIGHT LINES

1. Let P and Q be points on the line joining A ( -2, 5 ) and B ( 3, 1 ) such that AP = PQ = QB. Then the mid point of PQ isa) ( 1/2, 3 ) b)( -1/2, 4 ) c) ( 2, 3 ) d) ( -1, 4 )

2. If O be the origin and if P1 ( x1 , y1 ) and P2 ( x2 , y2 ) be two points, then OP1.OP2 cos ( )∠POP1 2 is equal to

a) x y x y1 2 2 1+ b) ( ) ( )x y x y12

12

22

22+ + c) ( ) ( )x x y y1 2

21 2

2− + − d) x x y y1 2 1 2+

3. The coordinates of the middle points of the sides of a triangle are ( 4, 2 ) ( 3, 3 ) and ( 2, 2 ), then the coordinates of its centroid area) ( 3, 7/3 ) b) ( 3, 3 ) c) ( 4, 3 ) d) none of these

4. If A and B are two fixed pionts, then the locus of a point which moves in such a way that the angle APB is a right angle isa) a circle b) an ellipse c) a parabola d) none of these.

5. If a variable line passes through the point of intersection of the lines x + 2y - 1 = 0 and 2x - y - 1= 0 and meets the coordinate axes in A andB, then the locus of the mid point of AB is

a) x y+ =3 0 b) x y+ =3 10 c) x y xy+ =3 10 d) none of these.

6. The image of the point ( -1, 3 ) by the line x - y = 0 isa) ( 3, -1 ) b) ( 1, -3 ) c) ( -1, -1 ) d) ( 3, 3 )

7. If ( - 4, 0 ) and ( 1, -1 ) are two vertices of a triangle of area 4 square units, then its third vertex lies on

a) y = x b) 5 12 0x y+ + = c) x y+ − =5 4 0 d) none of these

8. The equation of the line with gradient -3 / 2 which is concurrent with the lines 4 3 7 0x y+ − = and 8 5 1 0x y+ − = is

a) 3 2 2 0x y+ − = b) 3 2 63 0x y+ − = c) 2 3 2 0y x− − = d) none of these

9. The medians AD and BE of the triangle with vertices A ( 0, b ), B ( 0, 0 ) and C ( a, 0 ) are mutually perpendicular if

a) b a= 2 b) a b= 2 c) b a= − 2 d) a b= −2 2

10. The point A ( 2, 1 ) is translated parallel to the line x - y = 3 by a distance 4 units, If the new position 'A is in third quadrant, then thecoordinates of 'A are

a) ( , )2 2 2 1 2 2+ + b) ( , )− + − −2 2 2 1 2 2 c) ( , )2 2 2 1 2 2− − d) none of these

11. All points lying inside the triangle formed by the points ( 1, 3 ), ( 5, 0 ) and ( -1, 2 ) satisfya) 023 ≥+ yx b) 0132 ≥−+ yx c) 01232 ≤++ yx d) 02 ≥+− yx

12. If a line joining two pionts A ( 2, 0 ) and B ( 3, 1 ) is rotated about A in anti-clockwise direction through an angle 15°, then the equation of theline in the new position is

a) 3 2 3x y− = b) 3 2 3x y+ = c) x y+ =3 2 3 d) none of these.

13. Two sides of an isosceles triangle are given by the equation 7 3 0x y− + = and x y+ − =3 0. If its third side passes through thepoint ( 1, - 10 ), then its equations area) x y x y− − = + − =3 7 0 3 31 0or b) x y x y− − = + − =3 31 0 3 7 0or c) x y x y− − = + + =3 31 0 3 7 0or d) none of these.


14. The number of lines that are parallel to 2 6 7 0x y+ + = and have an intercept of length 10 between the coordinate axes isa) 1 b) 2 c) 4 d) infinitely many.

15. The ratio in which the line 3 4 2 0x y+ + = divides the distance between 3 4 5 0x y+ + = , and 3 4 5 0x y+ − = , isa) 7 : 3 b) 3 : 7 c) 2 : 3 d) none of these.

16. A ray of light coming from the point ( 1, 2 ) is reflected at a point A on the x - axis and then passes through the point ( 5, 3 ). The coordinatesof the point A are

a)135

0,FHGIKJ b)

513

0,FHGIKJ c) −7 0,b g d) none of these.

17. The equation(s) of the bisector(s) of that angle between the lines x y x y+ − = − − =2 11 0 3 6 5 0, which contains the point( 1, -3 ) isa) 3x = 19 b) 3y = 7 c) 3x = 19 and 3y = 7 (d) none of these.

18. The orthocentre of the triangle formed by the lines xy = 0 and x + y = 1 isa) ( 1/2, 1/2 ) b) ( 1/3, 1/3 ) c) ( 0, 0 ) d) ( 1/4, 1/4 )

19. If the sum of the distances of a point from two perpendicular lines in a plane is 1, then its locus isa) square b) circle c) straight line d) two intersecting lines.

20. The equations to a pair of opposite sides of a parallelogram are x x2 5 6 0− + = and y y2 6 5 0− + = . The equations to its diagonalsarea) x y y x+ = = −4 13 4 7and b) 4 13 7x y y x+ = = −and 4c) 4 13 4 7x y y x+ = = −and d) y x y+ x− = =4 13 4 7and .

21. The line which is parallel to x-axis and crosses the curve y x= at an angle of 45° isa) x = 1/4 b) y = 1/4 c) y = 1/2 d) y = 1.

22. The point ( 4, 1 ) undergoes the following two successive transformations :(i) reflection about the line y = x(ii) translation through a distance 2 units along the positive x-axis.Then the final coordinates of the point area) ( 4, 3 ) b) ( 3, 4 ) c) ( 1, 4 ) d) (7/2, 7/2).

23. Area of the quadrilateral formed by the lines | x | + | y | = 1 isa) 4 b) 2 c) 8 d) none of these.

24. If a line is perpendicular to the line 5 0x y− = and forms a triangle with coordinate axes of area 5 sq. units, then its equation is

a) x y+ ± =5 5 2 0 b) x y− ± =5 5 2 0 c) 5 5 2 0x y+ ± = d) 5 5 2 0x y− ± =

25. A point equidistant from the lines 4 3 10 0 5 12 26 0x y x y+ + = − + =, and 7 24 50 0x y+ − = isa) ( 1, -1 ) b) ( 1, 1 ) c) ( 0, 0 ) d) ( 0, 1 ).

26. The points ( , ), ( , ), ( , )k 1 k 2 k k 1 k 1 k− + + + are collinear fora) any value of k b) k = - 1/2 only c) no value of k d) integral values of k only

27. The straight lines x + y - 4 = 0, 3x + y - 4 = 0 and x + 3y - 4 = 0 from a triangle which isa) isosceles b) right angled c) equilateral d) none of these.


28. If the mid point of join of ( x, y + 1 ) and ( x + 1, y + 2 ) is ( 3/2, 5/2 ); then the mid point of join of ( x - 1, y + 1 ) and (x + 1, y - 1 ) isa) ( -1, -1 ) b) ( -1, 1 ) c) ( 1, -1 ) d) ( 1, 1 )

29. If x cos α + y sin α = p where p = - sin α tanα be the equation of a line, then the length of the perpendiculars on the line from thepoints ( a2, 2a ), ( ab, a + b ) and ( b2, 2b ) form

a) A.P. b) G.P. c) H.P. d) none of these

30. The locus of the point equidistant from the points ( a + b, b - a ) and ( a - b, a + b ) isa) ax + by = 0 b) ax - by = 0 c) bx - ay = 0 d) ( a + b ) x + ( a - b ) y = 0

31. The number of lines that can be drawn through the point ( 4, -5 ) at a distance 12 from the point ( -2, 3 ) isa) 0 b) 1 c) 2 d) infinite

32. If a, b, c from an A.P. with common difference d )1(≠ and x, y , z from a G.P. with common ratio r )1(≠ then the area of the triangle withvertices; ( a, x ), ( b, y ) and ( c, z ) is independent ofa) a b) d c) x d) r

33. If m1 , m2 are the roots of the equation x2 -ax - a - 1 = 0, then the area of the triangle formed by the three straight lines y = m1 x,y = m2 x and y )1( −≠= aa is

a) a aa

2 22 1

( )( )

++

if a > - 1 b)− +

+a aa

2 22 1

( )( )

if a < - 1 c) − ++

a aa

2 22 1

( )( )

if Ra∈ d) a aa

2 22 1

( )( )

++

if a < - 1

34. If the product of the intercepts made by the line x tan a + y sec a = 1 on the coordinate axis is sin a, then a is equal toa) π / 4 b) π / 6 c) π / 3 d) 2 π / 3

35. Area of the rhombus enclosed by the lines ax by c± ± = 0 is

(a) 2 2a bc/ b) 2 2b ca/ c) 2 2c ab/ d) none of these.

36. The distance between the orthocentre and circumcentre of the triangle with vertices ( 0, 0 ), ( 0, a ) and ( b, 0 ) is

a) a b2 2 2− / b) a + b c) a- b d) a b2 2 2+ /37. If a, b, c from a G.P. with common ratio r, the sum of the ordinates of the points of intersection of the line ax + by + c = 0 and the curve

x + 2y2 = 0 is

a) −r 2 2/ b) −r / 2 c) r / 2 d) r 2 2/

38. If a,b,c are in A.P, then the st. line 0=++ cbyax will always pass thro’ a fixed point whose co-ordinates are

a) ( )2,1 − b) (-1, 2) c) (1, 2) d) (-1, -2)39. The point (4,1) undergoes the following three transformations successively

i) reflection about the line y = xii) translation thro’ a distance of 2 units along the positive direction of x-axisiii) rotation thro’ an angle of 4/π about the origin in the counter-clockwise direction. The final position of the point is given by

a)

2

7,2

1b) ( )27,2− c)

−

27,

21

d) ( )27,2

40. The equations of the sides of a triangle are 11,0 cxmyx +== and 22 cxmy += . The area of the trinagle is

a)21

21mmcc

−−

b)( )

21

221

21

mmcc−−

c) ( )221

2121

mmcc

−

−d)

( )( )221

221

mmcc

−

−


ANSWERS

1. a 2. d 3. a 4. a 5. c 6. a 7. c8. a 9. b 10. c 11. a 12. a 13. c 14. b15. b 16. a 17. a 18. c 19. a 20. c 21. c22. b 23. b 24. a 25. c 26. a 27. a 28. d29. b 30. c 31. a 32. a 33. a 34. a 35. c36. d 37. c 38. a 39. c 40. b


CIRCLE

1. The centre of the circle passing through ( 0, 0 ) ( a, 0 ) and ( 0, b ) is

a) ( a, b ) b) a b2 2,FHGIKJ c) − −F

HGIKJ

a b2 2, d) ( , )− −a b

2. The straight line y = mx + c cuts the circle x2 + y2 = a2 in real points if

a) a m c2 21( )+ < b) a m c2 21( )− < c) a m c2 21( )+ > d) a m c2 21( )− >

3. The locus of the centre of a circle of radius 2 which rolls on the outside of the circle x y x y2 2 3 6 9 0+ + − − = is

a) x y x y2 2 3 6 5 0+ + − + = b) x y x y2 2 3 6 31 0+ + − − =

c) x y x y2 2 3 6 294

0+ + − + = d) none of these.

4. The circle described on the line joining the points ( 0, 1 ), ( a, b ) as diameter cuts the x-axis in points whose abscissae are roots of theequation

a) x ax b2 0+ + = b) x ax b2 0− + = (c) x ax b2 0+ − = d) x ax b2 0− − = .

5. A line is drawn through a fixed point P ( βα, ) to cut the circle x y r2 2 2+ = at A and B. Then PA . PB is equal to

a) ( )α β+ −2 2r b)α β2 2 2+ − r c) ( )α β− +2 2r d) none of these.

6. The centre of a circle passing through the points ( 0, 0 ), ( 1, 0 ) and touching the circle x y2 2 9+ = is

a)3212,FHGIKJ b)

1232,FHGIKJ c)

1212,FHGIKJ d)

12

2,−FHGIKJ

7. The equation x y gx fy c2 2 2 2 0+ + + + = will represent a real circle if

a) g f c2 2 0+ − < b) g f c2 2 0+ − ≥ c) always d) none of these.

8. The length of the chord cut off by y = 2x + 1 from the circle x y2 2 2+ = is

a)56 b)

65 c)

65

d)56

9. Equation of the circle with centre on the y-axis and passing through the origin and ( 2, 3 ) is

a) x y y2 2 13 0+ + = b) 3 3 13 02 2x y y+ − = c) x y x2 2 13 3 0+ + + = d) 6 6 13 02 2x y x+ − = .

10. ABCD is a square whose side is a. The equation of the circle circumscribing the square, taking AB and AD as axes of reference, is

a) x y ax ay2 2 0+ + + = b) x y ax ay2 2 0+ + − =

c) x y ax ay2 2 0+ − − = d) x y ax ay2 2 0+ − + =

11. Two perpendicular tangents to the circle x y a2 2 2+ = meet at P. Then the locus of P has the equation

a) x y a2 2 22+ = b) x y a2 2 23+ = c) x y a2 2 24+ = d) none of these


12. If the equation of a given circle is x y2 2 36+ = , then the length of the chord which lies along the line 3 4 15 0x y+ − = is

a) 3 6 b) 2 3 c) 6 3 d) none of these

13. The number of common tangents that can be drawn to the circles x y x y2 2 4 6 3 0+ − − − = and x y x y2 2 2 2 1 0+ + + + = isa) 1 b) 2 c) 3 d) 4

14. The condition that the chord x y pcos sinα α+ − = 0 of x y a2 2 2 0+ − = may subtend a right angle at the centre of the circleis

a) a p2 22= b) p a2 22= c) a p= 2 d) p a= 2 .

15. The coordinates of the point on the circle x y x y2 2 12 4 30 0+ − − + = , which is farthest from the origin area) ( 9, 3 ) b) ( 8, 5 ) c) ( 12, 4 ) d) none of these.

16. The circles x y x y2 2 0+ + + = and x y x y2 2 0+ + − = intersect at an angle ofa) 6/π b) 4/π c) 3/π d) 2/π

17. A circle of radius 5 units touches both the axes and lies in the first quadrant. If the circle makes one complete roll on x-axis along the positivedirection of x-axis, then its equation in the new position is

a) x y x y2 2 220 10 100 0+ + − + =π π b) x y x y2 2 220 10 100 0+ + + + =π π

c) x y x y2 2 220 10 100 0+ − − + =π π d) none of these

18. The number of the tangents that can be drawn from ( 1, 2 ) to x y2 2 5+ = isa) 1 b) 2 c) 3 d) 0

19. Equation of the chord of the circle x y x2 2 4 0+ − = whose mid point is ( 1, 0 ) is

a) y = 2 b) y = 1 c) x = 2 d) x = 1

20. The two circles x y2 2 5 0+ − = and x y x y2 2 2 4 15 0+ − − − =a) touch each other externally b) touch each other internally c) cut each other orthogonally d) do not intersect

21. If a circle passes through ( 1, 2 ) and cuts the circle x y2 2 4+ = orthogonally then the equation of the locus of its centre is

a) 2 4 9 0x y+ − = b)2 4 9 0x y+ + = c) 2 4 9 0x y− + = d) none of these

22. The lines 3 4 4 0x y− + = and 6 8 7 0x y− − = are tangents to the same circle. Then its radius isa) 4/1 b) 2/1 c) 4/3 d) none of these

23. If mm

iii

, , , , ,1 1 2 3 4FHGIKJ = are concyclic points, then the value of m m m m1 2 3 4 is


24. The angle between the tangents drawn from the origin to the circle ( ) ( )x y− + + =7 1 252 2 isa) π / 3 b) π / 6 c) π / 2 d) π / 8

25. The equation of a circle with origin as centre and passing through the vertices of an equilateral triangle whose median is of length 3 a is

a) x y a2 2 29+ = b) x y a2 2 216+ = c) x y a2 2 24+ = d) x y a2 2 2+ = .

26. The slope of the tangent at the point ( h, h ) of the circle x y a2 2 2+ = isa) 0 b) 1 c) -1 d) depends on h


27. The circles x y x2 2 10 16 0+ − + = and x y r2 2 2+ = intersect each other in two distinct points if

a) r < 2 b) r > 8 c) 2 < r < 8 d) 82 ≤≤ r

28. The number of common tangents to the circles x y x x y x2 2 2 20 0+ − = + + =, is

a) 2 b) 1 c) 4 d) 3

29. The equation ax hxy by gx fy c2 22 2 2 0+ + + + + = represents a circle ifa) a = h = 2, b = 0 b) b = h = 2, a = 0 c) a = b = 2, h = 0 d) none of these

30. The locus of the middle points of the chords of the circle x y a2 2 24+ = which subtend a right angle at the centre of the circle is

a) x y a+ = 2 b) x y a2 2 2+ = c) x y a2 2 22+ = d) x y x y2 2+ = +

31. Tangents are drawn from the origin to a circle with centre at ( 2, -1 ). If the equation of one of the tangents is 3x + y = 0, the equation of theother tangent is

a) 3x - y = 0 b) x + 3y = 0 c) x - 3y = 0 d) x + 2y = 0

32. A line meets the coordinate axes in A and B. A circle is circumscribed about the triangle OAB. If the distances from A and B of the tangentto the circle at the origin be m and n, then the diameter of the circle is

a) m ( m + n ) b) m + n c) n ( m + n ) d) m2 + n2

33. The circle passing through three distinct points ( 1, t ), ( t, 1 ) and ( t, t ) passes through the point (for all values of t .)

a) ( 1, 1 ) b) ( -1, -1 ) c) ( -1, 1 ) d) ( 1, -1 )

34. If OA and OB are tangents from the origin to the circle x y gx fy c2 2 2 2 0+ + + + = and C is the centre of the circle, then area of thequadrilateral OACB is

a)12

2 2c g f c+ −c h b) c g f c2 2+ −c h c) c g f c2 2+ −c h d)g f c

c

2 2+ −

35. The locus of the point of intersection of the tangents to the circle x = r cos θ , y = r sin θ at points whose parametric angles differ by π /3is

a) x y r2 2 24 2 3+ = −( ) b) 3 12 2( )x y+ =

c) x y r2 2 22 3+ = −( ) d) 3 42 2 2( )x y r+ =

36. If the two circles x y gx fy2 2 2 2 0+ + + = and x y g x f y2 21 12 2 0+ + + = touch each other, then

a) f g fg1 1= b) ff gg1 1= c) f g f g2 212

12+ = + d) none of these.

