12th magudam maths em onemark 123.pdf

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Dept. of School EducationKanyakumari Dist

+2 - Question Bank

www.tnschools.co.in

1 Mark Questions

2013-14

w

W. A. S. RADHAKRISHNANCHIEF EDUCATIONAL OFFICER

KANYAKUMARI DIST
http://www.tnschools.co.in/http://www.tnschools.co.in/


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2013-14Maths fB ook Back & PTA Quest ion s)

Unit 1 Matrix and Deter mina nts

Book back Questions

1. The ran k of the mat rix

1 )1

1 - 1 2

2 - 2 44 - 4 8

2)2

r - 1

is

3) 3 4 ) 4

2. The rank of the diago nal matr ix

1) 0 2) 2

If/4 =[ 2 0 1], the n the ran k of the A Ar

1) 1 2)2

11

If/4 = 2 the n the ra nk of the A A1

. 3 J1) 3 2) 0A - 1 0

If the ra nk of the ma trix

- 4

is

3) 3

3 ) 3

3 ) 1

is 2, the n A is0 A - 1

- 1 0 A

1) 1 2) 2 3) 3

IfAis a scalar matri x wit h sca lar k *0, of order 3, the n/4 - 1 is

r - i 3 2

1 k - 3

1 4 5

1) Ar is any real nu mb er

If the matri x

*1*

I X H '

If/4 = I ^ J ], th en ( ad jA)A =

has an inverse then the valuea^^.^

2) * = - 4 4) k * 4

1)

i 05

o i5

*

3) Ml" - 1

is

^ ;l

9. If/4 is a sq uar e matr ix of or de r n, A | is

i m | 2 " V z ) 4 %

10. The inverse of the matrix | ( j k F 0

/ x f W ' o o

1 0 0

0 1 0

0 0 >

11. If/4 is a ma e^ g^ f' ar de r 3, then dct (kA)1) fc3(det A) ,J 2) k2(dctA)

12. If I is- the uni t mat rix of or de r n,whe r e k 0 is a con stan t, the n adj{kl) =

1) & (a($&>6>0> ^(0


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16. In a syst em of 3 linear non -h omo ge nou s equ atio n with thre e unkno wns, if A - 0 and Ax - 0, Ay10 and A z= 0

then the system has

1) unique solution 2) two solu tion s 3) infinite ly man y solu tions 4) no so lu ti on

17. The syst em of equ ati ons ax +y + z = 0, x + by + z = 0, x +y + cz= 0 has a non-trivi al solution, then

18.

19.

1) 1 2) 2

Ifaa* +hey=c, pex+ qe> =dan d Ai =a b c b 1 a c

P q

,A2 =

d 'A

H p d

1 ) ( | , | ) 2 ) ( 3 ) ( log, l o g| ) 4 ) ( lo.q, l o g | )

If the equ at ion - 2x +y +z = x- 2y + z =m,x +y - 2z = n such that / +m + n =0, then the sys tem has Alf'%.

1) a non-zero unique solution 2) trivial solution 3) infinite ly man y solu tion

Created Questions

1. The ra nk of the matrix

1^1J A

2. The ra nk of the mat rix

1)9

4) No solutiona l / r

2 - 4

- 1 2

2^ 2

7 - 1

isa -p

n a i n

2is

"|b."%/ \ J

4 )Ri^R,+Ci

12 ) 2 3) 1

3. If/1 and B are mat rice s con fo rma ble to multipli cation then (AR)T

is1) A '7?1 2)BJAT 3) AB

4. (A1)"1 is equal to

1 ) / H 2) i4T 3)/I

5. If p(/l) =r then which of the following is correct?

1) all the minors of order r which do not vanish

2) A has atleast one minor of r which does not vanish and all hi

3) A has atleast one (r +1) order minor which vanishes

4) all (r +1) and higher order minors should not vanish

6. Which of the following is not elem enta ry transfo rmat ion?

1)/?/r Ayor Azis non -ze ro the n the syst em is

V. ^Incon sistent 2) incons istent

3) consistent and the system reduces to two equations 4) consistent and the system reduces to a single equation

13. In a syste m of 3 lin ea r eq ua ti on s wit h th re e u nk no wn s, if A = 0 an d all 2 x 2 mino rs of A, Ax, Ay, Ay. are zeros and atleast

one non -ze ro el eme nt is in A the n the syste m is

1) consistent 2) inconsistent

3) consistent and the system reduces to two equations 4) consistent and the system reduces to a s ingle equation

14. Every hom ogen ous system (linear)

1) i s always consistent 2) has only trivial solutio n

3) has infinitel y man y sol utio ns 4) need not be con sist ent

S},6>0> (D6ffrUU


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15. If p04J = pJVltfl th en Lhe sy st em is

1) consi stent and has infinitely many solutions 2} consis tent and has a unique solution

3) consistent 4j inconsist ent

16. If p(/4) =p[AB] = the nu mh er of unknow ns then the system is

Ij consis tent and has infinitely many so lutions 2) cons iste nt and has a uniq ue solution

3) consist ent 4) inconsi stent

1 ~. If p(/1) t p(/l, B) then the system is

1) consist ent and has infinitely many solutions 2) consis tent and has a uniq ue solution3) consistent 4) inco nsist ent

IS. In a sys tem of 3 linea r equ ati ons with thr ee unkn own s, p(A) = p[A, B) = 1 then the s yste m

1) has uniq ue solution 2) redu ces to 2 equa tio ns and has infinitely many soluti ons

3) re duc es to a single eq uati on and has infinitely man y solu tion s 4) is incon sist ent

19. In the homoge nous system with thre e unknowns , p(/TJ = num ber of unk now ns then the system has

1) only trivial solution 2) redu ces to 2 equat ions and has infinitely many so lu ti on s^ . 'v "

3) red uces to a single equ ati on and has infinitely many solu tion s 4) inco nsi sten t

10. In a syst em of 3 linea r equat io ns with th ree unkn own s, p(/l) = p(i4B) =2 the n the sys tem

II has unique solution 2) redu ces to 2 equa tion s and has infinite ly many solu tion s '?f:

3) red uces to a single equa tio n and has infinitely man y solu tion s 4) is in co ns is te nt '

-1. In the homo geno us system, p(/1) < the num ber of unk now ns then the system has .\ ij"'

1) only trivial solution 2) trivial solu tio n and infi nite ly many non-t rivial sol uti ons

3) only non- triv ial sol utio n 4) no solu tio n %/"% ICram er's rule is appl icab le only (with th ree un kn ow ns ) whe n if, if

1) A *0 2) A = 0 3) A = 0, Ax#0 4) A* = Ay= A z 4 p L

23. Which of the following is correc t regardi ng homog enou s system?

1) alwa ys inco nsi sten t 2) has only trivial solu tion 3) has only non-tr ivial sol uti ons

4) has only trivial sol ut ion on ly if rank of the coeff icie nt matrix is equal to the num be r of un kn ow ns

Unit 2 Vector A l p b i \ >Book back Ouestions

20. If a' is a non- zero vecto r and m is a non-zero scalar then rn'a is a uni t vector, if

1)m = 1 2) a = | m \ ' 4

) a =1

21. If cf , b are two unit vecto rs and 0 is the angle h e p ^ e M h e m , then (~a 4- b ) is a un it vec to r if

22. If a' an d b include an angle 120 and their.^i^jFiiUide are 2 and V3 then a . b is equ al to

1)VT 2)- s /a I \ 3 ] 2 4 ) - y

f ^ ^ I ^'''^1 23. If u' = a x( b x c' ] + b x ( c x a'J + c' x (a xb) then

1) t/is an unit vector : :2) v. "^ a + b + c 3) w '= 0 4) u t 0

24. If lT v b + ~c= 0, |~a' | - '3,\ b | = 4, \c'\ = 5 then the angle between 7T and b is

1 ) 7 4 ) ?3 3 2

25. The vect ors 2T + 3' /+ 4 k and a7+ b /+ ck are perpendicular when

1) a = 2, b c = - 4V 2) a =4,b =4,c= 5 3) a =4 ,b = 4 , c = - 5 4) a =- 2 ,b= 3 ,c = 4

26. The are a of the par all elo gra m having a diago nal 37' + j - k and a side T-3~f+ Ak is

2 ) 6 ^ 3 0 3 J ^ V 3 0 4 ) 3 ^ 3 0

V"-_r __ 2

27. If |a 4- b | = |a - b j then

parallel to b 2) a is perpendicular to b 3) |~a | = | T | 4) IT and~~h are unit vectors

28 . :if~p ,~cf an d J>'+~q'are vectors of magni tude A then the mag nitu de of \~p'- ~q'\is

1)2 A 2) V3A 3) V2A 4)1

29. If~a'x (b x T) + b x (7 ' x "a' ) +~c x (~a'x b ) - 7 x y then

l)~x = 0 2)~y= 0 3)~x a n d7 a r e parallel 4 ) T = Oo ry = 0 o r~x and y'are parallel

30. If PR =2T+ 7 + k , QS =- T + 37 + 2 k then th e area of the quadr ila ter al PQRS is

1J5V 3 2) 10 V3 3) 4) ^

SjyUj'Qd) 6&)U)6v uuj6m((p(b (ipgei) ^ ( 9 . erQgg) 60)snuu^6v ^tjibuLDn&yg).