37. The equation of the chord of the circle x y a2 2 2+ = passing through the point ( 2, 3 ) farthest from the centre is

a) 2x + 3y = 13 b) 3x - y = 3 c) x - 2y + 4 = 0 d) x - y + 1 = 0

38. A circle is given by x y x y2 2 4 7 12 0+ + − + = . The points P ( 0, 0 ) and Q ( -2, 4) are such that

a) both lie inside the circle b) both lie outside the circlec) one lies inside and the other outside the circle d) one lies on the circle and the other is outside the circle


39. If the equation of a circle is ax a y x2 22 3 4 1 0+ − − − =( ) then its centre is

a) ( 2, 0 ) b) ( 2/3, 0 ) c) ( -2/3, 0 ) d) none of these

40. The equation x y x y2 2 2 4 5 0+ − + + = represents

a) a point b) a pair of straight lines c) a circle of non-zero radius d) none of these

41. C1 is a circle of radius 1 touching the x-axis and the y-axis. C2 is another circle of radius > 1 and touching the axes as well as the circle C1. Thenthe radius of C2 is

a) 3 2 2− b) 3 2 2+ c) 3 2 3+ d) none of these

42. If p and q be the longest distance and the shortest distance respectively of the point ( -7, 2 ) from any point ( βα, ) on the curve whose equation

is x y x y2 2 10 14 51 0+ − − − = then GM of p and q is equal to

a) 2 11 b) 5 5 c) 13 d) none of these

43. A region in the x-y plane is bounded by the curve y x= −25 2 and the line y = 0. If the point ( a, a + 1 ) lies in the interior of the region then

a) a ∈ −( , )4 3 b) a ∈ −∞ − ∪ +∞( , ) ( , )1 3

c) a ∈ −( , )1 3 d) none of these

44. The equation of the diameter of the circle 3 2 6 9 02 2( )x y x y+ − + − = which is perpendicular to the line 2 3 12x y+ = is

a) 3 2 3x y− = b)3 2 1 0x y− + = c) 3 2 9x y− = d) none of these

45. The line 03 =+ yx is the diameter of the circle

a) 02622 =+++ yxyx b) 02622 =+−+ yxyx

c) 02622 =−−+ yxyx d) 02822 =−++ yxyx

46. The distance between the chords of contact of the tangent to the circle 02222 =++++ cfygxyx from the origin and the point (g, f)is

a) 22 fg + b) )(21 22 cfg ++ c)

)(2 22

22

fg

cfg

+

++d)

)(2 22

22

fg

cfg

+

−+

47. The condition for the two circles 02 21

22 =+++ kxkyx and 02 22

22 =+++ kykyx to touch each other externally is

a) 222

21 kkk =+ b ) 22

221 kkk =−

c) 22

21

22

21

2 )( kkkkk =− d) 22

21

22

21

2 )( kkkkk =+

ANSWERS

1. b 2. c 3. b 4. b 5. b 6. d 7. b8. c 9. b 10. c 11. a 12. c 13. c 14. a15. a 16. d 17. d 18. a 19. d 20. b 21. a22. c 23. a 24. c 25. c 26. c 27. c 28. d29. c 30. c 31. c 32. b 33. a 34. b 35. d36. a 37. a 38. c 39. b 40. a 41. b 42. a43. c 44. a 45. b 46. d 47. d


CONIC SECTIONS - PARABOLA1. If the focus of a parabola is at ( 0, -3 ) and its directrix is y = 3, then its equation is

a) x y2 12= − b) x y2 12= c) y x2 12= − d) y x2 12=

2. The equation of the directrix of the parabola x x y2 4 3 10 0− − + = is

a) y = −54 b) y =

54 c) y = −

34 d) y =

54.

3. The angle made by a double ordinate of length 8a at the vertex of the parabola y ax2 4= isa) π / 3 b) π / 2 c) π / 4 d) π / 6

4. The coordinates of a point on the parabola y x2 8= whose focal distance is 4, are

a) ( / , )1 2 2± b) ( , )1 2 2± c) ( , )2 4± d) none of these

5. If the parabola y ax2 4= passes through ( 3, 2 ), then the length of its latus rectum isa) 2 / 3 b) 4 / 3 c) 1 / 3 d) 4.

6. The coordinates of the focus of the parabola x x y2 4 8 4 0− − − = area) ( 0, 2 ) b) ( 2, 1 ) c) ( 1, 2 ) d) ( -2, -1 ).

7. The area of the triangle inscribed in the parabola y x2 4= the ordinates of whose vertices are 1, 2 and 4 is

a) 7 / 2 sq. units b) 5 / 2 sq. units c) 3 / 2 sq. units d) 3 / 4 sq. units

8. The length of the latusrectum of the parabola whose focus is ( 3, 3 ) and directrix is 3 4 2 0x y− − = , isa) 2 b) 1 c) 4 d) none of these.

9. The locus of the points of trisection of the double ordinates of the parabola y ax2 4= is

a) y ax2 = b) 9 42y ax= c) 9 2y ax= d) y ax2 9= .

10. If y mx c= + touches the parabola y a x a2 4= +( ), then

a) cam

= b) c am am

= + c) c a am

= + d) none of these.

11. The parametric representation ( 2 + t2 , 2 t + 1 ) representsa) a parabola with focus at ( 2, 1 ) b) a parabola with vertex at ( 2, 1 ) c) an ellipse with centre at ( 2, 1 ) d) none of these.

12. If ( , )at at2 2 are the coordinates of one end of a focal chord of the parabola y ax2 4= , then the coordinates of the other end are

a) ( , )at at2 2− b) ( , )− −at at2 2 c)at

at2

2,FHGIKJ d)

at

at2

2, −FHG

IKJ

13. If the vertex and focus of a parabola are ( 3, 3 ) and ( -3, 3 ) respectively, then its equation is

a) x x y2 6 24 63 0+ − + = b) x x y2 6 24 63 0− + − =

c) y y x2 6 24 63 0− + − = d) y y x2 6 24 63 0+ − + = .


14. The locus of the foot of the perpendicular from the focus upon a tangent to the parabola y ax2 4= isa) the directrix b) tangent to the vertex c) x = a d) none of these.

15. The slope of the normal at the point ( , )at at2 2 of the parabola, y ax2 4= is,

a)1t b) t c) - t d) -

1t

16. If 2x + y + k = 0 is a normal to the parabola y x2 8= − , then the value of k isa) - 16 b) - 8 c) - 24 d) 24.

17. The equation of the normal at the point of contact of a tangent am

am2

2,FHGIKJ is

a) y mx am am= − −2 3 b) m y m x am a3 2 22= − − c) m y am m x a3 2 22= − + d) none of these.

18. If the vertex of the parabola y x x c= − +2 8 lies on x-axis, then the value of c isa) - 16 b) - 4 c) 4 d) 16.

19. The locus of the point of intersection of the ⊥ tangents to the parabola x ay2 4= isa) y = a b) y = - a c) x = a d) x = - a

20. If PSQ is the focal chord of the parabola y x2 8= such that SP = 6. Then the length SQ isa) 6 b) 4 c) 3 d) none of these

21. The curve represented by x = 3 ( cos t + sin t ), y = 4 ( cos t - sin t ) isa) ellipse b) parabola c) hyperbola d) circle

22. The graph represented by the equations x = sin2 t y = 2 cos t isa) a portion of a parabola b) a parabola c) a part of sine graph d) a part of hyperbola.

23. The tangents at the points ( , ), ( , )at at at at12

1 22

22 2 on the parabola y2 = 4ax are at right angles if

a) t t1 2 1= − b) t t1 2 1= c) t t1 2 2= d) t t1 2 2= −

24. The two ends of latusrectum of a parabola are the points ( 3, 6 ) and ( -5, 6 ). The focus isa) ( 1, 6 ) b) ( -1, 6 ) c) ( 1, -6 ) d) ( -1, -6 ).

25. Three normals to the parabola y x2 = are drawn through a point ( C, 0 ), thena) C = 1 / 4 b) C = 1 / 2 c) C > 1 / 2 d) none of these

26. The circles on focal radii of a parabola as diameter toucha) the tangent at the vertex b) the axis c) the directrix d) none of these.

27. The focus of the parabola 4 12 20 67 02y x y+ − + = isa) ( -7/2, 5/2 ) b) (-3/4, 5/2 ) c) (-17/4, 5/2 ) d) (5/2, -3/4 )

28. The line x + y = 6 is a normal to the parabola. y2 = 8x at the pointa) ( 18, -12 ) b) ( 4, 2 ) c) ( 2, 4 ) d) ( 3, 3 )

29. The straight line x + y = k + 1 touches the parabola y = x ( 1 - x ). Ifa) k = -1 b) k = 0 c) k = 1 d) k takes any value


30. A line bisecting the ordinate PN of a point P ( at2, 2at ), t > 0, on the parabola y2 = 4ax is drawn parallel to the axis to meet the curve at Q. IfNQ meets the tangent at the vertex at the point T, then the coordinates of T area) ( 0, (4/3) at ) b) ( 0, 2 at ) c) ( (1/4) at2, at ) d) ( 0, at )

31. If perpendiculars are drawn on any tangent to a parabola y2 = 4ax from the points ( , )a k± 0 on the axis. The difference of their squaresisa) 4 b) 4a c) 4k d) 4ak

32. For the parabola y x y2 8 12 20 0+ − + = , which of the following are correcta) vertex ( 2, 6 ) b) focus ( 1 , 6 )c) Length of the latusrectum = 4 d) axis is y = 0

33. Equation of a tangent to the parabola y2 = 7x which is inclined at an angle of 45° to the axis of the parabola isa) 4 4 7 0x y− + = b) 4x + 4y - 7 = 0 c) 7 7 4 0x y− + = d) 7 7 4 0x y+ + =

34. The point on the parabola y2 = 36 x, whose ordinate is three times its abcissa area) ( 0, 1) b) ( 3, 9 ) c) ( 4, 12 ) d) ( 6, 18 )

35. The slope of tangent to the parabola y2 = 9x which passes through ( 4, 10 ) isa) 9 / 4 b) 5 / 4 c) 3 / 4 d) 1 / 3

36. The point of contact of the tangent in the above example isa) ( 4, 2 ) b) ( 36, 18 ) c) ( 4, 6 ) d) ( 1/4, 3/2 )

37. If the focus of a parabola is ( -2, 1 ) and the directrix has the equation x + y = 3 then the vertex isa) ( 0, 3 ) b) ( -1, 1/2 ) c) ( -1, 2 ) d) ( 2, -1 )

38. If the vertex and the focus of a parabola are ( -1, 1 ) and ( 2, 3 ) respectively then the equation of the directrix isa) 3 2 14 0x y+ + = b) 3 2 25 0x y+ − = c)2 3 10 0x y− + = d) none of these.

39. If the vertex = ( 2, 0 ) and extremities of the latus rectum are ( 3, 2 ) and ( 3, -2 ) then the equation of the parabola is

a) y x2 2 4= − b) x y2 4 8= − c) y x2 4 8= − d) none of these.

40. The equation of the parabola whose vertex and focus are on the positive side of the x-axis at distance a and b respectively from the originis

a) y b a x a2 4= − −( ) ( ) b) y a b x b2 4= − −( ) ( )

c) x b a y a2 4= − −( ) ( ) d) none of these.

41. The equation λ λx xy y x y2 24 3 2 0+ + + + + = represents a parabola if λ isa) - 4 b) 4 c) 0 d) none of these.

42. The equation of the axis of the parabola 9 16 12 57 02y x y− − − = is

a) 2x = 3 b) y = 3 c) 3y = 2 d) x + 3y = 3

43. The parametric equation of a parabola is x t y t= + = +2 1 2 1, . The Cartesian equation of its directrix is

a) x = 0 b) x + 1 = 0 c) y = 0 d) none of these.

44. If two tangents drawn from the point ( βα, ) to the parabola y2 = 4x be such that the slope of one tangent is double of the other then

a) β α=29

2 b) α β=29

2 c) 2 9 2α β= d) none of these.


ANSWERS

1. a 2. b 3. b 4. c 5. b 6. b 7. d8. a 9. b 10. b 11. b 12. d 13. c 14. b15. c 16. d 17. c 18. d 19. b 20. c 21. a22. b 23. a 24. b 25. c 26. c 27. c 28. c29. b 30. a 31. d 32. a 33. a 34. c 35. a36. b 37. c 38. a 39. c 40. a 41. b 42. c43. a 44. b


ELLIPSE AND HYPERBOLA

1. The equation ax hxy by gx fy c2 22 2 2 0+ + + + + = represents an ellipse if

a) ∆ = <0 2, h ab b) ∆ ≠ <0 2, h ab c) ∆ ≠ >0 2, h ab d) ∆ ≠ =0 2, h ab

2. The equation to the ellipse (referred to its axes as the axes of x and y respectively ) whose foci are ( , )±2 0 and eccentricity 1/2, is

a)x y2 2

12 161+ = b)

x y2 2

16 121+ = c)

x y2 2

16 81+ = d) none of these.

3. The eccentricity of the ellipse 9 5 30 02 2x y y+ − = isa) 1 / 3 b) 2 / 3 c) 3 / 4 d) none of these.

4. Equations )(sin,cos babyax >θ=θ= represent a conic section whose eccentricity e is given by

a) e a ba

22 2

2=+

b) e a bb

22 2

2=+

c) ea ba

22 2

2=−

d) ea bb

22 2

2=−

.

5. If A and B are two fixed points and P is a variable point such that PA + PB = 4, the locus of P isa) a parabola b) an ellipse c) a hyperbola d) none of these.

6. If the focal distance of an end of the minor axis of an ellipse ( referred to its axes as the axes of x and y respectively ) is k and the distancebetween its foci is 2h, then its equation is

a)xk

yh

2

2

2

2 1+ = b)xk

yk h

2

2

2

2 2 1+−

= c)xk

yh k

2

2

2

2 2 1+−

= d)xk

yk h

2

2

2

2 2 1++

=

7. The equation xa

ya

2 2

10 41

−+

−= represents an ellipse if

a) a < 4 b) a > 4 c) 4 < a < 10 d) a > 10.

8. The curve with parametric equations θ+=θ+= sin32,cos41 yx isa) an ellipse b) a parabola c) a hyperbola d) a circle

9. If S and 'S are two foci of an ellipse xa

yb

a b2

2

2

2 1+ = <( ) and P ( x1 , y1 ) a point on it, then SP + 'S P is equal to

a) 2a b) 2b c) a + e x1 d) b + e y1

10. The length of the latus-rectum of the ellipse 5 9 452 2x y+ = is

(a)53 b)

103 c)

2 55

d)53

11. A set of points is such that each point is three times as far away from the y-axis as it is from the point ( 4, 0 ). Then the locus of the pointsisa) hyperbola b) parabola c) ellipse d) circle.

12. The product of the perpendiculars drawn from the two foci of an ellipse to the tangent at any point of the ellipse isa) a2 b) b2 c) 4a2 d) 4b2

.


13. If the normal at one end of the latusrectum of an ellipse xa

yb

2

2

2

2 1+ = passes through one end of the minor axis, then

a) e4 - e2 + 1 = 0 b) e2 - e + 1 = 0 c) e2 + e + 1 = 0 d) e4 + e2 - 1 = 0

14. If a tangent having slope of −43

to the ellipse x y2 2

18 321+ = intersects the major and minor axes in points A and B respectively, then the

area of ∆ OAB is equal toa) 12 sq. units b) 48 sq. units c) 64 sq. units d) 24 sq. units

15. If e is the eccentricity of the ellipse xa

yb

2

2

2

2 1+ = ( a < b ) , then

a) b a e2 2 21= −( ) b) a b e2 2 21= −( ) c) a b e2 2 2 1= −( ) d) b a e2 2 2 1= −( )

16. The number of normals that can be drawn from a point to a given ellipse isa) 2 b) 3 c) 4 d) 1

17. The radius of the circle passing through the foci of the ellipse x y2 2

16 91+ = , and having its centre ( 0, 3 ) is

a) 4 b) 3 c) 12 d) 7/2.

18. The centre of the ellipse ( ) ( )x y x y+ −+

−=

29 16

12 2

is

a) ( 0, 0 ) b) ( 1, 1 ) c) ( 1, 0 ) d) ( 0, 1 )

19. Two perpendicular tangents drawn to the ellipse x y2 2

25 161+ = intersect on the curve

a) x ae

= b) x y2 2 41+ = c) x y2 2 9+ = d) x y2 2 41− =

20. The ellipse xa

yb

2

2

2

2 1+ = and the straight line y = mx + c intersect in real points only if

a) a m c b2 2 2 2< − b) a m c b2 2 2 2> − c) a m c b2 2 2 2≥ − d) c b≥ .

21. The eccentricity of the hyperbola whose latusrectum is 8 and conjugate axis is equal to half the distance between the foci, is

a) 4 /3 b) 4 / 3 c) 2 / 3 d) none of these.

22. If e and 'e be the eccentricities of a hyperbola and its conjugate, then 1 12 2e e+

′ =


23. The eccentiricity of the hyperbola 3 4 122 2x y− = − is

a)73

b)72

c) −73

d) −72


24. The locus of the point of intersection of the straight lines xa

yb

+ = λ and λ1

=−by

ax

(λ is a variable )is

a) a circle b) a parabola c) an ellipse d) a hyperbola.

25. The equation of the chord of the hyperbola x y2 2 9− = which is bisected at ( 5, -3 ) is

a) 5 3 9x y+ = b) 5 3 16x y− = c) 5 3 16x y+ = d) 5 3 9x y− =

26. The equation of a tangent parallel to y = x drawn to x y2 2

3 21− = is

a) x y− + =1 0 b) x y− + =2 0 c) x y+ − =1 0 d) x y− + =2 0

27. If the line y = 3 x +λ touches the hyperbola 9 5 452 2x y− = , then the value of λ is

a) 36 b) 45 c) 6 d) 15.

28. A point moves in a plane so that its distance PA and PB from two fixed points A and B in the plane satisfy the relation PA - PB = k )0( ≠k ,

then the focus of P is

a) a parabola b) an ellipse c) a hyperbola d) a branch of a hyperbola

29. The equation of the conic with focus at ( 1, -1 ), directrix along x - y + 1 = 0 and with eccentricity 2 is

a) x y2 2 1− = b) xy = 1

c) 2 4 4 1 0xy x y− + + = d) 2 4 4 1 0xy x y+ − − = .

30. The centre of the hyperbola 9 16 36 96 252 02 2x y x y− − + − = area) (2,3) b) ( -2, -3 ) c) ( -2, 3 ) d) none of these.

31. The eccentricity of the hyperbola with latusrectum 12 and semi-conjugate axis 2 3 , is

a) 2 b) 3 c)32

d) 2 3.

32. The equation of the hyperbola with vertices ( 3, 0 ) and ( -3, 0 ) and semi - latusrectum 4, is given by

a) 4 3 36 02 2x y− + = b) 4 3 12 02 2x y− + = c) 4 3 36 02 2x y− − = d) none of these.

33. The equation of the tangent to the conic x y x y2 2 8 2 11 0− − + + = at ( 2, 1 ) isa) x + 2 =0 b) 2x + 1 = 0 c) x - 2 = 0 d) x + y + 1 = 0.

34. The value of m for which y = mx + 6 is a tangent to the hyperbola x y2 2

100 491− = is

a)1720

b)2017

c)320

d)203


35. The equation of the tangent to the hyperbola 4 12 2y x= − at the point ( 1, 0 ) isa) x = 1 b) y = 1 c) y = 4 d) x = 4.

36. The number of normals to the hyperbola xa

yb

2

2

2

2 1− = from an external point is

a) 2 b) 4 c) 6 d) 5

37. If the eccentricity of a conic is 3/5, the conic isa) a parabola b) an ellipse c) a hyperbola d) none of these.

38. The point ( at2, 2bt ) lies on the hyperbola, x a y b2 2 2 2 1/ /− = for

a) all values of t b) t 2 2 5= + c) t 2 2 5= − d) no real value of t

39. y = mx + c is a normal to the ellipse x a y b2 2 2 2 1/ /+ = if c2 is equal to

a)( )a ba m b

2 2 2

2 2 2

−+

b)( )a ba m

2 2 2

2 2

−c)( )a b ma b m

2 2 2 2

2 2 2

−+

d)( )a b ma m b

2 2 2 2

2 2 2

−+

40. If ',ee are the eccentricities of hyperbolas 12

2

2

2=−

by

ax and 12

2

2

2=−

ay

bx , then

a) 'ee = b) 'ee −= c) 1'=ee d) 1'1122 =+ee

ANSWERS

1. b 2. b 3. b 4. c 5. b 6. b 7. a

8. a 9. b 10. b 11. c 12. b 13. d 14. d

15. b 16. c 17. a 18. b 19. b 20. c 21. c

22. b 23. a 24. d 25. c 26. a 27. c 28. c29. c 30. a 31. a 32. c 33. c 34. a 35. a

36. b 37. b 38. b 39. c 40. d


FUNCTIONS, LIMITS AND CONTINUITY

1. Let A and B be two finite sets having m and n elements, respectively. Then the total number of mappings from A to B isa) mn b) 2mn c) mn d) nm

2. The total number of injective mappings from a set with m elements to a set with n elements, nm ≤ , is

a) mn b) nm c)n

n m!

( )!− d) n !

3. The total number of injective mappings from a finite set with m elements to a set with n elements for m > n is

a)n

n m!

( )!− b)mm n

!( )!− c) nm d) none of these

4. If f (x) = cos (loge x ), then f x f y f x y f xy( ) ( ) [ ( / ) ( )]− +12

is

a) 0 b)12f x f y( ) ( ) c) f x y( )+ d) none of these

5. If f x a xn n( ) ( ) ,/= − 1 where a > 0 and n N∈ , then fof (x) is equal toa) a b) x c) xn d) an

6. Which of the following functions from Z to itself are bijections ?

a) f x x( ) = 3 b) f x x( ) = + 2 c) f x x( ) = +2 1 d) f x x x( ) .= +2

7. If f R R: → be a mapping defined by f x x( ) ,= +3 5 then f x−1( ) is equal to

a) ( ) /x +5 1 3 b) ( ) /x −5 1 3 c) (5 ) /− x 1 3 d) 5 - x

8. Let f R R: → , g R R: → be two functions given by f x x g x x( ) , ( ) .= − = +2 3 53 Then ( ) ( )fog x−1 is equal to

a)x +FHGIKJ

72

1 3/

b) x −FHGIKJ

72

1 3/

c)x −FHGIKJ

27

1 3/

d)x −FHGIKJ

72

1 3/

9. If the function f R A: → given by f xxx

( ) =+

2

2 1 is a surjection, then A =

a) R b) [ 0, 1 ] c) ( 0, 1 ] d) [ 0, 1 )

10. Which of the following functions is inverse of itself

a) f x xx

( ) = −+

11

b) f x x( ) log= 5 c) )1(2)( −= xxxf d) none of these

11. If f (x) is defined on [0, 1] by the rule

f xx xx x

( ),,

=RST

if is rational1- if is irrational.

Then for all x f f x∈[ , ], ( ( ))0 1 is

a) constant b) 1 + x c) x d) none of these

12. The function f R R: → defined by f x x x x( ) ( ) ( ) ( )= − − −1 2 3 isa) one-one but not onto b) onto but not one-one c) both one -one and onto d) niether one-one nor onto.


13. The composite mapping fog of the maps f R R: → , f x x( ) sin= and g R R: → , g x x( ) ,= 2 is

a) x2 sin x b) ( sin x )2 c) sin x2 d)sin .xx2

14. The number of surjections from A = { 1, 2, ...., n }, n ³ 2 onto B = { a, b } is

a) n P2 b) 2n - 2 c) 2n - 1 d) none of these

15. Let f e R:( , )∞ → be defined by f (x) = log [ log ( log x )], thena) f is one-one but not onto b) f is onto but not one-onec) f is both one-one and onto d) the range of f is the subset of its codomain.

16. The value of limx

xx→

+ −

+ −0

2

2

1 19 3

is

a) 3 b) 4 c) 1 d) 2

17. The value of lim....

x

nx x x nx→

+ + + −−1

2

1 is

a) n b)n +1

2 c)n n( )+1

2 d)n n( )−1

2

18. lim cosx

xe xx→

−0 2

2

is equal to

a) 3 / 2 b) 1 / 2 c) 2 / 3 d) none of these

19. If [x] denotes the greatest integer less than or equal to x, then lim[ ] [ ] [ ] ..... [ ]

n

x x x nxn→∞

+ + + +2 32 equals

a) x / 2 b) x / 3 c) x / 6 d) 0

20. The value of limcos

sin cosx

xx x x→

−0

31 is

a) 2 / 5 b) 3 / 5 c) 3 / 2 d) 3 / 4

21. lim cos ( )x

xx→

− −−1

1 2 11

a) exists and it equals 2 b) exists and it equals – 2c) does not exist because ( )x − →1 0 d) does not exist because left hand limit is not equal to right hand limit

22. The value of lim/

x

xxx→

++FHG

IKJ0

2

2

11 51 3

2

is

a) e2 b) e c) e-1 d) none of these


23. The value of lim cosm

mxm→∞

FHGIKJ is

a) 1 b) e c) e-1 d) none of these

24. The value of lim log log

xx x

→1 555b g is

a) 1 b) e c) e-1 d) none of these

25. lim sin(sin )

,( )x

n

mxx

m n→

<0

is equal to

a) 1 b) 0 c) n / m d) none of these

26. The value of limsin

| |x

xx

x

x→∞

FHGIKJ −

−

F

H

GGG

I

K

JJJ

2 1

1 is

a) 0 b) 1 c) -1 d) none of these

27. The value of lim sinx

xxa ba→∞

FHGIKJ is ( a > 1 )

a) b log a b) a log b c) b d) none of these

28. If lim ,x a

x a

x aa xx a→

−−

= −1 then the value of a is

a) 1 b) 0 c) e d) none of these

29. limx

xxx→∞

+++FHGIKJ

21

3

is equal to

a) 1 b) e c) e2 d) e3.

30. The value of limsin

x

xx→∞

is


31. lim sinx

xx→0

is equal to

a) 0 b) 1 c) −12 d) none of these

32. lim tantanx

x xx x→

−0 2 equals

a) 1 b) 1 / 2 c) 1 / 3 d) none of these


33. If f xx

g x xx

( ) , ( )=−

=−+

23

34

and h x xx x

( ) ( ) ,= −+

+ −2 2 1

122 then lim [ ( ) ( ) ( )]x

f x g x h x→

+ +3

is

a) -2 b) -1 c) −27 d) 0

34. The value of lim ( )x

xe xx→

− +0 2

1 is

a) 0 b) 1 / 2 c) 2 d) e

35. The function f xx

x x( ) sec

[ ],=

−

−1

where [x ] denotes the greatest integer less than or equal to x is defined for all x belonging to

a) R b) R n n Z− −{( , ) ({ | . })}1 1 c) R+ -(0, 1) d) R n n N+ −{ | . }.

36. The value of lim cos cos cos .....cosn n

x x x x→∞

FHGIKJFHGIKJFHGIKJ

FHGIKJ2 4 8 2

is

a) 1 b)sin xx c)

xxsin d) none of these

37. limtanx

xx→ −0 1 2

is equal to

a) 0 b)12 c) 1 d) ∞

38. lim tan/

x

x

x→

+FHGIKJ

RSTUVW0

1

4π

is

a) 1 b) -1 c) e2 d) e

39. If f f' ( ) , ' ' ( ) ,2 2 2 1= = then lim'( )

x

x f xx→

−−2

22 42

is

a) 4 b) 0 c) 2 d) ∝

40. The value of lim sinx

xx→0 2


41. The function f (x) = 4

4

2

3

−−x

x x is

a) discontinuous at only one point b) discontinuous exactly at two pointsc) discontinuous exactly at three points d) none of these

42. If f x xx

k x

x x x

( ) cos,

,=

− − +− +

≠

=

RS|T|

36 9 4 12 1

0

0 is continuous at x = 0, then k equals

a) 16 2 2 3log log b)16 2 6ln c)16 2 2 3ln ln (d) none of these


43. The function f defined by sin ,

,

xx

x

x

2

0

0 0

≠

=

RS|T|

is

a) continuous and derivable at x = 0 b) neither continuous nor derivable at x = 0c) continuous but not derivable at x = 0 d) none of these.

44. If f x x x( ) ( )cot= +1 be continuous at x = 0, then f (0) is equal toa) 0 b) 1 / e c) e d) none of these

45. Let f x

x xx x

x

xx

( )|( ) ( )|

, ,

,,

=

− +− −

≠

==

R

S||

T||

4 25 41 2

1 2

6 1

12 2Then f (x) is continuous on the set

a) R b) R - {1} c) R - {2} d) R - { 1, 2 }

46. If f xmx x

x n x( )

,

sin ,=

+ ≤

+ >

RS|

T|1

2

2

π

π is continuous at x = π

2, then

a) m = 1, n = 0 b) mn

= +π

21 c) n m= π

2 d) m n= =π2

.

47. The function f x xx

( ) tan{ [ ]}[ ]

,=−

+π π

1 2 where [x] denotes the greatest integer less than or equal to x, is

a) discontinuous at some x b) continuous at all x, but f ' (x) does not exist for some xc) f ' (x) exists for all x. d) f ' (x) exists for all x, but f '' (x) does not exist

48. If f (x) = [ x sin p x ], then f (x) isa) discontinuous at x = 0 b) continuous in ( -1, 0 ) c) differentiable at x = 1 d) nondifferentiable in ( -1, 1 )

49. 01,11

)( <≤−−−+

= xx

pxpxxf 10,

212

≤≤−+

= xxx

is continuous in the interval [-1,1], then p equals

a) -1 b)21

− c)21

d) 1

50. The function 11 )]x4[log(2

3|x|cos)x(f −− −+

−

= is defined for

a) ]5,1[]0,1[ ∪− b) ]4,1[]1,5[ ∪−−

c) ]}3{)4,1[]1,5[ −∪−− d) }3{)4,1[ −

51. The value of the parameter α , for which the function 0,x1)x(f ≠αα+= is the inverse of itself, isa) - 2 b) - 1 c) 1 d) 2


52. =

−+

∞→ xxxx

x cossinlim

a) 0 b) 1 c) - 1 d) none of these

53. The values of x where the function 3x4x

)2x(logxtan)x(f 2 +−−

= is not defined are given by

a) }3{)2,( ∪−∞ b) }1n:2/n,3{)2,( ≥π+π∪−∞

c) )2,(−∞ d) none of these

54. The domain of definition of the function )1xlog(e3)x(f 12x −= − is

a) ),1( ∞ b) ),1[ ∞

c) set of all reals different from 1 d) ),1()1,( ∞∪−−∞

55. 0;3

/2

>>>

++∞→

cbacbaLtxxxx

x is equal to

a) 0 b) 2a c) cba d) cba

56. If 2)bax(1x1xLt 2

3

x=

+−

++

∞→, then

a) 1b,1a == b) 2b,1a == c) 2b,1a −== d) none of these

57. Let 2)x2(xsin1)x(f

−π−

= , when 2/x π≠ and k)2/(f =π . The value of k which makes f continuous at 2/x π= is

a) 1/2 b) 1/4 c) 1/8 d) none of these

58.3

24

x |x|1

xx1sinx

Lt+

+

∞−→ equals

a) - 1 b) 1 c) zero d) ∞

59. =+−

−

→ 1x1xsinLt

1

0x

a) 1 b) - 1 c) 0 d) - 2

ANSWERS

1. d 2. c 3. d 4. a 5. b 6. b 7. b8. d 9. d 10. a 11. c 12. b 13. c 14. b15. c 16. a 17. c 18. a 19. a 20. c 21. a22. a 23. a 24. b 25. b 26. a 27. c 28. a29. b 30. b 31. a 32. c 33. c 34. b 35. b36. b 37. b 38. c 39. a 40. d 41. c 42. c43. a 44. c 45. d 46. c 47. c 48. b 49. b50. c 51. b 52. b 53. b 54. a 55. b 56. c57. c 58. a 59. d


DIFFERENTIABILITY AND DIFFERENTIATION

1. The set of points where the function f (x) = x | x | is differentiable is(a) ( , )−∞ ∞ (b) ( , ) ( , )−∞ ∪ ∞0 0 (c) ( , )0 ∞ (d) [ , )0 ∞

2. Let f x x x x( ) ( | |) | |.= + Then for all x

(a) f is continuous but 'f is not continuous (b) f is differentiable for some x(c) f ' is continuous (d) none of these

3. If f x a x be c xx( ) |sin | | || |= + + 3 and if f (x) is differentiable at x = 0, then

(a) a = b = c = 0 (b) Rcba ∈=+ ;0 (c) b c a R= = ∈0, (d) c a b R= = ∈0 0, ,

4. Let h (x) = min { x, x2 }, for every real number x. Then(a) h is continuous for all x (b) h is differentiable for all x(c) h' (x) = -1, for all x > 1 (d) h is not differentiable at 3 values of x

5. If f xx x

x x x x( )

| |,( / ) ( / ),

,=− ≥

− + + <RST

4 12 3 1 2 13 2

forfor

then

(a) f x( ) is continuous at x = 1 and at x = 4 (b) f x( ) is differentiable at x = 4

(c) f x( ) is continuous and differentiable at x = 1 (d) f x( ) is only continuous at x = 1.

6. If f x xx

x

x( ) sin ,

,,=

FHGIKJ ≠

=

RS|T|

2 1 0

0 0 then

(a) f and f ' are continuous at x = 0 (b) f is derivable at x = 0(c) f is continuous at x = 0 and f ' is not continuous at x = 0 (d) f ' is derivable at x = 0.

7. Let [ x ] denotes the greatest integer less than or equal to x and f (x) = [ tan2 x ]. Then,

(a) lim ( )xf x

→0 does not exist (b) f (x) is continuous at x = 0

(c) f (x) is not differentiable at x = 0 (d) f ' (0) = 1.

8. If f x y f x f y( ) ( ) ( )+ = for all 3)0(',, =∈ fRyx Then f '(5) equals

(a) 153e (b) 3 (c) 5 (d) none of these

9. Let f x y f x f y( ) ( ) ( )+ = for all x y R, .∈ Suppose that 3)3( =f , then f ' (3) is equal to

(a) 22 (b) 44 (c) 28 (d) 3log

10. Let g (x) be the inverse of the function f (x) and f xx

' ( ) .=+1

1 3 Then g x' ( ) is equal to

(a)1

1 3+ ( ( ))g x (b)1

1 3+ ( ( ))f x (c) 1 3+ ( ( ))g x (d) 1 3+ ( ( ))f x

11. If f x x x( ) min{tan ,cot },= then

(a) f x( ) is not differentiable at x = 0, 4/5,4/ ππ (b) f x dx( ) ln/

=z 20

2π

(c) f x( ) is periodic with period 2/π . (d) none of these


12. If y xx

=+FHGIKJ

−cos ,12

21

then dydx

is

(a)−+

21 2x for all x (b)

−+

21 2x for all | x | > 1 (c)

21 2+ x for all | x | < 1 (d) none of these

13. If f x x x( ) ,= − +2 5 6 then f '(x) equals

(a) 2x - 5 for 2 3< <x (b) 5 2 2 3− < <x xfor (c) 2 5 2 3x x− ≤ ≤for (d) 5 2 2 3− ≤ ≤x xfor

14. If y x x=

−FHG

IKJ

−cos cos sin ,1 2 313

then dydx

is

(a) zero (b) constant = 1 (c) constant 1≠ (d) none of these.

15. If 2 2 2x y x y+ = + , then the value of dydx

at x = y = 1 is

(a) 0 (b) -1 (c) 1 (d) 2.

16. The expression of dydx

of the function y a xax

=∞.......

is

(a)y

x y x

2

1( log )− (b)y y

x y x

2

1log

( log )−

(c)y y

x y x y

2

1log

( log log )− (d)y y

x y x y

2

1log

( log log ).

+

17. If y xx= log sin ,cos then dydx

is equal to

(a) (cot log cos tan log sin ) / (log cos )x x x x x+ 2 (b) (tan log cos cot log sin ) / (log cos )x x x x x+ 2

(c) (cot log cos tan log sin ) / (log sin )x x x x x+ 2 (d) none of these

18. If y e x=−sin 1

and u x= log , then dydu

is

(a) e xxsin /−

−1

1 2 (b) x e xsin−1(c)

x ex

xsin−

−

1

1 2(d) none of these

19. The differential coefficeint of f (x) = log ( log x ) with respect to x is

(a)xxlog

(b)log xx

(c) ( log )x x −1 (d) x log x

20. If x y x yp q p q= + +( ) , then dydx

is equal to


(a)yx (b) qx

px(c)

xy (d)

qypx

.

21. If x t y t= =φ ψ( ), ( ), then d ydx

2

2 is equal to

(a)φ ψ ψ φ

φ' ' ' ' ' '

( ' )−

2 (b)φ ψ ψ φ

φ' ' ' ' ' '

( ' )−

3 (c)φψ

' '' '

(d)ψφ

' '' '

22. If f x x x( ) /= 1 then f e' ' ( ) is equal to

(a) e e1 3/( )− (b) e e1/ (c) e e1 2/( )− (d) none of these.

23. If y f xx

=++

FHGIKJ

3 45 6

and f x x' ( ) tan= 2 then dydx

is equal to

(a) −++

FHG

IKJ ×

+2 3 4

5 61

6

2

2tan(5 )

xx x

(b) f xx

x3 35 6

2

2

22tan

tantan+

+FHG

IKJ

(c) tan x2 (d) none of these.

24. If f x g x' ( ) ( )= and g x f x' ( ) ( )= − for all x and f (2) = 4 = f ' (2), then f 2 (16) + g2 (16) is(a) 16 (b) 32 (c) 64 (d) none of these

25. If x = 2 sin t - sin 2t, y = 2 cos t - cos 2t, then the value of d ydx

2

2 at t = π2

is

(a) 2 (b) -1/2 (c) -3/4 (d) -3/2.