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31. The proj ectio n of OP on a unit vector OQequa ls thric e th e area of para lle log ram OPRQ. Then |POQ is

1) tan A f - ) 21cos-]( ) 3) sin-1! -L ^ 4) sirr1 ( -1V 3 J \ 10 ) V V1U > V 3 /

32. If the project ion of tt'on IT and the projec tion of b on 7f are equal then the angle between ~cfi b an d a'- b is

l ) f ^ J ) f 3 ) ^ 4 ) f

33. If (i x (/> x T ) = ( a x /.)) xc for non- copla nar vector s a , b, c thenl ) 7 f parallel to b 2) b parallel to ~~c 3) ~c'parallel to ~u 4) ~a + b + ~c' = 0

34. If a line make s 45, 60 with posit ive dire ction of axes x an d y th en the angl e it ma ke s with th e z axis is

1) 30 2) 90 3) 45 4) 60

35 . If f IT x ~/T, b x T, T xTT I = 64 t he n I ~~a, b ,c 1 is1 1 1 3 %

1) 32 2) 8 3) 128 4) 0 r i V

36. If [ a + b, b + ~c, c +~a ]= 8 the n f a , b, T ] is ^ ^ i ; '

1 ) 4 2 ) 16 3 ) 32 4 ) - 4 /*%) '37. The value of [T+7- 7+ !i> + '" ] is equal to

1 )0 2) 1 3) 2 4 ) 4 4'1 ? V

38. The shor test distance of the point (2 ,1 0,1 ) from the plane r .( 3 T - 7 + 4 k ) = 2 v^26 "'"H.

1) 2V2 6 2)x/ 26 3) 2 4) *

39. The vec to r f i f x ft ) x (T x ch isV J V. ' SK ~S jp

1) perpendicular to 7 f , b ,~c and d

2) parallel to the ve cto rs pT xb ) and ( T * d ) \

3) parallel to the line of int ers ec tio n of the pl ane c ont ain ing ~a* and bund the plane containing T an d d

4) perpend icula r to the line of intersec tion o f the plane cont aining If and band the plane containing T a n d d

40. If I f , b /F a r e a r ight handed tr iad of mutually perpend icular vectors of magnitu de a, b, c then the value of

\~a , b ,~c'] is

1) a2 b2c2 2)0 . S)Mal)c 4) ahc

( t , __ , ,

41. If a , b , carc non-coplanar and [~a' * b, b r

] = [ a' + T, b+7 , 7+ a ] then [~a , b ,7 J is1) 2 2) 3 A 3) 1 4) 0

42. 7 = s T + t7*is the equation of

1) a str aigh t line joini ng the poi nts ("and";' 2) xoy plane 3) y o z plane 4) zox plane

43. If the magnit ude of the mom ent ab out the point 7+ k of a fo rc cT +a~f-lc) acting thro ugh the point T+ ; 'is V8 then

the value ofa is

1) 1 3 ) 3 4 ) 4

X~~3 V + 3 2z 544. The equati on of the line parallel .^) = - 7 - = and pas sin g throu gh th e point (1, 3. 5) in vect or form is

1) T= (T + 5;*+ 37"+ 5 k) 2)~r =(T + 3; + 5 k ) + t(T+ 57*+ 3 k)

3) T = (T+ 57+ ^ T ) + i f f + 37 + S~k) 4) T = (T + 3 7 + 5 T ) + t ( T + 5 7 + ^ T )

45. The point dtinte rsection of th e l ine T= ( T - T ) + t (37 + 27+ 7~k) and the plancT. (T+7~~k ) = 8 is1) (8, 6, 22) 2) ( - 8 , - 6 , - 2 2 ) 3) (4, 3, 11 ) 4) (- 4, - 3, - 11)

46. The equati on of the plane passi ng thr oug h the point (2 ,1 ,- 1) and the line of inters ection o ft he planes

T. (7+V37- k) = 0a n dT . Q '+ 2k ) = 0 is

l ) x 4 y - z = 0 2 ) x + 9 y + l l z = 0 3 ) 2 x + y - z + 5 = 0 4 ) 2 x - y + z = 0

47. The work done by the force F = 7 + 7 + k acting on a particle, if the par ticl e is displ aced fro m A(3, 3, 3) to the p oin t

B(4, 4, 4) is

1) 2 units 2) 3 units 3) 4 uni ts 4) 7 uni ts

48. If If = 7- 2 7 + 3k an d b = 3 7 + 7 + 2 k then a unit vector perpen dic ula r to If and b is

, , T + 7 + T T - 7 + T o-, - T +7 + 2 T ^ T - 7 - Ti J 7? ZJ ~r, 3 J -7= 4Jy/3 J J3 ' V3 J s/3Ti . t c . ^ ... ,. x-6 y + 4 z - 4 , x + l y + 2 2 + 3 .

I he poi nt ot in te rse ct io n or th e lines = = an d = = is- 6 4 - 8 2 4 - 2

1) (0, 0, -4) 2) (1 ,0 ,0 ) 3) (0 ,2 ,0 ) 4) (1, 2, 0)

Sjy(p(D6ST 3^(p(p6S)6VU UCU6WU&>(Jfib(D S>rt6V(ty0j,6V.


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50. The point of inte rse ctio n of the lines

~r' = (-7*+2 j'+ 3k) i t (' 27* + T +k )an d r'= [2 7+37 + 5k ) + 5(T+ 2~f+ 3k )

1)(2, 1,1) 2) (1 ,2 ,1) 3) ( 1 , 1 , 2) 4) (1 , 1 ,1), , , .r -i y2 z3 . x-y. y-4 z 5 .

ol . The sh ort es t dis tanc e be tw ee n the lines - - - = and = p = - y - is

^ 2 ) ^ 3 ) ;

,, , ,. x-3 y-1 z-5 .x-1 y-2 z -3 .The shor test distance bet wee n the parallel lines = 1 = and = is4 2 - 3 4 2 3

1) 3 2) 2 3) 1 4 ) 0x~ 1. y 1 z x y 1 7 i

53. The following two lines are : - : = - a n d 7- = - = 2 1 1 3 5 2

1) parallel 2) interse cting 3) sk ew 4) pcrp cndi cula r

54. The cen tre and radius of the sp he re given byx2 +y2 + z2 - 6x + Sy -lOz +1 = 0 is

1) (- 3, 4, - 5), 49 2) ( - 6 , 8 , - 1 0 ) 1 3) (3, - 4 , 5), 7 4) (6, - 8, 10) , 7

Created Questions

1. The value of 7 f . b w h e n l f = T - 2 7 " + k an d b = 47 '- 47 '+ 7k is

1 ) 1 9 _ J U - - J ] ~ 1 9 4 ) 1 4

2. The value of a . b when a =) + 2 k an d b =2 F+ k is -4. %.... \1 ) 2 2 ) - 2 _ 3 ) 3 _ 4 ) 4 f \ V

3. The valu e of a', /j when a =~f-2~k and /> = 2 7 + 3 7 - 2 k is

1)7 2 ) -7 3 )5 T i ; r

4. Ifm~7+ 27'+ k and 4 i '- 9 / + 2 k are perpendicular then m is

1) - 4 2) 8^ 3 ) 4

5. I f 5 T - 9 ~ + 2 A : a n d w T + 2 X + k are perpendicular then m is

2 ) - ^ _ 3 ) t

6. If a and b are two vec tors such tha t | a' | = 4 and | b | = 3 and c7"."b = 6, the n the angle be tw ee n a and b is

^ 2)