26.ddx

ax x(log( ) ), where a is a constant is equal to

(a) 1 (b) log ax (c) 1/a (d) log (ax) + 1

27. If yx x x x x x

=+ +

++ +

++ +

+ +− − −tan tan tan .....12

12

12

11

13 3

15 7

upto n terms, then y' (0) is equal to

(a) − +1 12/ ( )n (b) − +n n2 2 1/ ( ) (c) n n2 2 1/ ( )+ (d) none of these

28. The number of points at which the function f x x x x( ) . tan= − + − +05 1 does not have a derivative in the interval (0, 2) is(a) 1 (b) 2 (c) 3 (d) 4

29. The function f xx x

( )[ ]

,=−1 where [x] is equal to the greatest integer not exceeding x, is

(a) continuous at all points (b) continuous at x = 0(c) discontinuous when x is an integer or zero (d) discontinuous at all points


30. The set of all points where the function f x xx

( ) =+1

is differentiable is

(a) ( , )−∞ ∞ (b) ( , )0 ∞ (c) ( , ) ( , )−∞ ∪ ∞0 0 (d) ( , ) ( , )−∞ − ∪ − ∞1 1

31. Which of the following statements is (are) necessarily true ?1. If a function is differentiable at x, then it is continuous at x2. If a function is continuous at x, then it is differentiable at x3. If a function is integrable on [ x - a, x + a ], then it is continuous at x(a) only 1 (b) only 2 (c) 1 and 2 only (d) 1 and 3 only

32. If y x x= + +10 5 54 610 5log log , then dydx

at x = 1 is equal to

(a) 0 (b) -2 (c) 10 (d) none of these

33. If x = f (t) cost - f ' (t) sin t, y = f (t) sin t + f ' (t) cos t, then dydt

dxdt

FHGIKJ + FHG

IKJ

2 2

is equal to

(a) [ ( ) ' ' ( )]f t f t− 2 (b) [ ( ) ' ' ( )]f t f t+ 2 (c) [ ( )] [ ' ' ( )]f t f t2 2+ (d) none of these

34. If x ey x y= − , then dydx

is

(a)log

( log )xx1 2+

(b) not defined (c)1

1+

+xxlog

(d)11−+

loglogxx

35. If y e x= 3log , then dydx

is equal to

(a) 3 x2 (b) 2 log x (c)3yx

(d) 3 xy

36. If y x x x xn

= + + + +( ) ( ) ( ).....( )1 1 1 12 4 2 then dydx

at x = 0 is

(a) 0 (b) -1 (c) 1 (d) none of these

37. If f (x) is a differentiable function on [0, 3] and f (2) = 3, then lim( )

x

f x tx

dt→ −z

2 3

52

(a) 5 f ' (2) (b) 3 f ' (2) (c) 15 f ' (2) (d) none of these

38. If yxx

xx

=+−

+−+

− −tan tan ,1 111

11

then dydx

is

(a)1

1 2+ x (b)1

1 2− x (c)2

1 2

xx+ (d) 0


ANSWERS

1. a 2. c 3. b 4. a 5. a 6. c 7. b8. a 9. d 10. c 11. a 12. a 13. b 14. b15. b 16. c 17. a 18. c 19. c 20. a 21. b22. d 23. a 24. b 25. d 26. d 27. b 28. c29. c 30. a 31. a 32. c 33. b 34. a 35. a36. c 37. c 38. d 39. b 40. a

39. The derivative of trwx

..12

1sec 21

−− 21 x− at

21

=x is

(a) 2 (b) 4 (c) 1 (d) -2

40. If ( ) ( )xxf aa loglog= , then ( )xf ' is

(a)xxe

e

alog

log (b)xxa

a

elog

log(c)

xaelog

(d)a

x

elog


APPLICATION OF DERIVATIVES

1. If there is an error of k% in measuring the edge of a cube then the percentage error in estimating its volume isa) k b) 3k c) k / 3 d) none of these

2. If the line ax + by + c = 0 is a normal to the rectangular hyperbola xy = 1 thena) 0>ab b) 0<ab c) 0,0 >> ba d) none of these

3. Let f (x) = ( x - p )2 + ( x - q )2 + ( x - r )2. Then f (x) has a minimum at x =λ where λ is equal to

a) ( ) /p q r+ + 3 b) ( ) /p q r 1 3 c)3

1 1 1p q r+ +

d) none of these

4. The critical point (s) of f xx

x( ) =

−22

is (are)

a) x = 0 b) x = 2 c) 0=x and 2=x d) none of these

5. If f x x x( ) / sin= and g x x x( ) / tan= where 0 1< ≤x , then in the interval

a) both f (x) and g(x) are increasing function b) f x( ) is an increasing function

c) both f (x) and g(x) are decreasing function d) g x( ) is an increasing function

6. If 4a + 2b + c = 0 then the equation 3ax2 + 2bx + c = 0 has at least one real root lying betweena) 0 and 1 b) 1 and 2 c) 0 and 2 d) none of these

7. The function f x x x x x( ) sin= + + +3 2 5 2λ will be an invertible function if λ belongs to

a) ( , )−∞ −3 b) ( , )−3 3 c) ( , )3 +∞ d) none of these

8. The difference between the greatest and the least values of the function f x x x( ) sin [ / , / ]= − −2 2 2on π π is

a)3 2

2+

b)3 2

2 6+

+π

c) π d)3 2

2 3+

−π

9. The minimum value of f x x x x( ) = − + + + −3 2 5 isa) 0 b) 7 c) 8 d) 10

10. If f x kx x( ) sin= − is monotonically increasing, then

a) k > 1 b) k > -1 c) k < 1 d) k < -1

11. If ax bx

c+ ≥ for all positive value of x and a, b, c are positive constants, then

a) ab c≥ 2 4/ b) ab c≤ 2 4/ c) bc a≥ 2 4/ d) ac b≥ 2 4/

12. The value of b for which the function f (x) = sin x - bx + c is decreasing in the internal ( , )−∞ ∞ is given bya) b < 1 b) 1≥b c) b > 1 d) 1≤b


13. The equation of the tangents to 2 3 362 2x y− = which are parallel to the straight line x y+ − =2 10 0 are

a) x + 2y = 0 b) x y+ + =2 28815

0 c) x y+ + =2 115

0 d) none of these

14. The slope of the tangent to the curve represented by x t t= + −2 3 8 and y t t= − −2 2 52 at the point M ( , )2 1− isa) 7 / 6 b) 2 / 3 c) 3 / 2 d) 6 / 7

15. The number of real roots of the equation e xx− + − =1 2 0 isa) 1 b) 2 c) 3 d) 4

16. If the two curves y a x= and y bx= intersect at an angle α , then tanα equals

a)log log

log loga ba b−

+1b)

log loglog loga ba b+

−1c)

log loglog loga ba b−

−1d) none of these.

17. For the curve 1,12 −=−= tytx the tangent line is perpendicular to x-axis where

a) t = 0 b) t =∞ c) t = 13

d) t = − 13

.

18. The normal to the curve x a= +(cos sin ),θ θ θ y a= −(sin cos )θ θ θ at any θ is such thata) it makes a constant angle with x-axis b) it passes through the originc) it is at a constant distance from the origin d) none of these

19. The tangent to the curve y e x= 2 at the point (0, 1) meets x-axis at

a) (0, 2) b) (2, 0) c) −FHGIKJ

12

0, d) none of these

20. The equation of the tangents at the origin to the curve y x x2 2 1= +( ) are

a) y x= ± b) x y= ± c) y x= ±2 d) none of these

21. If y x= −4 5 is a tangent to the curve y px q2 3= + at (2, 3), then

a) p q= = −2 7, b) p q= − =2 7, c) p q= − = −2 7, d) p q= =2 7, .

22. Tangents are drawn from the origin to the curve y x= cos . Their points of contact lie on

a) x y y x2 2 2 2= − b) x y y x2 2 2 2= + c) x y x y2 2 2 2= − d) none of these

23. The minimum value of ( ) ( )x a x b+ + is

a) ( )a b+ 2

4b) ( )a b− 2

4c) a b d) none of these

24. If f is an increasing function and g is a decreasing function such that gof is defined, thena) gof is increasing b) gof is decreasing c) gof may not be monotomic d) none of these


25. The function f x x x ax( ) = + + −3 224 10 attains its relative minimum value at x = 1. Then the value of a isa) 51 b) -51 c) -45 d) none of these

26. If an

an

an

a ann

0 1 2 1

1 1 20

++ +

−+ + + =−.... . Then the function ( ) n

nnno axaxaxaxf ++++= −− ..........2

21

1 has in ( 0, 1)

a) at least one zero b) at most one zero c) only 3 zeros d) only 2 zeros.

27. If x y u u2 2 1+ = − / and x y u u4 4 2 21+ = + / then

a) x y y3 1'= b) xy' = y c) x y y2 1'= d) none of these

28. The equation of the horizontal tangent to the graph of the function y e ex x= + − isa) y = 2 b) y = -2 c) y = 0 d) none of these

29. The image of the interval [ -1, 3] under f x x x( ) = −4 123 is

a) [ , ]−2 0 b) [ , ]−8 72 c) [ , ]−8 0 d) [ , ]8 72

30. The maximum value of x1 / x isa) 1 / e b) e c) e1 / e d) (1 / e )e

31. The number of solutions of the equation a g xf x( ) ( ) ,+ = 0 where a > 0, g (x)≠ 0 and g (x) has minimum value 1/4, isa) one b) two c) infinitely many d) zero

32. Let x and y be two real numbers such that x > 0 and xy = 1. The minimum value of x y+ isa) 1 b) 1 / 2 c) 2 d) 1 / 4

33. The maximum value of x xx x

2

2

11

− ++ +

for all real values of x is

a) 1 / 2 b) 1 c) 2 d) 3

34. The point of intersection of the tangents drawn to the curve x2y = 1 - y at the points where it is met by the curve xy = 1 - y is given bya) (0, -1) b) (1, 1) c) (0, 1) d) none of these.

35. If ∞++++= ......!3!2

132 xxxy then

dxdy

is equal to

a) x b) 1 c) y d) ∞

36. If ( ) ( ) ( ) ( ) ,2',1,1',2 =−=== agagafaf then the value of ( ) ( ) ( ) ( )

axxfagafxgLt

ax −−

→ is

a) -5 b) 1/5 c) 5 d) 0

37. If ( ),11 22 yxayx −=−+− then dxdy

is

a) 2

2

1

1

x

y

−

−b)

2

2

1

1

y

x

−

−c) 21 x− d) 21 y−


38. Let ( ) ( ) ( )yfxfyxf =+ for all x and y. Suppose that ( ) 33 =f and ( ) 110' =f then ( )3'f is given bya) 22 b) 33 c) 28 d) none of these

39. If ( ),2 xpy = a polynomial of degree 3, then

2

232dxydy

dxd is equal to

a) ( )xpxp ')(''' + b) ( )xpxp ''')('' c) ( )xpxp ''')( d) a constant

40. If ( ),34cos 31 xxy −= − then dxdy

is equal to

a) 21

3

x−b) 21

3

x−− c) 21 x− d) none of these

ANSWERS

1. b 2. b 3. a 4. a 5. b 6. c 7. b8. c 9. b 10. a 11. a 12. c 13. d 14. d15. a 16. a 17. a 18. c 19. c 20. a 21. a22. c 23. d 24. b 25. b 26. a 27. d 28. a29. b 30. c 31. d 32. c 33. d 34. c 35. c36. c 37. a 38. b 39. c 40. b


INDEFINITE INTEGRATION

1. The integral e x xx

dxxtan−z + ++

FHG

IKJ

1 11

2

2 is equal to

a) ex

Cxtan−

++

1

1 2 b) x e Cxtan− +1 c) x e

xC

xtan−

++

1

1 2d) none of these

2. If CxKdxxx

x+=

−+

∫ 4costancot14cos then

a) K = -1/2 b) K = -1/3 c) K = -1/5 d) none of these

3.e x

xdx

x( sin )cos11

++z is equal to

a) log tan x C+ b) e x Cx tan / 2+ c) e x Cx cot + d) sin log x C+

4.dx

a x b x2 2 2 2cos sin+z is equal to

a) sin ( ( / ) tan )− +1 a b x C b) tan ( ( / ) tan )− +1 b a x C

c) 1 1

abb a x Ctan ( ( / ) tan )− + d) none of these

5. The value of ∫+

αα

α dcos1

sin is

a) − +2 2 2cos /α C b) 2 2 2cos /α +C c) 2 2cos /α +C d)

− +2 2cos /α C

6. The value of dx

x5 4+z cos is

a)13 3

1tan tan− FHGIKJ +x C b)

13

23

1tan tan ( / )− FHG

IKJ +

x C

c)23

13

1tan tan− FHGIKJ +x C d)

23

13

21tan tan ( / )− FHG

IKJ +x C

7.dxx xsin cos+z is equal to

a) log tan / /π 4 8+ +x Cb g b) 2 2 8log tan / /x C+ +πb g

c)12

2 8log tan / /x C+ +πb g d) 2 4 8log tan / /x C− +πb g


8. The value of cossin cos

xx x

dx+z is

a)x x x C2

2+ + +log sin cosb g b) ( )( ) Cxxx +++ cossinlog

21

c) 2 log (sin cos )x x x C+ + + d) none of these

9. If dxx x

f x Csin cos

log ( )= +z then

a) f x x x( ) sin cos= + b) f x x( ) tan= c) xxf 2sec)( = d) none of these

10. If Nnxxxxxxf nn

nn

n∈<<

+−

= −

−

∞→,10,lim)( then (sin ) ( )−z 1 x f x dx is equal to

a) − + − +−[ sin ]x x x C1 21 b) x x x Csin− + − +1 21

c) constant d) none of these

11. If f x x x nx xn

n( ) lim[ ... ]( )= + + < <→∞

−2 4 2 0 13 2 1 then f x dx( )z is equal to

a) − −1 2x b) 11 2− x

c)112x −

d)1

1 2− x

12. 11 12 2+ −z x xc h

dx is equal to

a)12

21

1

2tan −

−

FHG

IKJ

xx

b)12

21

1

2tan −

+

FHG

IKJ

xx

c) 12

21

1

2tan −

−

FHG

IKJ

xx

d) none of these

13.x exdx

x

1 2+z b g is equal to

a) ex

Cx

++1

b) e x Cx + +1b g c) −+

+ex

Cx

1 2b gd) e

xC

x

1 2++

14. e e dxx a xlog .z is equal to

a) (ae)x b)( )logaeae

x

b g c) ea

x

1+ logd) none of these

15. If g x dx g x( ) ( ),=z then g x f x f x dx( ) ( ) ' ( )+z l q is equal to

a) g x f x g x f x C( ) ( ) ( ) ' ( )− + b) g x f x C( ) ' ( )+

c) g x f x C( ) ( )+ d) g x f x C( ) ( )2 +


16. cos log(sin )3 x e dxxz is equal to

a) − +sin4

4x C b) − +

cos4

4x C c)

e Cxsin

4+ d) none of these

17.tan

sin cosx

x xdxz is equal to

a) 2 tan x C+ b)2 cot x C+ c)tan x

C2

+ d) none of these

18. dxxexxx x cos

2sin1cossin sin∫ −

− is equal to

a) e Cxsin + b) e Cx xsin cos− + c) e Cx xsin cos+ + d) e Cx xcos sin− +

19. If e x dxx3 4 11log +

−z c h is equal to

a) log ( )x C4 1+ + b)14

14log x C+ +c h c) − +log x4 1c h d) none of these

20. 5 5 55 55x x x dx. .z is equal to

a)55

5

3

x

Clogb g + b) 5 55 35x

Clogb g + c) 55

5

3

5x

Clogb g + d) none of these

21. [ "( ) " ( ) ( )]f x g x f x g x dxb g −z is equal to

a)f xg x( )' ( ) b) f x g x f x g x' 'b g b g b g b g−

c) f x g x f x g xb g b g b g b g' '− d) f x g x f x g xb g b g b g b g' '+

22. If 4 69 4

9 42e ee e

dx A x B e Cx x

x xx+

−= + − +

−

−z log ,c h then

a) A B C= − = =32

3536

0, , b) A C C R= = − ∈3536

32

, ,

c) A B C R= − = ∈32

3536

, , d) none of these.

23. f ax b f ax b dxn

' + +z b g b gm r is equal to

a) 11

1

nf ax b C n

n

++ + ∀

+b gm r , except n = -1 b) 11

1

nf ax b C n

n

++ + ∀

+b gm r ,


c) 11

1

a nf ax b C n

n

( ),

++ + ∀

+b gm r except n = -1 d)11

1

a nf ax b C n

n

( ),

++ + ∀

+b gm r

24. ∫ dxxx 3cossin

1 is equal to

a)−

+2tan x

C b) 2 tan x C+ c)2tan x

C+ d) − +2 tan x C

25. The value of the integral 11

2

4

++z xxdx is equal to

a) tan − +1 2x C b)12

12

12

tan− −FHGIKJ

xx

c)12 2

2 12 1

2

2log x xx x

C+ +− +

FHG

IKJ + d) none of these

26. ( )( ) ,2

tan11log

41

32 122

2 xbxxadx

xxx −+

−+

=+−

+∫ then (a, b) is

a) (- 1/2, 1/2) b) (1/2, 1/2) c) (-1 , 1) d) (1, - 1)

27. ∫ =−

− dxxxxx22

88

cossin21cossin

a)12

2sin x b) −12

2sin x c) −12sin x d) − sin2 x

28. ∫ −−−

)3x()2x(dx)1x(

is equal to

a) )}2x/()3xlog{( 2 −− b) 2)}2x/()3xlog{( −− c) )}3x(2x{(log −− d) none of these

29. If ,b1)x1(

1)x1(loga

)x1(x

dx3

3

3+

+−

−−=

−∫ then =a

a) 1/3 b) 2/3 c) - 1/3 d) - 2/3

30. The value of the integral ∫ + xsin9xcosdxxcos

22

2

is

a) ]x)xtan3(tan3[81 1 −− b) ]x)xtan3(tan3[

91 1 +−

c)

−− x)xtan3(tan2718 1

d) none of these


31. dx)2x(x

xlog)2xlog(∫ +

−+ is equal to

a) cx2xlog

2

+

+

− b) cx2xlog2

2

+

+

−

c) cx2xlog

41

2

+

+

− d) none of the above

32. dxxcos1xcosxcos

3

3

∫ −− is equal to

a) ( ) cxcossin32 2/31 +− b) ( ) cxcossin

23 2/31 +− c) ( ) cxcoscos

32 2/31 +− d) none of the above

33. The function dx)t1tlog()x(fx

0

2∫ ++= is

a) an even function b) an odd function c) a periodic function d) none of the above

34. ∫ =+dx1x

x3

a) c2x

3x 23

+− b) c)1xlog(x2x

3x 23

++−+−

c) c)1xlog(x2x

3x 23

++++− d) c)1x(2

x2

4

++

35. =+

∫ dx)ex(cos)x1(ex2

x

a) c)ex(coslog2 x + b) c)ex(sec x + c) c)xe(tan x + d) c)ex(tan x ++

36. =∫ dxea xx

a) cea xx + b) calogea xx

+ c) c1x)ea( x

++

d) calog1

ea xx

++

37. ∫ =dxxcosxsin4

2

a) cxtan31 2 + b) cxtan

21 2 + c) cxtan

31 3 + d)

cxx +− cos42sin3

38. The value of ∫ +

−

6

312

x1xtanx

will be

a) 31 xtan61 − b) ( )231 xtan

61 − c) ( )331 xtan

61 − d) ( )431 xtan

31 −


39. The value of ∫ ++ dx

xcos1xsinx

is

a)2xsinx b)

2xcosx c)

2xtanx d) xcosxsinx +

40. The value of ∫ +dx

)1x(xlog2

a) )1x(logxlog1xxlog

+−++

−b) )1x(logxlog

1xxlog

+−++

c) )1log(log1

log+++

+xx

xx

d) )1(loglog1

log+−−

+− xxx

x

ANSWERS

1. b 2. d 3. b 4. c 5. a 6. d 7. c8. b 9. b 10. a 11. d 12. c 13. a 14. b15. c 16. b 17. a 18. a 19. b 20. c 21. c22. c 23. c 24. b 25. b 26. a 27. b 28. a29. a 30. a 31. c 32. c 33. a 34. b 35. c36. d 37. c 38. b 39. c 40. a


DEFINITE INTEGRATION

1. The value [ ]x dxn

0z (where [ x ] is the greatest integer function) is

a)n n+1

2b g

b)n n−1

2b g

c) n n( )−1 d) none of these

2. If the F x ddt

t dtx

b g= +FHGIKJz 2

3

cos then F ' (x) is equal to

a) 2− sin x b) 2+ sin t c) sin x d) 0

3. Let f be a positive function and l x f x x dx l f x x dxk

k

k

k

11

21

1 1= − = −− −z zb gc h b gc h,

then l l is1 2/a) 2 b) k c) 1/2 d) 1

4. The value of 0

2

31π /

tanz +dx

x is

a) 0 b) 1 c) π / 4 d) π / 2

5. The value of 3 2 1

0

4x dx+z is

a) 63

13 43log log

−FHG

IKJ b)

663log

c)66

313 5

3log log−FHG

IKJ d) none of these

6. The value of 0

1991z −x x dxb g is

a)1

10100b) 11

10100c)

10101 d) none of these

7. The value of −z − +1

3

2x x dx[ ] ,m r where [x] denotes the greatest integer less than or equal to x is

a) 7 b) 5 c) 4 d) 3

8. The value of e dxx x

n

n

n

−

−=z∑ [ ]

11

1000

is ( [ x ] is the greatest integer function )

a)e1000 11000

−b)ee

1000 11−

− c)e−11000

d) 1000 1e−b g


9. If f g: :R R R R→ →, are continuous functions then the value of the integral

( ( ) ( )) ( ( ) ( ))/

/f x f x g x g x+ − − −

−z ππ 2

2 is

a) π b) 1 c) -1 d) 0

10. The value of sin cosn mx x dx2 1

0

+zπ is

a)2 1

2

mn+b gb g

!!

b)2 1mn+b g!!

c) cos2 1

0

m x dx−zπ d) none of these

11. If dx

x xk

2 20 4 9+ +=

∞z c hc h π then the value of k is

a) 1/60 b) 1/80 c) 1/40 d) 1/20

12. If x x dx ksin ,/

π π=−z 2

1

3 2then the value of k is

a) 3 1π + b) 2 1π + c) 1 d) 4

13. If f f x f x x x f( ) , ' ,0 2= = = +b g b g b gφ (x) , then f x x dx( ) ( )φ0

1z is

a) e2 b) 2 2e c) 2e d) 2 3 2e− /

14. The value of limcos

x

xt dt

x→

z0

2

0 is

a) 0 b) 1 c) -1 d) 2

15. The value of the integral ddx x

dxtan−

−

FHG

IKJz 1

1

1 1 is

a) π / 2 b) π / 4 c) −π / 2 d) none of these

16. The equation of the tangent to the curve ydttx

x=

+z 1 22

3

at x = 1 is

a) 2 1y x+ = b) 3 1x y+ = c) 3 1 3x y+ + = d) none of these

17. sin/

x dx0

42πz =

a) 0 b) 1 c) 2 d) 4

18. The value of the integral x x

xdxsin

cos/

/

23

3

−zπ

π

is

a) π π/ log tan /3 3 2−b g b) 2 2 3 5 12π π/ log tan /−b gc) 3 2 12π π/ log sin /−b g d) none of these


19. The value of x x dx− +−z 21

3

[ ]c h is ([x] stands for greatest integer less than or equal to x)

a) 7 b) 5 c) 4 d) 3

20. The value of the integral sin [ ]x x dx−z b gπ0

100

is

a) 100 / π b) 200 / π c) 100π d) 200 π

21. If f x dx k f x dxn

(cos ) cos ,2

0

2

0

π πz z= c h then the value of k is

a) 1 b) n c) n/2 d) none of thes

22. The value of the integral −z −π

π

/

/

cos cos2

23x x dx is

a) 0 b) 2/3 c) 4/3 d) none of these

23. The value of the integral 1 2

20

+z cos xdx

π is

a) -2 b) 2 c) 0 d) -3

24. Let Ixdx1 21

2 11

=+z and I

xdx2 1

2 1= z . Then

a) I I1 2> b) I I2 1> c) I I1 2= d) I I1 22>

25. The value of the integral cos log/

/x x

xdx1

11 2

1 2 +−FHGIKJ−z is

a) 0 b)12

c) −12

d) none of these

26. The value of x x x x dx3 5

2

2

1+ + +−z cos tan

/

/

c hπ

π

is equal to

a) 0 b) 2 c) π d) none of these

27.sin

sin cos/ 2

0

2 xx x

dx+zπ is equal to

a)π2 b) 2 2 1log +d i c) 1

22 1log +d i d) none of these

28. The value of the integral xxdx a b

a

b, 0 < <z is

a) b - a b) a - b c) b + a d) none of these


29. The value of the integral sin cos/

x x dx−z0 2π is

a) 0 b) ( )122 − c) 2 2 d) 2 2 1+d i30. The value of the integral e dxx2

0

1z lies in the interval

a) ( , )0 1 b) ( , )−1 0 c) ( , )1 e d) none of these

31. The function f x xxdx

x( ) log=

−+FHGIKJz 1

10 is

a) an even function b) an odd function c) a periodic function d) none of these

32. If f a b x f x( )+ − = b g , then xf x dxa

b( )z is equal to

a)a b f b x dx

a

b+−z2b g b)

a b f x dxa

b+ z2b g c)

b a f x dxa

b− z2b g d) none of these

33. To find the numerical value of px qx s dx3

2

2+ +

−z c h it is necessary to know the values of the constants:

a) p b) q c) s d) p and s.

34. The value of the integralf x

f x f a xa b gb g b g+ −z 20

2

dx is equal to

a) 0 b) 2a c) a d) none of these

35. If f xb g and g xb g are continous functions satisfying f x f a xb g b g= − and g x g a xb g b g+ − = 2, then f x g x dxa b g b g

0z is

equal to

a) g x dxa

( )0z b) f x dx

a( )

0z c) 0 d) none of these

36. The value of sin log/

/x x dx+ +

−z 2

2

21e j{ }π

πis


37. The value of cos99

0

2x dx

πz is

a) 1 b) -1 c) 99 d) 0

38. The value of the integral x x dx[ ]0

2z is

a)72 b)

32 c)

52 d) none of these

39. The value of xxdx

3

80

1

1+z is

a)π4 b)

π8 c)



40. If f xx xx x

x f x dx( ),

,, ( )=

<− ≥RST zfor

for then

0

211 1

2 is equal to

a) 1 b)43 c)

53 d)

52

41. The value of the integral sin/ 6

0

2x dx

πz is

a)34π

b)5

32π c)

316

π d) none of these

42. lim ...n n n n n n→∞ +

++

++

+ ++

LNM

OQP

11

12

13

1 is equal to

a) loge 3 b) 0 c) loge 2 d) 1

43. The value of sin cos3 2

1

1x x dx

−z is

a) 0 b) 1 c)12 d) 2

44. 20

sin x dxπz =

a) 2 3π / b) −5 3π / c) −π d) −2π

45. If for every integer ∫+

=1

2)(,n

n

ndxxfn , then the value of ∫−

4

2

)( dxxf is


46. The value of the integral ∑∫=

+−n

k

dxxkf1

1

0

)1( is

a) ∫1

0

)( dxxf b) ∫2

0

)( dxxf c) ∫n

dxxf0

)( d) ∫1

0

)( dxxfn

47. The value of ∫ ∫ ++

+

x

e

x

e ttdt

tdtttan

/1

cot

/1 22 )1(1 is

a) 1 b) 2e c) 2/e d) none of these

48. ∫− =+−1

1 22log dxxx


49. ∫π

=−3/

0|1tan| dxx

a) 2log21

2+

πb)

68log+π

c)3

2log π− d) none of these


50. The value of the integral ∫π

π

2/

6/ 3sin(cos xx

dx is given by

a) 1 b) 2 c) )31(2 4/1−+ d) none of these

51. ∫ =−−

4

2 )}4()2{( xxdx

a) 2/π b) π c) 0 d) none of these

52. Assuming that f is continuous everywhere, ∫ =

bc

acdx

cxf

c1

a) ∫b

adxxf )( b) ∫

b

adxxf

c)(1

c) ∫b

adxxfc )( d) none of these

53. =

−+

−+

−∞→ 222 1.....

12

11lim

nn

nnn

a) 0 b) - 1/2 c) 1/2 d) none of these

54. If ∫π

=2/

0

2sin dxxA and ∫π

=2/

0

2cos dxxB and ∫π

=2/

0

3sin dxxC , then

a) 0=− BA b) 0=+ BA c) 0=+CA d) 0=−CA

55. If ∫π

=2/

0

1010 sin dxxxu , then the value of 810 90uu + is

a)8

29

π b)

9

2

π c)

9

210

π d)

9

29

π

56. If kdxx

log92

136

0

=+∫ , then k is equal to

a) 3 b) 9/2 c) 9 d) 81

57. The value of dxx

x

x

∫][

0][2

2 is

a)2log][x

b) 2log][x c)2log][

21 x

d) none of these

ANSWERS

1. b 2. a 3. c 4. c 5. d 6. a 7. a8. d 9. d 10. c 11. a 12. a 13. b 14. b15. a 16. a 17. c 18. b 19. a 20. b 21. b22. c 23. c 24. b 25. a 26. c 27. c 28. a29. b 30. c 31. a 32. b 33. c 34. c 35. b36. c 37. d 38. b 39. c 40. c 41. b 42. c43. a 44. a 45. c 46. c 47. a 48. a 49. d50. d 51. b 52. a 53. b 54. a 55. c 56. a57. a


AREA UNDER THE CURVENMENT

1. The area of the figure bounded by the curves y x= −1 and y x= −3 isa) 2 b) 3 c) 4 d) 1

2. Area lying in the first quadrant and bounded by the circle x y2 2 4+ = the line x y= 3 and x-axis is

a) π b)π2 c)


3. If A is the area lying between the curve y = sin x and x-axis between x = 0 and x = π / 2 . Area of the region between the curvey x= sin2 and x- axis in the same interval is given by

a) A/2 b) A c) 2A d) none of these

4. The area bounded by y x y x x= = + ≤2 1 1, , and the y-axis isa) 1/3 b) 2/3 c) 1 d) 7/3

5. The area enclosed within the curve x y+ = 1 isa) 1 b) 1.5 c) 2 d) none of these

6. The area bounded by the curve x 2 = 4 y and the straight line x y= −4 2 isa) 3/8 b) 5/8 c) 7/8 d) 9/8

7. The area between xa

yb

2

2

2

2 1+ = and the straight line xa

yb

+ = 1 is

a)12ab b)

12πab c)

14ab d) 1

412

πab ab−

8. The area cut off a parabola by any double ordinate is k times the corresponding rectangle contained by that double ordinate and its distancefrom the vertex. The value of k isa) 2/3 b) 3/2 c) 1/3 d) 3

9. Area bounded by the curve y = x sin x and x-axis between x = 0 and x = 2π isa) 2π b) 3π c) 4π d) none of these

10. The area of the region bounded by y x= −1 and y = 1 isa) 1 b) 2 c) 1/2 d) none of these

11. The area bounded by the curve y x x x= −, axis and the ordinates x x= = −1 1, is given bya) 0 b) 1/3 c) 2/3 d) none of these

12. The smaller area enclosed by the circle x y2 2 4+ = and the line x y+ = 2 is equal to

a) 2 2π −b g b) π − 2 c) 2 1π − d) none of these

13. Area lying in the first quadrant and bounded by the curve y x= 3 and the line y x= 4 isa) 2 b) 3 c) 4 d) 5


ANSWERS

1. c 2. c 3. b 4. b 5. c 6. d 7. d8. a 9. c 10. a 11. c 12. b 13. c 14. a15. b 16. b 17. b 18. b 19. c 20. a 21. b

14. In the interval 0 2, / ,π area lying between the curves y x y x= =tan , cot and x- axis is

a) log2 b)12

2log c) 2 12

logFHGIKJ d) none of these

15. Area bounded by the curve y x x x= − − −1 2 3b gb gb g and x-axis lying between the ordinates x = 0 and x = 3 is equal toa) 9/4 b) 11/4 c) 11/2 d) none of these

16. The area bounded by the curves y e y ex x= = −, and y = 2 is

a) log /16 eb g b) log /4 eb g c) 2 ( 2e2 - 1 ) d) none of these

17. The area of the figure bounded by y e y xx= = =−1 0 0, , and x = 2 isa) < 2 b) > 2 c) = 2 d) none of these

18. Area bounded by the curves y x y= − =1 0, and x = 2 is

a) 4 b) 5 c) 3 d) 6

19. The ratio of the areas between the curves xy cos= and xy 2cos= and x-axis from 0=x to 3/π=x is

a) 1 : 2 b) 2 : 1 c) 1:3 d) none of these

20. For π≤≤ x0 , the area bounded by xy = and xxy sin+= , is

a) 2 b) 4 c) π2 d) π4

21. Area of the region bounded by the parabola 1)2( 2 −=− xy , the tangent to it at the point with the ordinate 3, and the x-axis is given bya) 9/2 sq. units b) 9 sq. units c) 18 sq. units d) none of these


DIFFERENTIAL EQUATIONS

1. The general solution of the differential equation dydx

xy

=2

2 IS

a) x y C3 3− = b) x y C3 3+ = c) x y C2 2+ = d) x y C2 2− =

2. The order of the differential equation of all circles of radius r , having centre on y -axis and passing through the origin isa) 1 b) 2 c) 3 d) 4

3. The order of d the differential equation whose solution is y a x b x Ce x= + + −cos sin isa) 3 b) 2 c) 1 d) none of these

4. The slope of a curve at any point is the reciprocal of twice the ordinate at the point and it passes through the point ( 4 , 3 ). The equationof the curve is

a) x y2 5= + b) y x2 5= − c) y x2 5= + d) x y2 5= −

5. The curve in which the slope of the tangent at any point equals the ratio of the abscissa to the ordinate of the point isa) an ellipse b) a parabola c) a rectangular hyperbola d) a circle.

6. The equation of the curve whose subnormal is constant isa) y = ax + b b) y2 = 2ax + b c) ay2 - x2 = a d) none of these

7. A differential equation associated to the primitive y a be cex x= + + −5 7 is

a) y y y3 2 12 0+ − = b) 4 5 20 03 2 1y y y+ − = c) y y y3 2 12 35 0+ − = d) none of these

8. The solution of kbyhax

dxdy

++

= represents a parabola when

a) a b= =0 0, b) a b= =1 2, c) a b= ≠0 0, d) a b= =2 1,

9. The solution of the differential equation ydydx

x≠ = −1 satisfying y 1 1b g= is

a) y x x2 2 2 2= − + b) y x x2 22 1= − − c) y x x= − +2 2 2 d) none of these

10. The degree of the differential equation d ydx

dydx

x d ydx

2

2

2 2

2+ FHGIKJ = log is


11. The order of the differential equation of all tangent lines to the parabola y x= 2 isa) 1 d) 2 c) 3 d) 4

12. The particular solution of log dy dx x y y/ ,b g b g= + =3 4 0 0 is

a) e ex y3 43 4+ =− b) 4 3 33 4e ex y− =− c) 3 4 73 4e ex y+ =+ d) 4 3 73 4e ex y+ =−


13. If y f x= b g passing through (1,2) satisfies the differential equation y xy dx x dy1 0+ − =b g , then

a) f x xx

b g =−22 2

b) f x xx

b g = ++112

c) f x xx

b g = −−1

4 2d) f x x

xb g =

−4

1 2 2

14. The equation of the curve satisfying the differential equation y x xy22

11 2( )+ = passing through the point ( 0, 1 ) and having slope oftangent at x = 0 as 3 is

a) y x x= + +2 3 2 b) y x x2 2 3 1= + + c) y x x= + +3 3 1 d) none of these

15. Order and degree of differential equation

4/12

2

2

+=dxdyy

dxyd

are

a) 4 and 2 b) 1 and 2 c) 1 and 4 d) 2 and 4

16. Solution of the differential equation : 0)( =++ xydxdyxa is

a) )()2()3/2( xaxaAey +−= b) )()()3/2( xaxaAey +−−=

c) )()2()3/2( xaxaAey ++= d) )()2()3/2( xaxaAey +−−=

17. The integrating factor of )/1sin(2 22 xxydxdyx +=− is

a) 2x b) 2/1 x c) 2x− d) none of these

18. The solution of the differential equation 0sin.sin =− dxyxdy , is

a) cye x =2/tancos b) cye x =tancos c) cyx =tan.cos d) cyx =sin.cos

19. Equation of the curve passing through (3, 9) which satisfies the differential equation )/1(/ 2xxdxdy += is

a) 29636 2 +−= xxxy b) 62936 2 +−= xxxy c) 62936 3 −+= xxxy d) none of these

20. Solution of the differential equation 1,2/coscossin2/sin2 2 =π=−= yxatandxyxxdxdyxy , is given by

a) xy sin2 = b) xy 2sin= c) 1cos2 += xy d) none of these

21. The solution of differential equation 0)()( =−−+ dxyxdyyx

a) cxxyy =++ 22 2 b) cxxyy =−+ 22 2 c) 02 22 =++ xxyy d) cxxyy =+− 22 2

22. The solution of differential equation mxeaydxdy

=+ is

a) ceyma mx +=+ )( b) cmeye mxax +=

c) axmx ceey −+= d) axmx ceeyma −+=+ )(


23. The solution of the differential equation 32 =− ydxdyx represent

a) Straight line b) circles c) parabolas d) ellipses

24. Solution of the equation 02cos12cos1

=−+

+xy

dxdy is

a) cxy =+ cottan b) cxy =− cottan c) cxy =+ tancot d) none of these

25. The solution of the equation 32)2(

2

2

−+−−

=xxyy

dxdy

is

a) cxx

yy

+−+

=+−

13log

41

12log

31

b) cxx

yy

++−

=−+

31log

41

21log

31

c) cxx

yy

++−

=+−

31log3

12log4 d) none of these

ANSWERS

1. a 2. a 3. a 4. c 5. c 6. b 7. c8. c 9. a 10. d 11. a 12. d 13. a 14. c15. d 16. a 17. b 18. a 19. c 20. a 21. b22. d 23. c 24. b 25. c


DETERMINANTS

1. If

x x x xx x x x

x x x

2

2

2

2 3 7 2 42 7 2 33 2 1 4 7

− + + ++ − +

− − += + + + + + +ax bx cx dx ex fx g6 5 4 3 2 the value of g is


2. If A B C+ + = π then value of ∆ =+ +

− + ++ −

sin sin cossin tan tan

cos tan

A B C B CB A B C A

A B A

b gb g

b g 0 is

a) tan2 A b) sin2 B c) cos2 C d) none of these

3. If x y z, , are distinct and x x xy y yz z z

k k k

k k k

k k k

+ +

+ +

+ +

2 3

2 3

2 3

= − − − + +FHG

IKJx y y z z x

x y zb gb gb g 1 1 1 , then

a) k = −3 b) k = −1 c) k = 1 d) k = 3

4. If γβα ,, are the roots of the equation x px q3 0+ + = , then ∆ =α β γβ γ αγ α β

is equal to

a) − pq b) 2p c) 3q d) none of these

5. If ∆ =+ + ++ + ++ + +

=x x x ax x x bx x x c

1 22 33 4

0 , then a, b, c are in

a) A.P b) G.P c) H.P d) none of these

6. If ω ≠ 1b g is a cube root of unity and x

xx

++

+=

11

10

2

2

2

ω ωω ωω ω

, then

a) x = 1 b) x = ω c) 2ω=x d) none of these

7. If Unn N Nn N N

n = + ++

1 52 1 2 13 3 1

2

3 2

, then Unn

N

=∑1

is equal to

a) 21

nn

N

=∑ b) 2 2

1

nn

N

=∑ c) 1

22

1

nn

N

=∑ d) 0


8. If a, b, c are complex number, then zb c

b ac a

=− −

−0

00

is equal to

a) 0 b) a b c c) 3 a b c d) 4 a b c

9. If a, b, c are non-zero real numbers and ∆ =111

2

2

2

///

,a a bcb b cac c ab

then

a) ∆ is independent of a, b, c b) a b c c) a b c2 2 2 d) none of these

10. The system of equations x y z x ky z x ky z+ + = + + = + + =4 3 2 4, , is consistent if

a) k = 1 b) k = 2 c) k = −2 d) none of these

11. If a b c> > >0 0 0, , , are respectively the p q rth th th, , terms of a GP, then the value of determinant

logloglog

a pb qc r

111

is



1 1 1

11

12

1

21

22

2

m m m

m m m

C C CC C C

+ +

+ +

is equal to

a) 1 b) -1 c) 0 d) none of these13. If A is a square matrix of order n such that its elements are polynomial in x and its r - rows become identical for x = k, then

a) x k r−b g is a factor of A b) x k r− −b g 1 is a factor of A

c) x k r− +b g 1 is a factor of A d) x k r−b g is a factor of A

14. If x, y, z are in AP then the value of the determinant A is, where A

xyz

x y z

=

4 5 65 6 76 7 8

0



11

1

3 5

3 4

5 4

ω ωω ωω ω

, where ω is an imaginary cube root of unity, is

a) 1 2− ωb g b) 3 c) -3 d) none of these


16. If ∆1 =x b ba x ba a x

and ∆2 =x ba x

, then

a) ∆ ∆1 223= b g b) d

d x∆ ∆1 23b g = c)

dd x

∆ ∆1 223b g = d) ∆ ∆1 2

3 23= b g /

17. The factors of

x a ba x ba b x

are

a) x a x b− −, , and x a b+ + b) x a x b+ +, and x a b+ +

c) x a x b+ +, and x a b− − d) x a x b− −, , and x a b− −

18. Ifω is a cube root of unity, then 1

11

2

2

2

ω ωω ωω ω

=

a) 1 b) ω c) ω2 d) 0

19. Let a, b, c be positive real numbers. The following system of equations in x, y and z

xa

yb

zc

xa

yb

zc

xa

yb

zc

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

21 1 1+ − = − + = − + + =, , has

a) no solution b) unique solution c) infinitely many solutions d) finitely many solutions.