"i - iw"'1 4)

^7. The angle bet ween the vect ors 3 1 - 2 j -6k and 4 1- j + 8 k is

1)cos-1( | i ) 2) s in - ( - 1 | ) sin Q 4) cos - ( - f ~)8. The angle bet wee n the vec tor s 7- j and j -,-k 1%

l l f 2 3 ) - 4)2=

9, The proj ect ion of the vec to r 7T + $L- -4 o,n 2 ; + 67 + 3 /c is

10. a. b when a =2 F+ 2 /- k and 6' = 6 T - 37* + 2 k is

1 ) 4 2 ) - 4 _ 3 ) 3 4 ) 5

11. If th e vecto rs 2T + AT" + 4 and T- 27*+ /c are pe rp en di cu la r to each othe r, the n A isJ)i 2

H 3)l 4

H12. If th e vector ^, a = 3 i'+ 2j + 9 k an d b = i'+ m j + 3 A' are perp endi cula r to each other, then rnis

1 ) - 1 5 V w y f r 2)_15 ^ _3) 30 4 ) - 3 013. If the vect ors 'a. = 37*+ 27*+ 9 k an d b =7'+rrij*+3k are parallel, then m is

2 1 j

3H _

4H

14. I,f a , b , c are three mutually perp endi cula r unit vectors, then |a + b + c \ =

^ % 3 3 ^ _ 2 ) 9 3 ) 3 V3 4 ) V3'15. If [T + b | = 60, l a* - b | = 40 and |b | = 46 the n | 7f | = is

V1) 2 2 2 ) 2 1 3) 18 4) 1116. If Tf , 7T, and Iv 'b e vect ors such th at TT+lT+~w'= 0 . If |~u'\ =3, |7T| = 4 and |"w\ - 5, the n ~u\ ~v + ~v . ~w + ~W. ITis

1) 25 ^ 2 ) - 2 5 3) 5 4) V517. The proj ect ion of 7- / onz -axi s is

1) 0 _2)1 3) - 1 4) 2

18. The proj ecti on of 7 + 2 j'- 2 k on 27' ~"f+ 5k is1 0 io .... i ... vl o

30V - T B z ' v f o

6> 6)6V(Y)(r?uJ6GT $a#0J(D.


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19. The projection of th e vecto r 3Y + /- k on 4 Y- / + 2 /c is

2 ) - 9

VZT 3 ^V2l vTT

20. The wor k don e in movi ng a part ici e from the point A,with posi tion vec tor 2 T - 6J*+ 7 k to the point B,

with position vector 3T- ~f 5 k by a force "F* = 7 + 37*- ^ is

1) 25 2) 26 3) 27 4) 2 8

21. The wo rk don e by the force ~F - a~7+j'+ k in mov ing th e poin t of app lic ati on f ro m (1, 1, 1) to (2, 2, 2J

along a straight line is given to be 5 units. The value of a is

13 -3 2 ) 3 3 ) 8 4 ) - 8

22. If | T | = 3, |b | = 4 and IT. b =9 the n \~a * b | =

1 ) 3 / 7 2) 63 3) 69 4) Vfi9

4) - :S=

23. The angle betw een two vecto rs a 'a n d b i f | V x lT | = a . 6 i s

24, I f | T | = 2,n

2 ) f flfft | - 7 an d a' x b = 3 T - 2 ; + 6 A: the n tiie angle bet wee n a' and b is

2) - r 41

VVr'

25,

112* '4 - ' 3 6The d.c.s of a vector whose direction ratios arc 2, 3, - 6 are

4 ) f

k ( y

The ur.il normal vectors inthr> plane 2x y + 2z = 5 arc"(l-f-f) -fi.-v. 'l ] 2 T - T + 2 k 2 ) - [ 2 T - T + 2 f c ) 3 ) - j ( 2 T - J + 2 T ) \ $ t * ( 2 T - 7 + 2 7 )

27. The length of the perpen dic ula r from the origin to the pl an eT . (37+ 4^ 4J 2~ fc ') sf26 is

1 )2 6 2 ) ^ 3 ) 2

28. The distance from the origin to the pl an eT . (2 T - / + 5 k ) = 7 i;

>/30 2)f

4 ) ( I . 0 , - 1 0 )

4) (2,1, - 4) an d 5

4) 2 ) an d 5

29. The cho rdAB is a diameter oft he sphere | T - ( 2 T + 7 - 6 /c)J = VT8 with coordinate o f 4 as (3, 2, - 2).

The coordinate of B is -

1) (1, 0,10 ) 2) (- 1 , 0 , - 10) j a ( - 1 ,0,10)

30. The centr e and radius of th e sph ere | T - ( 2 T - ~~f 5 are

1) (2, - 1 ,4 ) and 5 2) (2, 1,4) and 5 3 H - 2, 1, 4) and 6

31. The cent re and radius of the sph ere | 2 T + + 4 k ) | = 4 are

32. The vectoi e quat ion of a plane passi ng through a point whe re P.F is d and perpe ndicu lar to a vect or n is

l ) 7 . " n = a . " n 2) x "rf 3) 7+ 7 = IT + 7 4) T- 7 = 7 - T

33. The vec tor equation of a plane wh os e distance from the origin isp and perpend icular to a unit vector n is

1) T . n = p ^ = q 3) T x T =p 4) 7 n = p

34. The non-parametri c vector equation of a plane passing through a point wh ose P. Vis T a n d parallel to T a n d T is

1 ) [ 7 - V , 7 , v ] = 0 , V\ J ) ' [ T T ? | = 0 3 ) [ 7 7 7 x 7 j = 0 4 ) [ T T V ] = 0

35. The non -pa ram e^f ^/e c^a equ ati on of a plane passing through the points whose P. Vsare T, ~band parallel to T is

1) [ 7 - a b - 7 7 ] * 0 2) [r ~b - T T ] = 0 3) f T T "T] = 0 4) f r 7 T ] = 0

36. The non-parametric vector equation of a plane passing through three non-collincar points wh ose P. l/s are T, ~b, T is

1)[ 7 - 7 T ^ ' T - T ] = 0 2) [ 7 7 T] = 0 3) [7 IT 71 = 0 4) ["a* T F ] = 01he vector eqtMtion of a plane passing through the line of intersection of the planes r .~n^= q\ and T. 7 T = c/2

*) ~ ?! )_+_A(T." n 7 - q i ) = 0 2 ) T . I f f + 7 . 7 T = qi + Xqz

3)"rx "^ + Tx n2' =91 + ^2 4)Tx77' - T * 7 T = qi+Q2

37.

&fhe angle b et wee n th e line r'= a +tb and the plane r .n = q is con nec ted by the relation

a,rV a . n^ ) c o s B = 2)cos 0 =

b Ti3) sin 0 =

a .b

T Ib 11n I ' In

39. The vecto r equat ion of a sph ere whos e centre is origin and radi us 'a' is

l ) r = T 2 ) 7 - c = T 3 ) | 7 | = | T' l

4) sin 0 = 7==b

4) r = a

b .n

I n

68Tg GTGBtuQe, 6W", Gtytjlb.