20. If every element of a third order determinant of value ∆ is multiplied by 5, then the value of new determinant is

a) ∆ b) 5∆ c) 25∆ d) 125∆

21. If ω is a cube root of unity, then a root of the following equation

01

11

2

2

2

=+

++

ωωωω

ωω

xx

x is

a) 1=x b) ω=x c) 2ω=x d) 0=x

22.222 234234111

equals

a) 2 b) -2 c) 1 d) 0


23. If ,543012321

−=∆ then 5123064361

− is equal to

a) ∆ b) ∆2 c) ∆6 d) none of these

24. Value of the determinant xxx

xxx

seccottan010tansinsec

=∆ is given by

a) 0 b) - 1 c) 1 d) none of these

25. If )(xf is a polynomial satisfying )/1(1)()/1()(

21)(

xfxfxfxf

xf−

= and 17)2( =f , then the value of )5(f is

a) 624 b) - 124 c) 626 d) 126

26. If 643610363234232 =+++++++++

zyxyxxzyxyxxzyxyxx

, then the real value of x is

a) 2 b) 3 c) 4 d) 6

27. The parameter, on which the value of the determinant xdppxxdpxdppxxdp

aa

)sin(sin)sin()cos(cos)cos(

1 2

+−+− does not depend upon, is

a) a b) p c) d d) x

28. For the equations : x y z x y z x y z+ + = + + = + + =2 3 1 2 3 2 5 5 9 4, , ,a) there is only one solution b) there exists infinitely many solutionc) there is no solution d) none of these.

29. If a,b,c, are different and 0111

32

32

32

=−−−

cccbbbaaa

then

a) 0=++ cba b) 1=abc c) 1=++ cba d) 0=++ cabcab

30. The value of baccbacbaacb

++

+

is

a) 0 b) cba ++ c) 4abc d) abc


ANSWERS

1. d 2. d 3. b 4. d 5. a 6. d 7. b8. a 9. a 10. d 11. b 12. a 13. b 14. a15. b 16. b 17. a 18. d 19. b 20. d 21. d22. b 23. c 24. c 25. c 26. c 27. b 28. a29. b 30. c


MATRICES

1. If A is a square matrix, then A + A' isa) symmetric matrix b) skew symmetric matrix c) scalar matrix d) diagonal matrix

2. If A = [ 2 -3 5 ], then A' A is a matrix of ordera) 1 × 1 b) 3 × 3 c) 1 × 3 d) 3 × 1

3. If A A=−L

NMOQP

−cos sinsin cos

,θ θθ θ

1 is given by

a) - A b) A' c) - A' d) A

4. If A and B are square matrices, then ( )A B A AB B+ = + +2 2 22 ifffa) A 2 = 0 or B 2 = 0 b) AB = BA c) AB = 0 d) BA = 0

5. If A =L

NMMM

O

QPPP

3 0 00 3 00 0 3

and Ba a ab b bc c c

=L

NMMM

O

QPPP

1 2 3

1 2 3

1 2 3

then AB is equal to

a) B b) 3B c) B 3 d) A + B.

6. If Aii

=LNMOQP

00

then A2 is equal to

a)1 00 1−LNMOQP b)

−−

LNM

OQP

1 00 1

c)1 00 1LNMOQP

d)−LNMOQP

1 00 1

7. If A A I=−LNM

OQP =

cos sinsin cos

,θ θθ θ

then 2 if

a) 0=θ b) 4/π=θ c) 2/π=θ d) none of these.

8. If Aa

A nn=FHGIKJ ∈

10 1

, )then (where equalsN

a)10 1naF

HGIKJ b)

10 1

2n aFHG

IKJ c)

10 0naF

HGIKJ d)

n nan0

FHGIKJ

9. A matrix A = [ aij ] is an upper triangular matrix ifa) it is a square matrix and aij = 0, i < j b) it is a square matrix and aij = 0, i > jc) it is not a square matrix and aij = 0, i > j d) it is not a square matrix and aij = 0, i < j

10. If A is any m × n matrix such that AB and BA are both defined, then B is ana) m × n matrix b) n × m matrix c) n × n matrix d) m × m matrix

11. If A is a square matrix of order n × n and k is a scalar, then adj ( kA ) is equal toa) k adj A b) kn adj A c) kn - 1 adj A d) kn + 1 adj A


12. If A = [ aij ] is a square matrix of order n × n and k is a scalar, then | kA | =

a) k An b) k A c) k An−1 d) none of these

13. If A and B are matrices such that AB and A + B both are defined, then

a) A and B can be any two matrices

b) A and B are square matrices not necessarily of the same order

c) A, B are square matrices of the same order

d) number of columns of A is same as the number of rows of B

14. If A = 5 23 1

1LNMOQP =−, then A

a)1 23 5

−−LNM

OQP b)

−−

LNM

OQP

1 23 5

c)− −− −LNM

OQP

1 23 5

d)1 23 5LNMOQP

15. If A + B = 1 01 1LNMOQP and A - 2B =

−−

LNM

OQP =

1 10 1

, then A

a)1 12 1LNMOQP

b)2 3 1 31 3 2 3

/ // /LNM

OQP c) none of these d)

1 3 1 32 3 1 3

/ // /LNM

OQP

16.3 12 5

43

−LNMOQPLNMOQP= −LNMOQP

xy

a) x = 3, y = - 1 b) x = 2, y = 5 c) x = 1, y = - 1 d) x = - 1, y = 1

17. With 1, 2,ωω as cube roots of unity, inverse of which of the following matrices exists ?

a)1

2

ωω ωLNM

OQP b)

ωω

2 11LNMOQP c)

ω ωω

2

2 1LNM

OQP d) none of these

18. Matrix theory was introduced bya) Cauchy - Riemann b) Caley - Hamilton c) Newton d) Cacuchy - Schwarz

19. If A A=L

NMMM

O

QPPP

=−

abc

0 00 00 0

1then

a)abc

0 00 00 0

L

NMMM

O

QPPP

b)a

abac

2 0 00 00 0

L

NMMM

O

QPPP

c)1 0 0

0 1 00 0 1

//

/

ab

c

L

NMMM

O

QPPP

d)

−−

−

L

NMMM

O

QPPP

ab

c

0 00 00 0


20. If A A=− − −

−−

L

NMMM

O

QPPP

=1 2 2

2 1 22 2 1

then adj

a) A b) AT c) 3 A d) 3 AT

21. If In is the identity matrix of order n, then ( In ) -1 =a) does not exist b) In c) 0 d) n In

22. If A and B are square matrices of order 3 such that 3||,1|| =−= BA , then =|3| ABa) -9 b) -81 c) -27 d) 81

23. Let

ααα−α

=α1000cossin0sincos

)(F , then )'()( αα FF is equal to

a) )'( ααF b)

αα

'F c) )'( α+αF d) )'( α−αF

24. The value of x for which 0111

012120201

]11[ =

x is

a) 2 b) - 2 c) 3 d) - 3

25. If

=

abba

A and

αββα

=2A , then

a) abba 2,22 =β−=α b) abba 2,22 =β+=α

c) 2222 , baba −=β+=α d) 22,2 baab +=β=α

26. Let

−

=53

21A and

=

2001

B and X be a matrix such that BXA = . Then X =

a)

− 5342

b)

− 5342

.21

c)

−5342

.21

d) none of the above

27. If

=

1111

A and ,Nn∈ then nA is equal to

a) An2 b) An 12 − c) An d) none of these


28. If

=

dcba

A such that 0≠− bcad , then 1−A is

a)

−− ac

bdbcad

1b)

−

−acbd

c)

−

−− ac

bdbcad

1d) none of these

ANSWERS

1. a 2. b 3. b 4. b 5. b 6. b 7. a8. a 9. b 10. b 11. c 12. a 13. c 14. b15. d 16. c 17. d 18. b 19. c 20. d 21. b22. b 23. c 24. b 25. b 26. b 27. b 28. c


VECTORS

1. If OCBOOBAO +=+ then A, B, C area) coplanar b) collinear c) Non-collinear d) none of these.

2. If | | | |,a b a b+ = − then

a) a is parallel to b b) a b⊥ c) | | | |a b= d) none of these.

3. If a and b are two unit vectors inclined at an angle θ such that a b+ is a unit vector, then θ is equal to

a)π3 b)

π4 c)

π2 d)

23π .

4. The pojection of the vector i j k− +2 on the vector 4 4 7i j k− + is

a)5 610

b)199 c)

919 d)

619

5. If a and b are two vectors such that a b. = 0 and a b× = 0 , then

a) a b|| b) a b⊥

c) either a or b is a null vector d) none of these.

6. The number of vectors of unit length pependicular to vectors a i j= + and b j k= + is

a) one b) two c) three d) none of these.

7. The unit vector perpendicular to vector i j− and i j+ forming a right handed system is

a) k b) −k c) 12

( )i j− d)12

( )i j+

8. If the constant forces 2 5 6i j k− + and − + −i j k2 act on a particle due to which it is displaced from a point A ( 4, -3, -2 ) to a point B (6, 1, -3 ), then the work done by the forces isa) 15 units b) -15 units c) 9 units d) -9 units.

9. The work done by the force F i j k= − −2 in moving an object along the vector 3 2 5i j k+ − is

a) -9 units b) 15 units c) 9 units d) none of these.

10. If forces of magnitudes 6 and 7 units acting in the directions i j k− +2 2 and 2 3 6i j k− − respectively act on a particle which is displacedfrom the point P ( 2, -1, -3 ) to Q ( 5, -1, 1), then the work done by the forces isa) 4 units b) -4 units c) 7 units d) -7 units.


11. If ,n n1 2 are two unit vectors and θ is the angle between them, then cos θ / 2 =

a)12 1 2| |n n+ b)

12 1 2| |n n− c) 1

2 1 2| . |n n d)| || | | |n nn n1 2

1 22×

12. The projection of the vector a i j k= − +4 3 2 on the axis making equal acute angles with the coodinate axes is

a) 3 b) 3 c)3

1d) none of these.

13. a b c a b c.( ) ( )+ × + + =

a) 0 b) 2[ ]a b c c) [ ]a b c d) none of these.

14. A unit vector in xy-plane making an angle of 45° with the vector i j+ and an angle of 60° with the vector 3 4i j− is

a) i b)i j+

2c) i j−

2d) none of these.

15. If the unit vectors a xi y j z k= + + and b j= are such that a c b, and form a right handed system, then c is

a) z i x k− b) 0 c) y j d) − +z i x k .

16. If C is the middle point of AB and P is any point outside AB, then

a) PCPBPA =+ b) PCPBPA 2=+ c) OPCPBPA =++ d) OPCPBPA =++ 2

17. The value of b such that the scalar product of the vector i j k+ + with the unit vector parallel to the sum of the vectors 2 4 5i j k+ −

and bi j k+ +2 3 is one, isa) -2 b) -1 c) 0 d) 1

18. Volume of the parallelopiped whose coterminal edges are 2 3 4 2 2 3, , ,i j k i j k i j k− + + − − + isa) 5 units b) 6 units c) 7 units d) 8 units

19. .( ) .( ) .( )i j k j k i k i j× + × + × =a) 1 b) 3 c) -3 d) 0

20. Each of the angle between vectors a b c, and is equal to 60°. If | | , | | | ,a b c= = =4 2 6and | then the modulus of a b c+ +isa) 10 b) 15 c) 12 d) none of these

21. If a = + + ,i j k2 3 b = − + +i j k2 and c = +3 ,i j then t such that a + t b is at right angle to c will be equal toa) 5 b) 4 c) 6 d) 2

22. If | a | = 2, | b | = 5 and | a × b | = 8 then a . b can be equal toa) 4 b) 6 c) 5 d) none of these.

23. If a . b = b . c = c . a = 0, then [ a, b, c ] is equal toa) 0 b) 1 c) -1 d) | a | | b | |c |.


24. Let a, b and c be three non-coplanar vectors, and let p, q and r be the vectors defined by the relations

p b cabc

q c aabc

r a babc

=×

=×

=×

[ ],

[ ] [ ]and

Then the value of the expression ( a + b ) . p + (b + c ) . q + ( c + a ) . r is equal to

a) 0 b) 1 c) 2 d) 3

25. The volume of the parallelopiped whose sides are given by OA = 2 3 ,i j− OB = ,i j k+ − OC = 3i k− isa) 4 / 13 b) 4 c) 2 / 7 d) none of these.

26. Let a, b, c be distinct non negative numbers. If the vectos ai aj ck ,+ + i k ci cj bk+ + +and lie in a plane, then c is

a) the arithmetic mean of a and b b) the geometric mean of a and bc) the harmonic mean of a and b d) equal to zero

27. Given a b= + − = − + +,i j k i j k2 and c = − + − .i j k2 A unit vector perpendicular to both a + b and b + c is

a)2

6i j k+ +

b) j c) k d)i j k+ +

328. A unit tangent vector at t = 2 on the curve x = t 2 + 2, y = 4t - 5, z = 2t 2 - 6t is

a)13

( )i j k+ + b)13

2 2( )i j k+ + c)16

2( )i j k+ + d) none of these.

29. A particle moves along a curve so that its coordinates at time t are x = t, y = 12

t2, z = 13

3t . The acceleation at t = 1 is

a) j k+ 2 b) j k+ c) 2 j k+ d) none of these.

30. If a and b are two unit vectors and φ is the angle between them, then 12

| |a b− is equal to

a) 0 b) π / 2 c) | sin φ / 2 | d) | cos φ / 2 |

31. The value of ( ) ( ) ( )i i j j k k× × + × × + × ×a a a is

a) a b) 2a c) 0 d) 3a

32. The value of [ a × b, b × c, c × a ] is

a) 2 [ a b c ] b) [ a b c ] c) [ a b c ]2 d) 0

33. If a = + +xi j k5 7 , b = + − ,i j k c = + +i j k2 2 are coplanar then the value of x is

a) 1 b) -2 c) -1 d) none of these.

34. If a and b are position vector of A and B respectively, then the position vector of a point C in AB produced such that ABAC 3= is

a) ba −3 b) ab −3 c) ba 23 − d) ab 23 −

ANSWERS

1. b 2. b 3. d 4. b 5. c 6. b 7. a8. b 9. c 10. a 11. a 12. b 13. a 14. d15. a 16. b 17. d 18. c 19. b 20. a 21. a22. b 23. d 24. d 25. b 26. b 27. c 28. b29. a 30. c 31. b 32. c 33. d 34. d


THREE DIMENSIONAL GEOMETRY

1. The distance of the point P ( a, b, c ) from the x-axis is

a) b c2 2+ b) a c2 2+ c) a b2 2+ d) none of these.

2. If γβα ,, are the angles which a directed line makes with the positive directions of the coordinate axes, then sin2 α + sin2 β + sin2 γ isequal toa) 1 b) 2 c) 3 d) none of these.

3. Equation of the line passing thro’ (1,1,1) and perpendicular to 0532 =+++ zyx is

a)11

11

11 −

=−

=−− zyx

b)11

31

21 −

=−

=− zyx

c)11

31

31 −

=−

=− zyx

d)21

31

11 −

=−

=− zyx

4. The value of λ for which the lines x y z−

=−

=+−

11

2 11λ

and x y z+−

=+

=−1 1

221λ

are perpendicular to each other is

a) 0 b) 1 c) -1 d) none of these.

5. The number of spheres of radius r touching the coordinate axes isa) 4 b) 6 c) 8 d) none of these.

6. The equation of the sphere which passes through the points ( 1, 0, 0 ), ( 0, 1, 0 ), ( 0, 0, 1 ) and having radius as small possible, is

a) 3 2 1 02 2 2( ) ( )x y z x y z+ + − + + − = b) x y z x y z2 2 2 1 0+ + − − − − =

c) 3 2 1 02 2 2( ) ( )x y z x y z+ + − + + + = d) none of these.

7. The angle between the lines x y z−=

+−

=23

12

2, and x y z−

=+

=+1

12 33

52

is

a) π / 2 b) π / 3 c) π / 6 d) none of these.

8. The DCs of the line 6x - 2 = 3y + 1 = 2z - 2 are

a)13

1313

, , b)114

214

314

, , c) 1, 2, 3 d) none of these.

9. A line passes through two points A ( 2, -3, -1 ) and B ( 8, -1, 2 ). The coordinates of a point on this line at a distance of 14 units from A are

a) ( 14, 1, 5 ) b) ( -10, -7, -7 ) c) ( 86, 25, 41 ) d) none of these.

10. The position vector of the point in which the line joining the points i j k− +2 and 3 2i j− cuts the plane through the origin and the

points 4 j and 2 ,i k+ is

a) 6 10 3i j k− + b) kji ˆ43ˆ2ˆ

23

+− c) − + −6 10 3i j k d) none of these.

11. The coordinate of the middle point of the line joining the points ( -1, -1, 1 ) and ( -1, 1, -1 ) area) ( 0, 0, 0 ) b) ( -1, 0, 0 ) c) ( 0, -1, 1 ) d) ( 0, 1, -1 ).


12. The direction cosines of the line joining the points ( 1, 2, -3 ) and ( -2, 3, 1 ) are

a) -3, 1, 4 b) -1, 5, -2 c)−326

126

426

, , d)− −130

530

230

, ,

13. The equation of the XOY plane isa) x = 0 b) y = 0 c) z = 0 d) 0, ≠= ccz

14. The equation of z-axis isa) z = 0, x = 0 b) z = 0, y = 0 c) x = 0, y = 0 d) x = k, y = -k., ( k ≠ 0)

15. The ratio in which the yz plane divides the line joining the points ( -2, 4, 7 ) and ( 3, -5, 8 ) isa) 2 : 3 b) 3 : 2 c) 4 : 5 d) -7 : 8

16. The coordinates of the point equidistant from the points ( a, 0, 0 ), ( 0, a, 0 ), ( 0, 0, a ) and ( 0, 0, 0 ) area) ( a/3, a/3, a/3 ) b) ( a/2, a/2, a/2 ) c) ( a, a, a ) d) ( 2a, 2a, 2a, )

17. The coordinate of the foot of the perpendicular from the point ( a, b, c ) on z-axis isa) ( a, 0, 0 ) b) ( 0, b, 0 ) c) ( 0, 0, c ) d) ( a, b, 0 )

18. l = m = n = 1 represents the direction cosines ofa) x-axis b) y-axis c) z-axis d) none of these.

19. The direction cosines of a line equally inclined with the co-ordinate axes are

a)121212

, , b)12

12

12

, , c)13

13

13

, , d)32

32

32

, ,

20. A line makes an angle of 60° with each of x and y axis, the angle which it makes with z axis isa) 30° b) 45° c) 60° d) none of these.

21. If l m n1 1 1, , and l m n2 2 2, , be direction cosines of two perpendicular lines, then

a) l l m m n n1 2 1 2 1 2 0+ + = b) l l m m n n1 2 1 2 1 2 1+ + = c) ll

mm

nn

1

2

1

2

1

2

1+ + = d) ll

mm

nn

1

2

1

2

1

2

0+ + =

22. Algebraic sum of the intercepts made by the plane x + 3y - 4z + 6 = 0 on the axes isa) - 13/2 b) 19 / 2 c) - 22/3 d) 26/3

23. The equation of the plane passing through the point ( 1, -1, 2 ) and parallel to the plane 3x + 4y - 5z = 0 isa) 3 4 5 11 0x y z+ − + = b) 3 4 5 11x y z+ − = c) 6 8 10 1x y z+ − = d) 3 4 5 2x y z+ − =

24. The reflection of the point A ( 1, 0, 0 ) in the line x y z−

=+−

=+1

213

108

is

a) ( 3, -4, -2 ) b) ( 5, -8, -4 ) c) ( 1, -1, -10 ) d) ( 2, -3, 8 )

25. The centre of the sphere ( ) ( ) ( ) ( ) ( ) ( )x x y y z z+ − + − + + − + =1 1 2 2 3 3 0 is

a) ( -1, 2, 3 ) b) ( 1, -2, -3 ) c) ( 0, 0, 0 ) d) ( 1, 2, 3 )


26. A sphere of constant radius k passes through the origin and meets coordinate axes in A, B, C. The centroid of the triangle ABC lies on

a) x y z k2 2 2 2+ + = b) 4 92 2 2 2( )x y z k+ + =

c) 2 32 2 2 2( )x y z k+ + = d) 9 42 2 2 2( )x y z k+ + =

27. The ratio in which the plane

( ) 1732. =+− kjir divides the line joining the points kji 742 ++− and kji 853 +− isa) 1 : 5 b) 1: 10 c) 3 : 5 d) 3 : 10

28. The direction ratios of the diagonals of a cube which joins the origin to the opposite corner are (when the 3 concurrent edges of the cube areco-ordinate axes

a)32,

32,

32−

b) 1,1,1 c) 2, -2, 1 d) 1, 2, 3

ANSWERS

1. a 2. b 3. b 4. b 5. c 6. a 7. a8. b 9. a, b 10. b 11. b 12. c 13. c 14. c15. a 16. b 17. c 18. d 19. c 20. b 21. a22. a 23. a 24. b 25. c 26. d 27. d 28. b


PROBABILITY

1. Three of the six vertices of a regular hexagon are chosen at random. The probability that the triangle with these vertices is equilateral, equalsa) 1/2 b) 1/5 c) 1/10 d) 1/20

2. The probability of India winning a test match against West Indies is 1/2. Assuming independence from match to match the probability that in a 5 matchseries, India's second win occurs at the third test isa) 1/8 b) 1/4 c) 1/2 d) 2/3

3. If two events A and B are such that P A P B( ') . , ( ) .= =03 04 and P A B( ' ) . ,∩ = 05 then P B A B( / ' )∪ equalsa) 3/4 b) 5/6 c) 1/4 d) 3/7

4. If the probabilities that A and B will die within a year are x and y respectively then the probability that exactly one of them will be alive at the endof the year is

a) x y xy+ + 2 b) x y xy+ + c) x y xy+ − d) x y xy+ − 2

5. The probability of three persons having the same date and month for the birthday isa) 1/365 b) 1/(365)2 c) 1/(365)3 d) none of these

6. A speaks truth in 70 percent cases and B speaks the truth in 80 percent cases. The probability that they will say the same thing while describinga single event isa) 0.56 b) 0.62 c) 0.38 d) 0.94

7. One mapping is selected at random from all the mappings from the set S = {1,2,3,..,n} into itself. The probability that the selected mapping is one-to-one is

a) l nn/ b) l n/ ! c) n nn− −1 1b g!/ d) none of these

8. Two non-negative integers are chosen at random. The probability that their product is divisible by 5 isa) 7/25 b) 8/25 c) 9/25 d) 2/5

9. A man is known to speak the truth 3 out of 4 times. He throws a die and reports that it is a six. The probability that it is actually a six isa) 3/8 b) 1/5 c) 3/4 d) none of these

10. Three six-faced fair dice are thrown together. The probability that the sum of the numbers appearing on the dice is k k3 8≤ ≤b g is

a) ( ) ( ) /k k− −1 2 432 b) k k( ) /−1 432 c) k 2 432/ d) none of these

11. A determinant is chosen at random from the set of all determinants of order 2 with elements 0 or 1 only. The probability that the determinantchosen is non-zero isa) 3 / 16 b) 3 / 8 c) 1 / 4 d) none of these

12. Three persons A, B and C are to speak at a function along with 5 other persons. If the persons speak in random order, the probability thatA speaks before B and B speaks before C is

a) 3/8 b) 1/6 c) 3/5 d) none of these

13. Three identical dice are rolled. The probability that the same number will appear on each of them is


a)16 b)

136 c)

118 d)

328

14. If there are 6 girls and 5 boys who sit in a row, then the probability that no two boys sit together is

a)6!6!2 11! ! b)

7 52 11! !! ! c)

6!72 11

!! ! d) none of these

15. Two dice are rolled one after the other. The probability that the number on the first is smaller than the number on the second isa) 1 / 2 b) 7 / 18 c) 3 / 4 d) 5 / 12

16. A four figure number is formed of the figures 1, 2, 3, 5 with no repetitions. The probability that the number is divisible by 5 isa) 3/4 b) 1/4 c) 1/8 d) none of these

17. In the above question the probability that the number is odd isa) 3/4 b) 1/4 c) 1/8 d) none of these

18. Suppose n ( )≥ 3 persons are sitting in a row. Two of them are selected at random. The probability that they are not together is

a) 12

−n b)

21n − c) 1

1−n d) none of these

19. Two events A and B have probabilities 0.25 and 0.50 respectively. The probability that both A and B occur is 0.14. Then the probability thatneither A nor B occurs

a) 0.39 b) 0.25 c) 0.11 d) none of these

20. An integer is chosen at random from the first 200 positive integers. The probability that the integer chosen is divisible by 6 or 8 isa) 1 / 3 b) 1 / 4 c) 1 / 5 d) none of these

21. If two squares are chosen at random on a chess board, the probability that they have a side in common isa) 1 / 9 b) 1 / 18 c) 2 / 7 d) none of these

22. The probability that the 13th day of a randomly chosen month is a Friday, isa) 1/12 b) 1/7 c) 1/84 d) none of these.

23. If the letters of the word 'REGULATION' be arranged at random, the probability that there will be exactly 4 letters between R and E isa) 1/10 b) 1/9 c) 1/5 d) 1/2

24. A coin is tossed three times. The probability of getting head and tail alternately isa) 1/8 b) 1/2 c) 1/4 d) none of these

25. A box contains 100 bulbs out of which 10 are defective. A sample of 5 bulbs is drawn. The probability that none is defective, is

a)110

5FHGIKJ b)

12

5FHGIKJ c)

910

5FHGIKJ d)

910

26. A carton contains 20 bulbs, 5 of which are defective. The probability that, if a sample of 3 bulbs is chosen at random from the carton, 2 will be


ANSWERS1. c 2. b 3. c 4. d 5. b 6. b 7. c8. c 9. a 10. a 11. b 12. b 13. b 14. c15. d 16. b 17. a 18. a 19. a 20. b 21. b22. c 23. b 24. c 25. c 26. c 27. a 28. a29. b 30. b 31. a 32. a 33. a

defective, isa) 1 / 16 b) 3 / 64 c) 9 / 64 d) 2 / 3

27. In a class of 125 students 70 passed in Mathematics, 55 in Statistics and 30 in both. The probability that a student selected at random from the class,has passed in only one subject isa) 13 / 25 b) 3 / 25 c) 17 / 25 d) 8 / 25.

28. 3 mangoes and 3 apples are in a box. If 2 fruits are chosen at random, the probability that one is a mango and the other is an apple isa) 3 / 5 b) 5 / 6 c) 1 / 36 d) none of these

29. There are 4 white and 4 black balls in a bag and 3 balls are drawn at random. If balls of same colour are identical, the probability that none of themis black, isa) 1 / 4 b) 1 / 14 c) 1 / 2 d) none of these

30. A father has 3 children with at least one boy. The probability that he has 2 boys and one girl isa) 1 / 4 b) 3 /7 c) 2 / 3 d) none of these

31. If x∈[ , ],0 5 then what is the probability that x x2 3 2 0− + ≥ ?a) 4/5 b) 1/5 c) 2/5 d) none of these

32. An almirah stores 5 black and 4 white socks well mixed. A boy pulls out 2 socks at random. The probability that 2 are of the same colourisa) 4/9 b) 5/8 c) 5/9 d) 7/12

33. If x ∈[ , ],15 then the probability that 0452 <+− xx isa) 3/4 b) 4/4 c) 2/4 d) 1/4

Quest Tutorials 107Head Office : 44C,Kalusarai,New Delhi-16; Ph.(011) 46080363

STATISTICS

1. Which of the following is not a measure of central tendencya) Mean b) Median c) Mode d) Range

2. The mean of first n natural numbers is

a)( )21+nn

b) ( )1+nn c)21+n

d) ( )1+n

3. For a continuous series the mean is computed by the following formula

a)nf

AMean ∑+= b) ∑∑+=fd

AMean c) ∑∑+=df

AMean d) ∑∑+=fdf

AMean

4. The weighted mean of first n natural numbers whose weights are equal to the squares of corresponding number is

a)21+n

b) ( )( )122

13++nnn c) ( ) ( )

6121 ++ nn d)

( )21+nn

5. The mean of a set of numbers is x . If each number is increased by λ , the mean of the new set is

a) x b) λ+x c) xλ d) none of these

6. The relationship between mean, median and mode for a moderately skewed distribution isa) Mode = Median -2 Mean b) Mode = 2 Median - Meanc) Mode = 3 Median -2 Mean d) Mode = 2 Median -3 Mean

7. If in a moderately asymmetrical distribution mode and mean of the data are λ6 and λ9 respectively, then median isa) λ8 b) λ7 c) λ6 d) λ5

8. If a variable takes the discrete values

21,2,3,

25,

27,4 +−−−−+ αααααα , ( )05,

21

>+− ααα , then the median is

a)45

−α b)21

−α c) 2−α d)45

+α

9. Geometric mean of n2,2,2,2 32 ⋅⋅⋅⋅⋅ is

a) n2

2 b) 22n

c) 21

2−n

d) 21

2+n

10. The harmonic mean of 3,7,8,10,14 is

a)5

1410873 ++++d)

141

101

81

71

31

++++ c)4

141

101

81

71

31

++++d)

141

101

81

71

31

5

++++

11. If each observation of a raw data, whose variance is 2σ , is multiplied by λ , then the variance of the new set is

a) 2σ b) 22σλ c) 2σλ + d) 22 σλ +


12. The harmonic mean of the numbers 2,3,4, is

a) 3 b) ( )31

24 c)1336

d)3613

13. Let s be the standard deviation of n observations. Each of the n observations is multiplied by a constant c. Then the standard deviation ofthe resulting numbers isa) s b) cs d) cs d) none of these

14. The S.D of the first n natural numbers is

a)21+n

b)( )21+nn

c)1212 −n

d) none of these

15. Which of the following, in case of a discrete data, is not equal to the median ?a) 50th percentile b) 5th decile c) 2nd quartile d) lower quartile

16. For a frequency distribution, 7th decile is computed by the formula

a) if

cn

lD ×

−

+= 77 b) i

f

cn

lD ×

−

+= 107

c) if

cn

lD ×

−

+= 107

7 d) if

cn

lD ×

−

+= 710

7

17. For a continuous series, the mode is computed by the formula

a) ifff

florCfff

flmmmm

m ×

−−

+×−−

++−

−

21

1

11

1b) i

ffffflorC

fffffl

m

m

mmm

mm ×−−

−+×

−−−

++−

−

21

1

11

1

c) ifff

fflorCfff

fflm

m

mmm

mm ×−−

−+×

−−−

++−

−

21

1

11

122 d) i

ffffflorC

fffffl

m

m

mmm

mm ×−−

−+×

−−−

++−

−

21

1

11

1 22

18. The standard deviation for the set of numbers 1,4,5,7,8 is 2.45 nearly. If 10 are added to each number, then the new standard deviation willbea) 2.45 nearly b) 24.45 nearly c) 0.245 nearly d) 12.45 nearly

19. For a frequency distribution, lower quartile is computed by

a) if

CN

lQ ×

−

+= 41 b) i

f

CN

lQ ×

−

+= 21

c) if

CN

lQ ×

−

+= 43

1 d)( ) ifCNlQ ×

−+=1


20. The upper quartile for the following distribution 2378542Frequency7654321itemsofSize is given by the size of

a)th

+4131

item b)th

+41312 item c)

th

+41313 item d)

th

+41314 item

21. The A.M of a set of 50 numbers is 38. If two numbers of the set, namely 55 and 45 are discarded, the A.M of the remaining set of numbersis

a) 38.5 b) 37.5 c) 36.5 d) 36.

22. An automobile driver travels from plane to a hill station 120 km distant at an average speed of 30 km per hour. He then makes the returntrip at an average speed of 25 km per hour. He covers another 120 km distance on plane at an average sped of 50 km per hour. His averagespeed over the entire distance will be

a)3

502530 ++ km/hour b) ( ) 3/150,25,30

c)

501

251

301

3

++km/hour d) none of these

23. If nxxxx ⋅⋅⋅⋅⋅⋅⋅321 ,, be n observations, then the quantity ( ) nnxxxx /1

321 ,, ⋅⋅⋅⋅⋅⋅ is calleda) H.M b) A.M c) G.M d) none of these

24. If the mean of 1,2,3,.........n is 116n

then n is

a) 10 b) 12 c) 11 d) 13

25. If the standard deviation of 0,1,2,3,........9 is K, then the standard deviation of 10,11,12,13.......19 isa) K b) 10+K c) 10+K d) 10K

26. If in a moderately asymmetrical distribution mode and mean of the data are λ6 and λ9 respectively, then median isa) λ8 b) λ7 c) λ6 d) λ5

27. If a variable takes the discrete values )0(5,2/1,2/1,2,3,2/5,2/7,4 >α+α−α+α−α−α−α−α+α , then the median isa) 4/5−α b) 2/1−α c) 2−α d) 4/5+α

28. For a symmetrical distribution 25P and 75P are 40 and 60 respectively. The value of median will bea) 50 b) 40 c) 60 d) none of these

29. The standard deviation of first n natural numbers is

a)6

)12()1( ++ nnnb)

1212 −n

c)

−1212n

d) none of these


30. If S.D of a variate x is σ , then S.D of cbax + where cba ,, are constants is

a) σac

b) σ22

ac

c) σcb

d) σca

31. If the variance of observations nxxx ,........., 21 is 2σ , then the variance of 0.,......., 21 ≠ppxpxpx n is given by

a) 2σp b) 22σp c) 22 / pσ d) none of these

ANSWERS

1. d 2. c 3. d 4. b 5. b 6. c 7. a8. a 9. d 10. d 11. b 12. c 13. b 14. c15. d 16. c 17. c 18. a 19. a 20. c 21. b22. c 23. c 24. c 25. a 26. a 27. a 28. a29. c 30. d 31. b


CORRELATION AND REGRESSION ANALYSIS

1. If a linear relation aX + bY + c = 0 exists between the variables X and Y and ab < 0, then the coefficients of correlation between X and Y isa) 1 b) -1 c) 0 d) any number between -1 and 1.

2. Let X and Y be two variables with the same variance and U and V be two variables such that U = X + Y, V = X - Y. Then Cov ( U, V ) isequal toa) Cov ( X, Y ) b) 0 c) 1 d) -1

3. If X and Y are independent variables, then the two lines of regression area) x = 0, y = 0 b) x = 0, y = const c) x = const, y = 0 d) x = const, y = const.

4. If bYX and bXY are both positive, then

a) rbb XYYX

211≤+ b) rbb XYYX

211≥+ c) 2

11 rbb XYYX

≤+ d) none of these.

5. The line of regression of X on Y referred to the means of X and Y as the origin is

a) x X r XYy Y− = −

σσ

( ) b) x r XYy Y= −

σσ

( ) c) x r XYy=

σσ

d) none of these.

6. If the regression of Y on X is 43

, then the regression coefficient of X on YY

a) is 34

b) is less than or equal to 34

c) is less than 1 d) can take any value

7. Let X and Y be two variables with the same mean. If the lines of regressions of Y on X and X on Y respectively y = α x + b andx = α y + b, then the value of the common mean is

a)ba1− b)

1− ab c)

β1− a d)

b1−α

8. The arithmetic mean of the first n odd natural numbers isa) n b) ( n + 1 ) /2 c) ( n - 1 ) d) none of these.

9. The arithmetic mean of the squares of the first n natural numbers isa) ( n + 1 ) / 6 b) ( n + 1 ) ( 2n + 1 ) / 6 c) ( n2 - 1 ) / 6 d) none of these.

10. The arithmetic mean of the series 1, 2, 22 , ....., 2n-1 is

a) 2n n/ b) ( ) /2 1n n− c) ( ) /2 1n n+ d) none of these.

11. The arithmetic mean of the data given byVariate (x) 0 1 2 3 ... n

Frequency (f ) nC0 nC1nC2

n C3 ... nnC

a) ( n + 1 ) / 2 b) n / 2 c) 2n / n d) none of these.


12. The mean height of 25 male workers in factory is 61 cms, and the mean height of 35 female workers in the same factory is 58 cms. The combined

mean height of 60 workers in the factory is

a) 59.25 b) 59.5 c) 59.75 d) 58.75.

13. The mean weight of 9 items is 15. If one more item is added to the series the mean becomes 16. The value of 10th item is

a) 35 b) 30 c) 25 d) 20.

14. If X and Y are two random variables with the same standard deviation and coefficient of correlation r, then the coefficient of correlation

between X and X + Y is

a)1

2+ r

b)1

2+ r

c)1

2− r

d)1

2− r

.

15. Suppose x y xy n=∑ =∑ =∑ =15 36 110 5, , , , then cov ( x, y ) isa) 0.4 b) 0.6 c) 0.8 d) 0.9.

16. If the correlation coefficient between X and Y is 0.6 and if UX V Y

=−

=−5

23

3and ,

then the coefficient of correlation between U and V isa) 0.8 b) 0.6 (c) 0.3 d) 0.2

17. Suppose the correlation coefficient between X and Y is -0.7 and if U X V Y=

−−

=−3

24

5and , then the correlation coeff.

between U and V isa) 0.7 b) - 0.7 c) 0.8 d) none of these.

18. If cov (X, Y ) = -13.5, Var ( X ) = 2.25 and Var (Y ) = 100, then the coefficient of correlation between X and Y is

a) 0.8 b) - 0.8 c) -0.9 d) 0.9.

19. If two lines of regressions are 3x + 12y = 19 and 9x + 3y = 46, the correlation coefficient is

a) 1 2 3/ b) −1 2 3/ c) 0.52 d) - 0.52.

20. The two lines of regressions intersect in

a) ( , )σ σx y b) ( , )x y c) ( / , / )x yx yσ σ d) none of these.

ANSWERS

1. a 2. b 3. d 4. b 5. c 6. b 7. a8. a 9. b 10. b 11. b 12. a 13. c 14. b15. a 16. b 17. a 18. c 19. b 20. b


SETS, RELATIONS AND MAPPINGS FUNCTIONS

1. A survey shows that 63% of the Americans like cheese whereas 76% like apples. If x% of the Americans like both cheese and apples, thena) x = 39 b) x = 63 c) 6339 ≤≤ x d) 39≤x

2. If X n n Nn= − − ∈{ | }4 3 1 and Y n n N= − ∈{ ( ) | },9 1 then X Y∪ is equal toa) X b) Y c) N d) none of these

3. If sets A and B are defined as A x y yx

x R= = ≠ ∈RSTUVW( , ) | , ,1 0 B x y y x x R= = − ∈( , ) | , ,l q then

a) A B A∩ = b) A B B∩ = c) A B∩ = φ d) none of these

4. If a N = { a x | x ∈N } and b N c N d N∩ = , where b, c ∈N thena) d = bc b) c = bd c) b = cd d) none of these

5. If A and B are two sets, then A A B∩ ∪( ) equals

a) A b) B c) φ d) none of these

6. Which of the following is the empty set ?a) { x | x is a real number and x2- 1 = 0 } b) { x | x is a real number and x2 + 1 = 0 }c) { x | x is a real number and x2- 9 = 0 } d) { x | x is a real number and x2 = x + 2}.