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Unit 3 Complex Nu mb ers

. , , r - i + ^ i1 0 0

f -1-1V 3 ]1 U 0

.d5. T he value o i l - j + | - j i s

1) 2 2) 0 3) - 1 4) 1

56. The modu lus and amplitu de of the complcx numb er [e:i ]3 arc respectively

l ) * * , f 2 ) < * . - f 3 )

57. II (m - 5) +i(n +4) is the comple x con jug ate of [2m + 3) + /(3/? - 2] t he n (ri, m) are

1)( I -8 ) 2) ( - i s ) 3)(i , -8 ) 4) (1. 8), , . 1+x+ ty, i " ' ' '

58. I fx 2+y2 - 1 then th e va lue ol is " ,r r>l+x- ry J V

I)x-iy 2) 2x 3) - 2 iy 4) x + iy

59. The mo du lu s of th e com plc x n um b er 2 + i>/3 is ,


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75. The conj uga te of i'13+ Z14+ /'' + /" is

1.) 1 2) - 1 3) 0

76. If - / + 2 is one root of liu : equat ion ax2 - bx* c = 0, then the other root is

1) - / - 2 2 J / - 2 3 ) 2 + /

77. The quadra tic equation wh os e roots are iV7 is

1) x2 + 7 = 0 2) x2 - 7 - 0 3) x* + x + 7 = 0

78. The equat ion ha ving 4 - 3/ and 4 + 3/ as roots is1) *2 + 8* + 25 = 0 2) x

2 + 8 x - 25 = 0 3 ) x

2 - 8x + 25 = 0

79. 'f ~~ is the roo t of th e equat ion ax2 + bx +10, wher e a, b are real then (a, b) is

1)(1 ,1 ) 2 ) (1 , - 1 ) 3 ) (0 ,1 )

80 . If - / + 3 is a roo t of x2 - 6x +k =0, then the va lue of k is

1 ) 5 2) V5 3) vTO

81. If to is a cub e ro ot of uni ty the n th e valu e of (1 - to + to*)4+ ( 1+ - to 2)4 is

1) 0 2) 32 3 ) - 16

82. It to is th e nth root of unity then

1)1+ a>2+ a)4 + ... = a) + u> < + a/ ' + ... 2) at" = 0 3) a)n = 1

83. If oo is th e cube ro ot of uni ty th en t he va lue of (1 - go) (1 - oj2) (1- to1) (1 - to8) is

4) - /

4)2/ + /

4) x2 - x - 7 = 0

4) x 2 - 8x - 25 = 0

4) (1,0)

4) 10

4 ) - 3 2

2) in G.P. wit h co mm on di ffe ren ce a>2

3) in A.P. wit h co mm on dif fere nce a) 4) in A.P. with co mmo n dif fere nce with to2

16. The argum ent s of nth roots of a complex nu mbe r differ by1 ) - 2)- 3 ) - 4 )12

n n ' n ' n

17. Which of th e following sta tem ent is correct ?1) negative complex nu mb ers exist 2) ord er relatio n does not exist in real nu mb ersT ) n r H p r r e l a t i o n p v i c t i n r n m n l P Y n n m h p r c 4 .1 f 1 + f \ > f3 - 7i1 i s m p a n i n f f l p ^

ms>


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%_

A*

p"

10. Which of the following st at em en t are corrc ct?

a) Re (z) < | z | b) Im (z ) > | z | c) |T \ = |z | d) (zn ) = (~z )

n

1.)(a), (h j _ 2) (bj, (c ) 3) (b), (c) and (d) 4) (a), (c) and (d)

19. The val ue s of T" + T is

1) 2 Rc (z) 2) Re (*) 3) Im (z) 4) 2 Im (z)

20. Th e va lu es of z ~z is

1) 2 Im (z ) 2) 2/ /m (z ) 3) Im (z) 4) /Im (z)21. The valu es of z l T is

1 ) M 2 ) | * P 3 ) 2 | z | 4 ) 2 1 21? +

22. If | z - zi | = | z - Z2 | th en th e lo cu s of z is

1) acirclc wi th ce nt re at th e ori gin 2) a circl c wi th ce nt re at zi "%

3) a straigh t line passin g thr ou gh the origin 4) is a perpendicular bisector of the line joining zi and Z2

23 . If a> is a cub e roo t of unit y th en U ' 1

1) a>2 = 1 2) 1 + = 0 3) 1 + o) + (O2= 0 4) 1 - w + oo2= 0

24. The prin cipa l valu e of arg z lies in the interval

l ) [ 0 , f ] 2) C- 7T, 7T] 33 [0, TTj 4) (- 71 ,0 ]

25. If zi and Z2 are any two com plex n um be rs then which one of the following is false?

1) Re (zi + Zi] - Re (zi) + Re (7.2) 2) Im [z\ + z?) = Im (zi) +Im [7.2) .[ f-

3) arg (z i + z2) =arg (zi) +arg (Z2) 4) | zi Z21 = | zi | | zi \

26. The fou rth roo ts of uni ty ar e ^4 |1) 1 i,- 1 / 2) /, 1 1 3) 1, i 4) % i i 'V

27. The fourth root s of unity form the vertic es of

1) an equila teral triang le 2) a square 3) a hexag on 4p rectan gle

20. Cube roo ts of uni ty ar e

^ - l i V 3 , tV5 , i i V 3 \ V , . l i V 3m . j 2 ) i , - i

3H - ^ . ^ \ j

4H r ~

29. The n um be r of val ues of {cos 0 + i sin 0)p/6vr6ff)i) tyeGrGVrt&QU).11


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42. Which one or the followi ng is inc orr ect ?

1J Multiplying a complex num be r by / is equiv alen t to rotati ng the nu mb er c ount er clock wise

about the origin through an angle 90

2) Multiplying a complex number by - i is equivalent to rotating the number clockwisc


3) Dividing a complex number by iis equi valen t to rotating the num ber co unt er clockwi se


4) Dividing a complex number by / is equivalent to rotating the number clockwisc about theorigin through an angle 90

43. Which one oft he following is incor rect regar ding nth roots of unity?

1) The number of distinct roots is n

2) The root s ar e in G.P. with co mmo n rat io cis /rt _ s

r

3) The arguments arc in A.P. with common difference

4) product of the roots is 0 and the sum of the roots is 1

44. Which of th e following arc tr ue?

i) Ifn is a posit ive in te ge r th en (cos B + / sin 0)" = cos n0 + / sin nO)

iij Itn is a negative integer then (cos 0 + / sin 0)n = cos n0 - / sin nWj

iii) Ifn is a fraction then cos n0 +is in nfl is one o f t h e val ues of (cos 0 + / sin 0)n

iv) Ifn is a negative integer then (cos 0 +i sin 0)n = cos n0 + / sin nO)

l)(i ) , ( i i ) , ( i i i ) , ( iv) 2) (i), (iii), (iv) 3) (i). (iv) 4) (ij45. If 0(0 , 0), B[Z2),B'(- ZL) are the complex nu mb er s in a arga nd plane then ^hif eh% fjHe folic.ving are corr ect?

i) In the paral lelogr am OACB, Crep rese nts zi +zi '>'

ii) In the arg and pla ne Erepresents zi zi where OE = OA.OBa nd OEmak es an angle ar g(zi ) + arg(z2) with positive real

axis

iii) In the argand parallelogram OB'DA, Drepresen ts zi - zi

angle arg(zi) - arg(z2) with positive real axis

. 4

"il

v%)

p%.

z OAiv) In the argand plane Freprese nts where OF = and OF r

2 OB

1)(>)()> (i) (iv)46. If Z= 0 then arg(Z) is

1)0

Book back Questions84.

2) (0, (HO, (iv)

2) TT

3) (0, (iv. 4) (i) only

4) indeterminate

Unit 4 Analytical Geom etry

85.

86.

The axis ofthe parabolay 2 -2y + 8x- 23 = 0j $

l ) y = - 1 2) x = - 3

16x 2 - 3y2 - 32x - 12y - 44 = 0 reprcsQj

1) an elli pse 2) a circl,

The line 4x + 2y - c is a tangent td tmyrarabolay 2 = 16x then

i ) - i 3 ) 4

3) x = 3

3) a parabola

c is

4)y=l

4) a hyperbola

4 ) - 4

87. The point of inter sectio n of the tang ents at ti = ta nd tz= 3tt o the parab ola^ 2 = 8x is

1 ) ( 6 f 2 , 8 1 ) % J (8 c < 6 f 2 ) 3) ( t 2 ,4 f ) 4 ) (4 t , t 2 )

The length of the latus rectum of the parabola y2 - 4x +4y +8 = 0

1 ) 8 ri. ' % i ^ ' 2 ) 6 3 ) 4 4 )2

The diretrix o|,th&parabola y2 = x + 4 is

is * 1 '

88.

89.

3 ) * = " T 4 ) X = Hl )x =

90. The leiigth ot the latus rect um o ft he para bola wh os e vertex is (2, -3 ) and the diretr ix is x = 4 is

2 V ' 2) 4 3) 6 4) 8

91. The focus of the para bol a x2= 16y is

\ '" f )'(4 , 0) 2) (0, 4 ) 3) ( - 4, 0) 4) (0, -4 )

92. The vertex of th e parabola x2 =Q y -1 is

U ( - j . o ) 2 ) ( i , 0 ) 3 ) ( 0 , | ) 4 ) ( 0 , - i )

93. The line2x + 3y+ 9 = 0 touches the pa rab olay 2 =8x at the point

2 ) [ 2, 4 ) 3 ) ( - 6 , | ) ( I ' - 6 )

94. The tangent s at the end of any focal chor d to the para bolay2 = 12x is intersect on the line

1) x - 3 = 0 2) x + 3 = 0 3)y +3 = 0 4 ) y - 3 = 0

2-,6rterT$$6BT 'S-.JJ/Jib gam12


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95. The angle bet wee n the two tange nts dra wn from the poi nt (- 4, 4) to y 2 - 16x is

1) 45" 2) 30 3) 60 4) 90

96. The ecce ntric ity of the coni c 9x2 i 5y 2 - 54x - 40y + 116 = 0 is

>! 3, 1 ^

y-2 y?.