7. A set contains n elements. The power set contains.a) n elements b) 2n elements c) n2 elements d) none of these

8. Let A and B be two non-empty subsets of a set X such that A is not a subset of B, thena) A is a subset of complement of B b) B is a subset of Ac) A and B are disjoint d) A and the complement of B are non-disjoint.

9. If A = { 1, 2, 3 } and B = { 3, 8 }, then ( ) ( )A B A B∪ × ∩ isa) {( 3, 1 ), ( 3, 2 ), ( 3, 3 ), ( 3, 8 )} b) {( 1, 3 ), ( 2, 3 ), ( 3, 3 ), ( 8, 3 )}c) {( 1, 2 ), ( 2, 2 ), ( 3, 3 ), ( 8, 8 )} d) {( 8, 3 ), ( 8, 2 ), ( 8, 1 ), ( 8, 8 )}

10. For any set A, (A' )' is equal to

a) A' b) A c) φ d) none of these

11. The symmetric difference of A = { 1, 2, 3 } and B = ( 3, 4, 5 } isa) { 1, 2 } b) { 1, 2, 4, 5 } c) { 4, 3 } d) { 2, 5, 1, 4, 3 }.

12. Let U be the universal set containing 700 elements. If A, B are sub-sets of U such that n (A) = 200, n (B) = 300 and n A B( ) .∩ = 100Then n A B( ' ' )∩ =


13. If A x R x= ∈ < <{ | }0 1 and B x R x= ∈ − < <{ | },1 1 then A × B is the set

a) of all points lying inside the rectangle having vertices at (1, 1), (0, 1), (0, -1) and (1, -1)b) of all points lying inside the rectangle having vertices at (1, 0), (1, 1), (0, 1) and (0, 0)


c) of all points lying on the sides of the rectangle whose vertices are at (1,1), (0,1), (0,-1) and (1,-1)d) none of these

14. If A = { a, b, c }, B = { c, d, e }, C = { a, d, f }, then A B C× ∪( ) is

a) {( a, d ), ( a, e ), ( a, c )} b) {( a, d ), ( b, d ), ( c, d )} c) {( d, a ), ( d, b ), ( d, c )} d) none of these

15. If R is a relation from a finite set A having m elements to a finite set B having n elements, then the number of relations from A to B isa) 2mn b) 2mn -1 c) 2mn d) mn

16. Let R be a reflexive relation on a set A and I be the identity relation on A. Thena) R I⊂ b) I R⊂ c) R = I d) none of these

17. Let A be the non-void set of the children in a family. The relation 'x is a brother of y' on A isa) reflexive b) symmetric c) transitive d) none of these

18. The relation R defined in N as bbRa ⇔ is divisible by a is

a) reflexive but not symmetric c) symmetric but not transitive c) symmetric and transitive d) none of these

19. Given the relation R = {( 1, 2 ), ( 2, 3 )} on the set A = { 1, 2, 3 }, the minimum number of ordered pairs which when added to R make it anequivalence relation isa) 5 b) 6 c) 7 d) 8

20. Let R1 be a relation defined by R1 = {( a, b ) | a ³ b, a, b ∈ R }. Then R1 isa) an equivalence relation on R b) reflexive, transitive but not symmetricc) symmetric, transitive but not reflexive d) neither transitive nor reflexive butsymmetric

21. Let P = {( x, y ) | x2 + y2 = 1, x, y ∈ R }. Then P isa) reflexive b) symmetric c) transitive d) antisymmetric

22. For real numbers x and y, we write x Ry x y⇔ − + 2 is an irrational number. Then the relation R is

a) reflexive b) symmetric c) transitive d) none of these.

23. Let n be a fixed positive integer. Define relation R on the set Z of integers by, || banRba −⇔ Then R is

a) reflexive b) symmetric c) transitive d) equivalence.

24. Let L denote the set of all straight lines in a plane. Let a relation R be defined by α β α β α βR L⇔ ⊥ ∈, , . Then R is

a) reflexive b) symmetric c) transitive d) none of these.

25. If R = {( x, y ) | x, y ∈ + ≤Z x y, }2 2 4 is a relation in Z, then domain of R is

a) { 0, 1, 2 } b) { 0, -1, -2 } c) {-2, -1, 0, 1, 2} d) none of these


26. Let R be a relation defined in the set of real numbers by a Rb ab⇔ + >1 0. Then R is

a) equivalence relation b) transitive c) symmetric d) anti-symmetric.

27. Which of the following is not an equivalence relation on Z ?

a) a Rb a b⇔ + is an even integer b) a Rb a b⇔ − is an even integer

c) a Rb a b⇔ < d) a Rb a b⇔ =

28. The relation R defined on the set A = { 1, 2, 3, 4, 5 } by R x y x y= − <{( , ):| | }2 2 16 is given by

a) R1 = {( 1, 1 ), ( 2, 1 ), ( 3, 1 ), ( 4, 1 ), ( 2, 3 )} b) R2 = {( 2, 2 ), ( 3, 2 ), ( 4, 2 ), ( 5, 2 )}c) R3 = {( 3, 3 ), ( 4, 3 ), ( 5, 4 ), ( 3, 4 )} d) none of these

29. Sets A and B have 3 and 6 elements each. What can be minimum number of elements in BA∪ ?a) 3 b) 6 c) 9 d) 18

30. Two finite sets have m and n elements respectively. The total number of subsets of first set is 56 more than the total number of subsets of the secondset. The values of m and n respectively area) 7, 6 b) 6, 3 c) 5, 1 d) 8, 7

31. Let S be the set of integers. For a b S a b, ,∈ R if and only if | a - b | < 1, then

a) R is not reflexive b) R is not symmetric c) R = {( a, a ); a I∈ } d) R is not an equivalence relation.

32. A and B are two sets having 3 and 4 elements respectively and having 2 elements in common. The number of relations which can be defined fromA to B isa) 25 b) 210 - 1 c) 212 - 1 d) none of these

33. If f (x) is a polynomial satisfying ( ) ( ) ( ) ( )xfxfxfxf /1/1 += and f ( ) ,3 28= then f ( )4 is given bya) 63 b) 65 c) 67 d) 68

34. The function f : [ -1/2, 1/2 ] → −[ / , / ]π π2 2 defined by f x x x( ) sin ( )= −−1 33 4 isa) both one-one and onto b) neither one-one nor onto c) onto but not one-one d) one-one but not onto

35. Given f xx x

( )| |

=−1

and g xx x

( )| |

=−1

. Then

a) dom f ≠ φ and dom g = φ b) dom f = φ and dom g ≠ φc) dom φ≠f and dom φ≠g d) dom f =φ and dom g = φ

36. Which of the following functions is not onto

a) f f x x: , ( )R R→ = +3 4 b) 2)(,: 2 +=→ + xxfRRf

c) f f x x: , ( )R R+ +→ = d) none of these

37. Which of the following functions is not one - one


a) f f x x: , ( )R R→ = +2 5 b) f f x x:[ , ] [ , ], ( ) cos0 1 1π → − =

c) f f x x:[ / , / ] [ , ], ( ) sin− → − = +π π2 2 1 1 2 3 d) f f x x: [ , ], ( ) sinR→ − =1 1

38. Let f x x( ) = 2 and g x x( ) = 2 then the solution set of fog(x) = gof (x) isa) R b) { 0 } c) { 0, 2 } d) none of these

39. The range of the function ]cos[)( xxf = for - 2/2/ π<<π x containsa) { -1, 1, 0 } b) { cos 1, 1, cos 2 } c) { cos 1, -cos 1, 1 } d) [ -1, 1 ].

40. If f xxx

( ) ,=−+11 the domain of f -1 (x) contains

a) ( , )−∞ ∞ b) ( , )−∞ −1 c) ( , )− ∞1 d) ( ) ( )∞−−∞− ,11, U

41. Let g x x x( ) = − −2 4 5 then

a) g is one - one on R b) g is one - one on ( , )−∞ 2c) g is not one - one on ( , )−∞ 4 d) none of these

42. Let f xxx

( ) sinsin

.=+1 3

Then dom f does not contain

a) ( 0, π ) b) ( , )− −2π π c) ( , )2 3π π d) ( , )4 6π π

43. The inverse of the function yx x

x x=−+

−

−

10 1010 10

is

a) log ( )10 2 − x b)12

1110log +−xx c)

12

2 110log ( )x − d)14

22

log xx−

44. Let the function }1{}{: −→−− RbRf be defined by f x x ax b

a b( ) ( ).=++

≠ Then

a) f is one - one but not onto b) f is onto but not one - onec) f is both one - one and onto d) none of these

45. If f x xx x

( ) sin([ ] ) ,=+ +

π2 1

where [x] denotes the integral part of x, them

a) f is one-one b) f is not one-one and non-constant functionc) f is a constant function d) none of these

ANSWERS1. c 2. b 3. c 4. a 5. a 6. b 7. b8. d 9. b 10. b 11. b 12. c 13. a 14. d15. a 16. b 17. c 18. a 19. c 20. b 21. b22. a 23. d 24. b 25. c 26. c 27. c 28. d29. b 30. b 31. c 32. d 33. b 34. a 35. a36. b 37. d 38. c 39. b 40. d 41. b 42. d43. b 44. c 45. c


LINEAR PROGRAMMING

1. The solution set of the inequation 2x + y > 5 isa) half plane that contains the origin b) open half not containing the originc) whole xy-plane except the points lying on the line 2x + y = 5 d) none of these

2. If a point ( h, k ) satisfies an inequation 4≥+ byax , then the half plane represented by the inequation isa) the halfplane containing the point ( h, k ) but excluding the points on ax + by = 4b) the halfplane containing the point ( h, k ) and the points on ax + by = 4c) whole xy-planed) none of these.

3. In equation y - x ≤ 0 representsa) the half plane that contains the positive x-axis b) closed half plane above the line y = x which contains positivey-axisc) half plane that contains the negative x-axis d) none of these.

4. In equations 33 ≥− yx and 4x - y > 4a) have solution for positive x and y b) have no solution topositive x and yc) have no solution for negative x d) have no solution for negative y.

5. The maximum value of P = 3x + 5y, subject to x y x y x≤ ≤ + ≤ ≥2 3 4 0, , , and y ≥ 0, isa) 15 b) 16 c) 18 d) 20

6. The maximum value of P = x + y, subject to 2 4 2 4 0 0x y x y x y+ ≤ + ≤ ≥ ≥, , and isa) 2 b) 3 c) 4/3 d) 8/3

7. The minimum value of P = x + 3y , subject to 0,,4,62 ≥≤≥+≥+ xyxyxyx and 0≥y , isa) 8 b) 7 c) 6 d) none of these.

8. If S is a convex subset of the plane bounded by lines in the plane, then a linear function z = cx + dy, ( x, y ) ∈S , where c, d are scalarsattains its extreme value at the

a) boundary points b) points inside S c) vertices of S d) none of these.

9. A firm manufacturing two types of electrical items A and B can make a profit of Rs 160 per unit of A and Rs. 240 per unit of B. Both A andB make use of two essential components, a motor and a transformer. Each unit of A requires 3 motors and 2 transformers and each unit ofB requires 2 motors and 4 transformers and total supply of components per month is restricted to 210 motors and 300 transformers. If x andy denotes the number of items A and B which the firm should produce in order to maximize the profit then the objective function isa) z = 3x + 2y b) z = 2x + 4y c) z = 210x + 300y d) none of these.

10. The minimum value of z = 2x + y subject to the contraints 4,1,50105 ≤≥+≤+ yyxyx and 0, ≥yx isa) 1/3 b) 1 c) 1/2 d) 5/2

11. A firm manufactures three products A, B and C. The profits are Rs 3, Rs 2 and Rs 4 respectively. The firm has 2 machines and belowis the required processing time in minutes for each machine ProductsMachine A B C C 4 3 5 D 2 2 4


Machines C and D have 2000 and 2500 machine minutes. The firm must manufacture 100 A's, 200 B's and 50 C 's but not more than 150 A's.The number of constraints the firm has is

a) 6 b) 7 c) 5 d) 8

12. A manufacturer of a line of patent medicines is preparing a production plan on medicines A and B, there are sufficient ingredients availableto make 20,000 bottles of A and 40,000 bottles of B but there are only 45,000 bottles into which either of the medicines can be put. Furthermore it takes 3 hours to prepare enough material to fill 1000 bottles of A. It takes one hour to prepare enough material to fill 1000 bottles ofB and there are 66 hours available for this operation. The number of constraints the manufacturer has isa) 4 b) 5 c) 6 d) 7

13. The maximum value of z = 5x + 7y subject to 0,,35710,2483,4 ≥≤+≤+≤+ yxyxyxyx isa) 22 b) 24.8 c) 26.4 d) 25

14. The maximum value of z = 3x + 5y subject to 0,,600,1500,20002 ≥≤≤+≤+ yxxyxyx isa) 5000 b) 5300 c) 6000 d) 6500

15. A firm manufacturer trucks and buses. Inputs available are 720 man year 900 machines weeks and 1900 tons of steel production of a truckrequires 1 man - year, 3 machine-week and 5 tons of steel. Production of a bus requires 2 man-years, 1 machine-week and 4 tons of steel.The price of a truck is Rs. 600,000 and the price of a bus is Rs. 800,000.

If a manufacturer produces x trucks and y buses then his objective function isa) z = x + 3y b) z = 3x + 5y c) z = 600000 x + 800000 y d) z = 720 x + 900 y

16. Maximum value of 80 x + 120 y subject to 3605020,3,2,9 ≤+≥≥≤+ yxyxyx isa) 928 b) 840 c) 960 d) 1140

17. The maximum value of P = 4x + 2y subject to 0,0,243,4624 ≥≥≤+≤+ yxyxyx isa) 42 b) 48 c) 52 d) 46

18. Maximum 21 116 xxZ +=

Subject to 1042 21 ≤+ xx , 762 21 ≤+ xx and 0,0, 21 ≥≥ xx , isa) 240 b) 540 c) 440 d) none of these

19. Maximum 21 75 xxZ +=

Subject to 2483,4 2121 ≤+≤+ xxxx

35710 21 ≤+ xx and 0, 21 ≥xx isa) 14.8 b) 24.8 c) 34.8 d) none of these

20. Minimum yxZ +=

Subject to 1223 ≥+ yx , 113 ≥+ yx and 0,0 ≥≥ yx isa) 5 b) 6 c) 4 d) none of these

21. For Maximum value of yxZ 25 +=

Subject to 632 ≥+ yx , 22 ≤− yx , 2446 ≤+ yx , 323 ≤+− yx and 0,0 ≥≥ yx the values of x and y area) 18/7, 2/7 b) 7/2, 3/4 c) 3/2, 15/4 d) none of these


22. For Maximum value of 21 26 xxZ −=

Subject to 3,22 121 ≤≤− xxx and 0, 21 ≥xx , the values of 1x and 2x area) 3, 4 b) 2, 3 c) 1, 2 d) none of these

23. Minimum 21 2xxZ +−=

Subject to 103 21 ≤+− xx , 621 ≤+ xx , 221 ≤− xx and 0, 21 ≥xx , isa) - 4 b) - 2 c) 2 d) none of these

24. For 3,3,55,42,5 ≤≥≥+≥+≤+ yxyxyxyx , then minimum value of yxz 3+= isa) 19/2 b) 11 c) 11/3 d) none of these

ANSWERS

1. b 2. b 3. a 4. a 5. c 6. d 7. a8. c 9. d 10. b 11. a 12. a 13. b 14. b15. c 16. c 17. d 18. c 19. b 20. a 21. b22. a 23. b 24. c


NUMERICAL METHODS

1. In decimal system, the number ( )1623C is equivalent to

a) ( )10962 b) ( )10862 c) ( )16864 d) ( )10860

2. Newton’s method for finding the root of the equation, ( ) 0=xf , is

a)( )( )nn

nn xfxfxx '

1 −=+ b)( )( )nn

nn xfxfxx '

1 +=+ c)( )( )nn

nn xfxfxx'1 −=+ d) none of these

3. The root of the equation ( ) 0=xf lying in the interval ( )ba, is

a)( ) ( )

ababfbaf

−−

b)( ) ( )

abbafabf

−−

c)( ) ( )

)()( afbfabfbaf

−−

d) none of these

4. In decimal form, the number ( )16362 is equivalent toa) 866 b) 867 c) 868 d) 869

5. In normalized floating point respesentation, 02E2562002E86420 ⋅÷⋅ gives

a) 0E37313 ⋅ b) 73333 ⋅ c) 04E37313 ⋅ d) none of these

6. Gauss-Elimination method is used for solvinga) algebraic equations b) exponential equations c) trignometric equations d) Linear simultaneous equations

7. The number of significant digits in the number 004520000 ⋅ isa) 3 b) 5 c) 8 d) none of these

8. In decimal systems, the number 2)11011101( is equivalent toa) 221 b) 222 c) 2021 d) none of these

9. Using the successsive bisection method, the approximate value of a root of the equation 043 =−− xx lying between 1 and 2 at the endof third iteration isa) 1.875 b) 1.796 c) 1.8125 d) none of these

10. If the equation 033 =+− kxx has all real roots, thena) 22 <<− k b) 11 <<− k c) ∞<< k0 d) none of these

11. The round off error when the number 8.987652 is rounded to five significant digits isa) 0.00048 b) -0.00048 c) -0.000048 d) none of these

12. By false positioning, the second approximation of a root of equation ( ) 0=xf is (where 10 , xx are initial and first approximationsrespectively)

a)( )

( ) ( )01

00 xfxf

xfx−

− b)( ) ( )( ) ( )01

0110xfxfxfxxfx

−−

c)( ) ( )( ) ( )01

1100xfxfxfxxfx

−−

d)( )

( ) ( )01

01 xfxf

xfx−

−

13. Let ( )xf be a polynomial. Then if ( ) ( ) 021 <xfxf , then ( ) 0=xf has

a) an odd number of roots in ( )21, xx b) any number of roots in ( )21, xx

c) no root or an even number of roots in ( )21, xx d) only one root in ),( 21 xx


14. Trapezoidal rule for evaluation of ( )∫b

a

dxxf requires the interval ( )ba, to be divided into

a) n2 sub-intervals of equal width b) 12 +n sub-intervals of equal widthc) any number of sub-intervals of equal width d) n3 sub-intervals of equal width

15. By Simpson rule, taking 4=n , the value of the integral dxx∫ +

1

021

1 is equal to

a) .785 b) .788 c) .781 d) none of these

16. By the method of successive bisection, root of the equation 013 =−− xx ; lying between 1 and 2, correct to 3 places of decimal isa) 1.432 b) 1.323 c) 1.324 d) 1.325

17. If a and ba + are two consecutive approximate roots of the equation ( ) 0=xf as obtained by Newton’s Method, then b is equal to

a)( )( )afaf' b)

( )( )afaf '

c)( )( )afaf '

− d)( )( )afaf'

−

18. Let ( ) ( ) 72.21,10 == ff , then the trapezoidal rule gives approximate value of ( )∫1

0

dxxf as

a) 3.72 b) 1.86 c) 1.72 d) 0.86

ANSWERS

1. a 2. c 3. c 4. a 5. b 6. d 7. d8. a 9. c 10. a 11. c 12. b 13. a 14. c15. a 16. c 17. d 18. b

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