97. The length of the se mi -m aj or and the length ol se mi -m in or axis of th e ellipse + = 1

1) 26, 12 2) 13 ,2 4 3) 12, 26 4) 13 ,12

98. The distanc e be tw ee n the foci of the ellipse 9x2 + 5y- =180

1) 4 2 ) 6 3) 8 4) 2

99. If the length of ma jor and sem i-m inor ax es of an ellipse are 8, 2 and th eir corr esp ond ing eq uat ion s arc y - 6 = Ojf i&x

4 = 0 then the equations of the ellipse is

, ( y - 6 ) " = 1 2 1 ( * + 4 )2 ( y - 6 ) 2 = (x+4)

2 __ ( . y - 6 ) " 2 = (x+4)? __ ( y - 6 ) 2 = .

4 16 16 4 16 4 J

4 16 1

100 The stra igh t line 2x-y+c = Qis a ta ng en t to the ellipse Ax2 + 8 y2 = 32, ifc is v

2) 6 J " " ' ' - *' ...

101. The sum of the dist anc e of any p oint on the ellipse 4x2 +9y 2 = 36 f ro m (_V5, 0) and (-V 5, 0) i | l

1 ) 4 2 ) 8 3 ) 6 4) 18 ifc

102. . he radi us of the di re cto r circle of the coni c 9x2

+ 16y2

= 144 i s M \. i \ 2) 4 3) 3 4) S " >

"it..

/ W'103 The locus foot of the p er pe nd ic ul ar fro m the focu s to a tan ge nt of the c urve 1 6xj[jfc 2.5y2= 400 is

l ) x : + y 2 = 4 2 ) x 2 + y2 = 2 5 3 ) x 2 + y 2 = 1 6 4 } * 2 + y 2 = 9

104. The eccentri city of th e hype rbo la 12y2 - Ax2 - 24x + 4 8 y - 127 = 0

D 4 2 ) 3 3 ) 2 . / y W * ' 4 ) 6

105. The eccentricity of the hyper bola wh ose l atus rec tum is equal t of ft al r^ 'i ^ con jug ate axis is

1 ) 7 2 ) | 3 ) j \ j

y2

106. The differ enc e be tw ee n the focal dist ance of any point on the hy per bola = 1 is 24

and the eccen tricity is 2. Then th e equat ion of the hyp erb ola is

y- v2 y?1 V ^ y2 ..'21 ) 1 4 4 " 4 3 2 = 1 2) 4 3 2 " 7 4 4 = \ V 9 " 1 2 " T z V f = 1 ^ W3 ~ 12 = 1

j "J"'107. The direct rices of the hyp erbo la x2 - 4(y -7.3) | ' i J % a r e

i ) y = i 2 ) , = i ^ V " 3 ) y , f 4 ) x = f

108. The line 5x - 2y + 4/c = 0 is a tangent ; to 4^ - y 2 = 36 then k is

a2

Wx2 yl

109. The equ ati on of the c h o r d | act of ta nge nts fro m (2, 1) to the hy pe rbo la - = 1 is

1) 9x - 8y - 72 = 0 f V M ) 9 x + 8 y + 7 2 = 0 3 ) 8 x - 9 y - 7 2 = 0 4 ) 8 x + 9 y + 7 2 = 0

8 T-&, 2 y 2

110. The angle betwe en the asym pto te s to the hyperbol a - = 1 is

1)tr - 2t a? i t j , 0) ^ 2) 7r - 2t a n- 1

( j ) 3 ) 2tan_1

4 ) 2 t an_ 1

111. The asymptotesTO the hyperbola 36y2 - 25x2 + 900 = 0 arc

l J y ^ J V ' 2)y = ^x 3 ) y = x 4 ) y = ^ x

a x2 y2

112. The pro duc t of the perp end icu lar s draw n fro m the point (8, 0) on the hyperbola to its asy mpt ote s i s 1 = 1 is%, 64 36

- - 2 5 5 7 6 6 25

" S 3 5 7 6 2 5 2 5 ^ 6r 2 2

x y113. The locus of the point of interse ction of per pen dic ula r tange nts to the hyperbo la 1 = 1 is

16 9

1) x2 + y2 = 25 2) x2+y 2 = 4 3 ) x 2 + y 2 = 3 4 ) x 2 + y 2 = 7

114. The eccen tric ity of the hyp erb ola with a sy mp to te s x + 2y - 5 = 0, 2x -y + 5 = 0

13 3 2 ) V2 3) V3 4) 2

115. Length of the semi-trasverse axis of the rectangular hyperbola xy = 8 is

1) 2 2 ) 4 3) 16 4 ) 8

6T6G)&, wrtfm-U6ST (pftQma ^6S)$(2LU s)rp)6V6K)(_ fyd-uj&rpfsm.13


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116. The asymptotes ofthe rectangular hyperbolaxy = c2are

l)x~c,y-c 2)x = 0,y = c 3) x = c,y = 0 4 ) x = 0,y = 0

117. The co ordinate of the vertices of the rectangular hyperbola xy ~ 16 arc

1) (4, 4), (-4, -4) 2) (2, 0), f- 2, -8 ) 3) (4, 0), (- 4, 0) 4) (8, 0), (- 8, 0)

1 18. One oft he foci of th e rectan gul ar hy perbol a xy = 18 is

1 )( 6 ,6 ) " 2) (3 ,3 ) 3) (4, 4) 4) (5, 5)

119. The length oft he latus rectu m of th e rectangul ar hyperbola xy =32 is

1) 1) 8V2 2) 32 3) 8 4) 16

120. The area of the triangle form ed by the tangent at any point on the rectan gula r hyperb ola xy = 72 and its asymptotes4s

1) 36 2) 18 3) 72 4) 14 4

121. The normal to the rectangular hyperbola xy = 9 at mee ts the curve again atii'VV

2 ) ( - 2 4 , - | ) 3 ) ( - | , - 2 4 )

Created Questions1. The axis oft he para bola y 2 = 4xis

1) x = 0 2 ) y = 0

2. The vertex of th e para bola y ? =4x is

1 ) ( 1 , 0 ) 2 ) ( 0 , 1 )

3. The focus of th e parab olay 2 = 4x is1 ) ( 0 , 1 ) 2 ) ( 1 , 1 )

4. The directrix of th e parabola y 2 = 4x is

l ) y = - 1 2) x = - 1

5. The equation of the latus rectu m of y 2= 4x is

1) x = 1 2 ) y = 1

6. Th e leng th of th e L.R. o fy 2 = 4x is

1)2 2)3

7. The axis of the par abo la x2 =- Ayi s

l ) y =1 2) x = 0

8. The vert ex of the par abo la x2 =- 4y is

D ( 0 , 1 ) 2 ) ( 0 , - 1 )

9. The focus of the para bol a x2 = - 4y is

1) (0 ,03 2) ( 0 , - 1 )10. The directrix of th e parabo la x2= - 4y is

l ) x = l 2) x =0

11. The equati on o ft he L. R. of x 2 =- 4y is

1 ) * = - 1 2 ) y y

12. The length of th e L. R. of x 2=- 4 y is -

1) 1 2) 2 j " V

13. The axis of the para bol a y2

1)* = 0

14. The vertex of the par abo lay 2 = - 8x is

1) (0, 0) (2, 0)

15. The focus of the para bo la^ 2 =- 8x is

1) (0, - 2) ^ 2) (0 ,2 )

16. The equati on of th e directri x of the parabola y

2

= - 8x isl ) y + 2 = 0 2) x - 2 = 0

17. The equati on of th e latus rectu m ofy 2 = - 8x is

l ) y - # % ^ 2 )y + 2 = 0

18. The length of th e latus re ct um y 2 =- 8x is

l 9. \T h ^ axis of the parabola x2= 20y is

\ > ) y - 5 2 ) ^ = 5

207 The vertex of the parabola x 2= 20y is

13(0,5) 2) (0 , 0)

21. The focu s of the par abo la x2= 20y is

1) ( 0 ,0 ) 2) (5 ,0)

22. The equation ofth e dirct r ixof the parabola x2 = 20y is

l ) y - 5 = 0 2) x + 5 = 0 3 ) x - 5 = 0

23. The equati on oft he latus rect um of the para bola x2= 20y is

1) x - 5 =0 2 ) y - 5 = 0 3 ) y + 5 = 0

4 ) ( 2 4 , | )

4)y = l

4 X

4) (0


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33.

34.

The length of the latus re ctu m of the parab ola x 2 = 2 0y is

1) 20 2) 10 ' 3 ) 5

if tiie centre ofthe ellipse is (2, 3) one ofthe foci is (3, 3) then the other focus

1) (1 , 3 ) 2 ) ( - 1 ,3 ) 3 ) (1 , -3 )

x2 y

2

The equati ons of th e majo r and mi nor axes of + = 1are

l)x = 3,y = 2 2) x ~ - 3, y ~ - 2 3) ,v =(),y = 0

The equations of the major and minor axes of Ax

2

+ 3y

2

=12 are(1) v = v 3 , y = 2 2) x = 0,y =0 3) x = -\[3,y = - 2

x2 y2

The leng ths of mi no r and ma jo r axe s of + = 1 ar e

1) 6 ,4 2 ) 3 ,2 3 ) 4 , 6

The lengths of ma jo r and m ino r axes of 4x2 + 3y2 =12 are

1 ) 4 , 2 \ 3 2 ) 2 , \ /3 3 ) 2 V 3 , 4

x2 y2

I he equ at ion of the d ire ct ric es of -f = 1 are

1) v= = 2 ) x = i |w v '

The equation of the directr ices of 25x2 + 9y 2 = 225 are

l ) X - 2 ) X - ^x2 y7

The equat ion of the lat us rec tu m of + =1 arc

16 9

1)y = v' 7 2 ) x = V 7

The equation ofthe I.. Rs of 25x 2 + 9y 2 = 225 are

1) >' = 5 2) x = 5

The length of t he L. R o f + = 1 is16 9

9 2

1) n 2) "2 ' 9

The length of th e L. Ro f2 5x 2 + 9y 2 = 225 is

! 2

' tx2 y2

The eccentricity of the ellipse + = 1 is

x2 y

2

The ecce ntric ity of th e ellipse + = 1 is

uf V tThe eccentr ici ty of the

is

4) 4

4 ) ( - 1 , - 3 )

4) y = 0, x = 0

4 ) y 0, x = 0

45.

46.

1) ( 5/ 0) 2) (0, V5 )

The foci of the ellipse 16x2 + 25y2 = 400 are

3, 0) 2) (0, 3)

x 2 y 2

The vertices of the ellipse + = 1 ore

1) (0 , 5) 2) (0, 3) 3) ( 5, 0)

x2 y2

The vert ic es of the elli pse + = 1 ar e

1) (0 , 3) 2) ( 2, 0) 3) ( 3, 0)

The vertices ofthe ellipse 16x2 + 25y 2 = 40 0 are

1) (0, 4) 2) ( 5, 0) 3) ( 4, 0)

4) (V5, 0)

4) ( 5,0)

4) ( 3, 0)

4) (0, 2)

4) (0, 5)

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47.

48.

49.

50.

51.

52.

53.

54.

55.

56.

57.

58.

59.

60.

61.

62.

63.

64.

65.

66.

67.

68.

If the ce nt re of the ellip se is (4, - 2) an d on e of the foci is (4, 2), the n t he ot he r focu s is

1) (4, 6) 2) (6 , - 4 ) 3) ( 4 , - 6 ) 4) (6, 4)X2 y2

Tiie eq ua ti on s of tra nsv erse: aim lun ju ga tc axes of Live iiyp erbu M = 1 2 ) - o r ' 3 3

3^t

The point of contact of the tangent y =mx + cand th e pa rabola y2 = 4ax is

" % . 1 - X 2 y2

69. The point of contact of the tan ge nt y =mx + c and the ellipse + = 1 is

2)-a2m b2\

c ' c )

70. The point of cont act of the tan ge nt y =mx + cand the hype rbol a - ^ - 1

3 , ( = = 4 )

is

4) 25x2 - 36y2 = 0

4) J 3

(=*-=*)

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71. The tru e sta tem ent s of th e following are

(a) Two tangents and 3 normals can be drawn to a parabola from a point

(bj TWO tangents and 4 normals can be drawn to an eiiipsefrom a point

(c) Two tangents and 4 normals can be drawn to an hyperbola from a point

(d) Two tangents and 4 normals can be drawn to an R. H. from a point

1) (a), (b), (c) arid (d ) 2) (a), (b) only 3) fc"), (d) only 4) (a), (b) and (c)

72. If't.i 't.2 are the extremities of any focal chord of a parabolay 2= 4ox then ti t? is

1 ) - 1 2) 0 3) 1 4) 1/ 2

73. The nor mal at 'V on the pa ra bol ay 2 = 4 ax meets the parabola at 't>'the n (i + ) is

l ) - h 2) t2 3)Li+ tz 4)f

x2 y2 :

'P'74. The condi tion tha t the lineIx + my + n= 0 may be a nor mal to th e ellipse + = 1 is ;;;; " "

i-> II ? I > > n -- a' I 1)2 {a.2\ b2Y a2 b2 (a 2-/ j2)2 a2 b2 (zVb2)21) a/ J+ 2alm2 + m'n =0 2 ) + = - 3 ) - 7 + 1 4 ) = v v

j

lJ m2 n2 J

I2 mz nl I2 m2 nfg 2 y2 . \

75. The condit ion that the line Ix + my + n =0 may be a nor mal to the hype rbo la - = 1 is i 'V

1 n ; 7 7 A rt"' I 1,2 (a2+b2)2 a2 b2 (a2-b2)2 a2 " ' 7 fc2 V ( a 2 + f c 2) 2

1) o/3+ 2alm2 + mln = 0 2) + = ^ ^ 3) + = 41 1

I2 m2 n2 I2 m2 n2 l\ ";ijn2 n

76. The condit ion that the lineIx + my + n - 0 may be a normal to the parabolay 2 = 4ax isu n , - , n (i2 . b2 (a2+b2)2 a2 , b2 b2 (a2+ h2)21) a/3 * cami" + m2n = 0 2) + = ^ 3) + - v j, .K. 4) =-r - = -L-

l- in n2 I2 m2 I? /"%, i m h77. The chord of contact of tang ent s from any point on the directrix of th e parab ola y 2 = 4ax passes thro ugh its

1) vert ex 2) focus 3) direc trix 4) latu s rec tumx2 -y2

78. The chord of cont act of tan gen ts from any point on the direc trix of the ellipseg4- = 1 pas se s thr ough its

1) vertex 2) focus 3) directr ix \ , J 4) latus rectumJ&j, ^S jf %. X 2 y 2

79. The chor d of cont act of ta nge nt s from any point on the directrixjjof th e?t|y |er bol a - = 1 pas ses through its

1) vert ex 2) focus 3) direc trix 4) latu s rec tum

80. The point of intersecti on of tang ent s at 't\ an d 'tz to the pa^bolay 2 = 4ax is

1) (a(ti+ ti), ati tz) 2) (ati t2, a(ti + ti)) 3H fc la l> ) 4) (at i ti, a{t\ - t2))

81. If the norm al to the R.H.xy = c2 at 'ti' meets the curve alaPh a?'^' then ti3 ti =

1 ) 1 2 ) 0 3 J - 1 4 ) - 2

82. The locus of th e point of intersection of pe rp e^ e^ la rt an ge nt s to the parab olay 2 = 4axis

1) latus rectum 2) directrix c3^ t|^ gent at the vert ex 4) axis of the para bola

83. The locus of th e foot of per pen di cul ar from the focus on any tan gen t to the ellipse ^ = 1 is

1) x2 +yz = a2- b2 2)x2&yfy=%2 3) x2+y2 = a2+ b2 4) x = 0J ? ' X 2 y2

84. The locus of the foot of per pen dic ula r from the focus on any tan gen t to the hyp erb ola - = 1 is

1)x2 +y2 = a2- b2 + y z= a 2 3) x2+ y 2=a2 + b2 4) x = 0

{T6rTfT6V &>rf68T 6T6V6Vrt$ (L)U) 6>Qfi)$fpUO

17
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Unit 5 Differe ntial Calculas and its Appl icati ons I

Book back Questions

1. The grad ient of the ciu v s y -2x' +3x + [j a I x 2 is

1) -20 2]2 7 3 ) - 1 6 4 ) - 2 1

The rate of change of are a A of a circle of radius r is~ dr ->dr d. dr

l)2nr 2 ) 2 n r - 3 ) ^ - 4 ) t t -

The velocity v of a particle moving along a strai ght line wh en at a distanc e x from the ori gin is given by a + bv2= x2

where aand b are constants. Then the acceleration is !>

2.

3.

i ) -x

4.

5.

6.

A spherical snowball is melting in such a way that its volume is decreasing at a rate of 1 cm 3 / min. The rate at which

the diameter is decreasing when the diameter is 10 cm is

2W

m/,nin i i3 ) - c m / m i n1 ) c m / m i n

The slope of the tangent to the curve y = 3x2+ 3sin x atx = 0

i ) 3 2) 2 3) 1 4) - i

The slope of the normal to the curvey = 3x2 at the point whose x coordinate is 2 is

4 ) - c m / m i l V >

$'%y-

2) 3 ) - ^ 4 ) i A

8.

9.

13 ' 14 " ' 12 ' J

12T

he point on the curvey =2K-- 6 X - 4 at wnid> the tangent is parallel to the x- axis ia %. T

(;=?) (f=r) (t-t)The equation of the tangen t to the curve y = at the point (-1, -1 / 5) if

1) 5y + 3x =2 2} 5y - 3x = 2 3 ) 3 x - 5 y = 4

The equation of the normal to the curve 0 = - at the point (-3, - f / 3 ) i

1) 30 = 27t - 80 2) 50 = 2 7t - 8 0

2 2 2

10. The angle bet wee n the cur ves + = 1 and - y2 5 9 8

2) -J

311. The angle bet wee n the cu rv ey =e

mx an dy = e '1

11 tan~i-h) 2)can-'(2m' 3) ^ { t B ), / 2m \

4) tan-HJ \ m 2 + l /

/3=a 2 / 3are12. The par amet ric equa tio ns of the cj,

1)x = asin30; y =acos 30 a co s3 0;y =a s in 30

3) x = a 3sin 0; y = a 3 cos 0 vH 4) X= a3cos 0; y = a 3 sin 0

13. If the normal to the c u r v e ^ " ^ | = a2/3 makes an angle 0 with the x- axis then the slope of the normal is

1) - cot 0 2} tan 0 3) - tan 0 4) cot 0

14. If the length of the diagdi'1^1'bf'a squa re is incre asing at the ra te of 0.1 cm /sec. Wha t is the rat e of increas e of its are a

when the side is

1) 1.5 cm 2 / s e c 2 )3c m 2 / s e c 3 )3V 2c m 2 / s e c 4) 0.15 cm 2 /sec

15. Wha t is the surfac e are a of a sp he re whe n the volume is incre asing at the sa me ra te as its rad ius

X)1W

2)^

3}4n 4)?

16. For wha t values of x is the rate of incr ease of x 3 - 2x2+3x +8 is twice the rate of increase ofx

r \ ^ H ' -3

) 2

) ( f (T6MTL-. ID6VTU) 6T6V)&,6V)(qti)18
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19/35

www. tnschools . co. i n

20.

21.

22.

23.

24.

The gradient ofthe tangent to the curvey =0 + 4x - 2x2 at the point where the curve cuts they-axis is

1 ) 0 2 ) 4 3) 0 4 ) - 4

The angle between the parabolas/ 2 = x and x- =y at the origin is

1 ) 2 t a i r 1 ( l ) 2 ) t a / r ' 0 3) n - 4)^

For the curve x - e rcost?,y - e ' s in L the tangent line is parallel to the x-axis when Lis equ al to

3)0z i ; 4 j fIf the norma l make s an angle 0 with positive x-axis then t he slope of th e curve at the point wher e the normal is dra wn

't-

is1) - co t 0 2) tan 0 3) - tan 0

The value of a' so that the curvesy = 3e x an dy = - e_xinterse ct orthogon ally is

ID - 1 2 ) 1 3 ) ;

2f>. Ifs =t:i - 4C2 + 7, the velocity when the acceleration is zero is. . 3 2 ,

ij y m / sec 2 ) - y ni/'scc16

6) in/ sec

4) cot 0

4 )3

v 32j - - - m/ sec

* A

26. If the velocity of a particle mov ing along a strai ght line is directly p rop ort ion al to t he s qua re

27.

28.

29.

30.

31.

32.

33.

po in t on th e line . The n its ac ce le ra ti on is pr op or ti on al to

l )s 2) s2 3) s3

The Rolle's constant for the fu nct ion y = x2 on [- 2, 2]is

*. \

i > f 2)0

Th e 'c of Lagranges Mean Value Theorem for the functionJ[x) = x2 + 2x ^ l f%=-^,b= 1 is

1) - 1 2) 1

The value of'c'in Rolle's Theorem for the function/[x) = cos - onwr, 31

n 3n1 ) 0 2 ) 2 n

' 4 )

TThe value 'c' of Lagra nges Mean Value Th eor em f or the fu ncti on /(x) = Vx wh en a= land b= 4 is

lim 1X - coex

1)2

lim ax-bx __

x - 0cx-dx

1 ) CO

2)!2)0

vs3) co

ab

4 )1

4)log (a/b)

log (c/d)

If/ to) = 2; /' (a ) = lif lfa ) = 2 then the value of = js/%?

! ) 5

4 ^ i . r 2) - 5 3) 3 4) ~ 3

3) - x 2

3) (4, oo)

3) (0, oo)

4) x - 2

4) everywhere

4) (-2# oo) .

34. Which of the "following fu nc ti on is inc reasi ng in (0, oo)

l ) e * " " V 2 ) ;

35. The function o ff [ x )= x2- 5x + 4 is incr easi ng in

l ) ( - 4 l f 2 ) (1,4)

36. 4' he func tion of/ [x) = x2 is decre asing in: %# f -o o , co ) 2 ) ( -0 0,0)

37i . The fu nc ti on y = tan x - x is

1) an incr easi ng functi on in (o, ^

3) increasin g in (o, and decr easing in

38. In a given semi circle of dia met er 4 cm a rec tang le is to be inscri bed. The maxi mum a rea of the recta l.gle is

1 )2 2 ) 4 3 ) 8 4) 16

39. The least possible peri met er of a rectangle of area 100 m2

is1) 10 2) 20 3) 4 0 4) 60

2) a decreasing function in (o,

4) dec rea sin g in (o, 0 and incre asin g in

GT$g,6)6V


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40. 1 f/ ( x) = xl 4x + 5 on [0, 3j then Che abs olu te m axi mum vaiu e is

1) 2 2) 3 3) 4 4) 5

4.1# Tho curv e y is

1) conc ave upw ar d (or x >0 2) conca ve do wn wa rd for .Y >0

3") everywhe re concave upwa rd 4) ev er yw he re con cave do wn wa rd

42. Which of the following curve s is concave down wa rd ?

l ) y =- x2 2) y =x2 3) y = ex 4 )y = x2+ 2x- 3

43. The point of inflexion of the cu rv ey = x4is at

l ) x = 0 2 )x = 3 3 ) x = 1 2 4 ) n o wh er e

44. The cu rv ey = ax3 + bx 2 +cx + dhas a point of inflexion at x = 1 the n

1) a + b= 0 2) c7+ 3b= 0 3) 3a + b = 0 4) 3o + b=l V

..J "III,

Created Questions

1. Let "h"he the height of the tank. Then the rate of change of pressure "p"of the tank with respect to height is

2) 3)? 4)ui ui up u ,i

2. If the temp erat ure 6C of the certain metal rod of"/" metres is given by I= I + 0.0 00 05 0 + O.OOOOOO402 then the rate

of chan ge of / in m/C wh en the te mp er at ur e is 100C is \ tlf'"

1) 0.00013 m/C 2) 0.000 23 m/C 3) 0.000 26 m/C 4) 0.0003 3 m/C

3. The following grap h gives the functio nal rela tionsh ip betwee n distanc e and time of a nioving car in

m/sec. The speed of the car is i(lj "f " i j r %7 m A 2) J m/s 3 ) ^ m / s 4) ^ 'm/s

4. The distance - time rela tion ship of a moving body is given by y = F f tf t hen the accelerat ion of the body is the

1) gradient of the velocity / time graph 2 ) g rad i e ^ c M f e distance / time graph

3) grad ient of the accele ration / time grap h 4) grad ient of the velocity / dista nce graph

5. The dist anc e travel led by a car in"t"seconds is given by x = 3t 3 4t - 1. Then the initial velo city an d initial

acceleration respectively are

1) (- 4m/s, 4m/s2) 2) (4 m/s, - 4 m/s2) 3) (0, 0) 4) (18. 25 m/s, 23 m/s2)

6. The angu lar displa ceme nt of a fly wheel in radi ans is given IifO = 912- 2r3. The time when the angular accelerationzero is i j

1 )2 .5s 2) 3, 5 s 3) 1 .5 s 4) 4.5 s7. Food pock ets were dropp ed from a helicopte iidurifig the flood and distance fallen in "t" seconds is given by

y ~ 2^ tZ = 98 m/ s 2 )- Then the speed of the food pocket after it has fallen for "2" seconds is

1 ) 1 9 . 6 m/sec 2) 9.8 m/sec. 3) - 19.6 m/sec 4 ) -9 .8 m/sec- | -%r l

An object dro pped from the sky follows the law of motion x = -gt2 [g= 9.8m/sec2). The accelera tion of the object

when t = 2 isi

1) - 9.8 m/sec2 2) 9*8 m/ se c 2 3) 19.6 m/sec2 4) - 19.6 m/sec2

9. A missile fired fro m grou nd level rises x met re s vertically upwa rd s in "t"sec ond s and x = t(1 00 - 12.5 t). The n the

maximum height reached by the missile is

1) 10 0 m , J r 2) L50 m 3) 25 0 m 4) 2 0 0 m

10. A cont inuous g r a j j f y k ^ $ i s such thatf'[x) - co as x - xi at (xi,yi). Theny =f[x) has a

1) vertical Un ge nt y = xi 2) horizo ntal tan gen t x = xi

3) vertical tangent x = xj 4) horizontal tangenty =yi11. The cu rv ey =j f f ) and y = g(x) cut orthogonall y if at the point of interse ction

1) slope of/J x) = slope of g( x) 2) slope of fix) +slope ofg{x) = 0

3) slop e of /f x) / slop e of.g(x) = - 1 4) [s lope of/(x )J [slope of ^f x) ] = - 1

12. The law'Of the me an can also be pu t in the for m

%J(cMh)=Aa)-hf'(a + dh) 0 < d < 1 2)f[a + h)=f[a) + hf\a + 0h) 0


21/35

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15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

26.

27.

28.

29.

30.

31.

32.

33.

34.

35.

36.

lim *

X - 0 tan x11 1

IS

2) - 1 3 ) 0 4) oo

/ is a real valued func tion defi ned on an inter val I aR[R being the set of real nu mbe rs ) incre ase s on /. Then

l )/ [x i) f[x2) wh en ev er xi > X2 xi,xi 0 2 ) 2 * 0 3 ) ^ < 0 4 ) ^ 0; dxdx " ' dx ~J dx

f is a differentiable function defined on an interval / with positive derivative. Th en /i s

1) increasing on I 2) decr easin g on / 3) strictly incr easi ng on / 4) strictly dec rea sing on /

The function/[x) =x 3 is

1) increasing 2) decr easin g 3) strictly decreasing 4) strictly incr easin g

If the gradi ent of a curv e chan ges f rom positive just befo re P to negative just after then "P"is a

1) minimum point 2) max imu m point 3) inflexion poi nt 4) disc onti nuo us point

The funct ion/[x ) = x2has

1) a maximum value at x = 0 2) min imu m valu e at x =0

3) funic no. O i i i i d A i i i i L i i u va l ues

3) local maximum

4 ) / = 0

4) 110 extrema

The funct ion/(x ) = x:i has

1) absolute maxim um 2) absolute mini mum

If/has a local extremum at u and if/ ' (a) exists then

0 2}f\a)> 0 3) f'(d) = 0In the following figure, the curvey =J[x) is

1) conc ave upw ard 2) convex upward

3) changes from cancavity to convexity 4) changes from convexity and concavity

The point that sepa rat es the convex par t of a contin uous c urve from the concave part is

1) the maxi mum poi nt 2) the minimum point 3) the in fl ex io n point 4) critical point

/ i s a twice different iable funct ion on an interval / and i f/" (x) > 0 A a ^ x y n t h e domain I o f / th en / i s

1) conc ave upw ard 2) convex upward 3) incr easin g %/' 4) decreasing

x = xo is a root of even or der for the eq ua ti on / ' (x ) = 0 then x - x o^ aJ -

1) maximum point 2) minimum point 3) infle xtion point 4) critical point

1 f xo is the x-c oor din ate of th e poi nt of inflection of a curve y,- J[x) then (second derivative exists)

l) A*o) =0 2 ) /' ( xo ) = 0 3 ) / " { x o * =t a > 4 ) / " ( x 0 )#o

The stateme nt "I f/ i s cont inuous on a closed interval ' ! ^ th en /a t t ai ns an absolute maximum valuef[c) and an

absolute minimum valuef[d) at some n u m b e ^ x j % \ p [u, h\" is1) The extrem e value the ore m 2) F e rn ^ rk t l f eo ' r em 3) Law of Mean 4) Rolle's theorem

The state ment: " If /h as a local ext rem um (m inimu m or maximum) at c and if / ' (c ) exists th en /' (c ) = 0" is

1) The extreme value theorem 2) JFermat's th eo re m 3) Law of Mean 4) Rolle's theorem

Identify the false sta te men t: 1 V

1) all the stationary numbers are cjriticaNiumbers

3) at critical numbers the first derivative need not exist

Identify the cor rec t;

(a) a continu ous funct ion

(b) a continuo us func tion

(c) a continuous function

(d) a continuous t

1) (a) an d (b) 2) (a) and (c)

Identify the correct statements(a) Every con stan t func tion is an incre asing func tion

(c) Every identity function is an increasing function

1) (a), (b) and (c) 2) (a) and (c)

Which of the following sta tem ent is incorre ct

1) Initial velocity means velocity at t = 0

2) initial acceleration means acceleration at t = 0

3) If the mot ion is upward , at the ma xim um height, the velo city is not zero

4) If the motio n is hori zonta l, v = 0 wh en the par ticle c ome s to rest

Which ofthe following statements are correct (mi and tm are slopes of two lines)

(a) If the two lines are p erpe ndic ular then m\ nn = - 1 (b) I f mirm =- 1 then the two lines are p erp end icu lar

(c) Ifm\ =1772 th en the tw o line s are par all el fd) If mi = - - th en th e tw o lines ar e pe rp en di cu la r'71-,

2) at the statio nar y point the first derivativ e is zer o

4) all the critical numbers are stationary numbers

eal maxim um then it has absolute maximu m

s local minimum then it has absolute minimum

s absolute maximum then it has local maximum

as,absolute minimum then it has local minimum

3) (c) and (d) 4) (a), (c) and (d)

(b) Every constant function is a decreasing function

(d) Every identity function is a decreasing function

3) (c) and (d) 4) (a), (c) an d (d)

1) (b), (c) and (d) 2) (a), (b) and ,(d ) 3) (c) and (b)

One of th e conditions of Rolle's theor em is

l ) / i s defined and continuous on (a,b) 2) / i s different iable on \a, b\

ft f f f l )= f(h) 41 f is rlif fcmi rifl hlp nn (a, ftl

4) (a) and (b).

#6U(T6i)&6)f 0,(T68T (D6^6^)6m ((fi((p6V)(D(jU(T&(g,(D.21


22/35

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37.

38.

39.

40.

Ifa and h are two roots of a polynomialJ[x] = 0 then Rolle's Llieorem says tha t ther e exist s at least.

1) one root between u and b f o r / ' ( x ) =0 2) two roo ts bet wee n a and b for/ ' (x) = 0

3) one root between a and b for/"(A-) = 0 4) two ro ot s be tw ee n a and bf or /""fx) =0

A real valued function which is continuous on [a, b]and d i f f e re nt i a t e on (a , b) then there exists atleast one c in

1) [a, b]such tha t./ '(c ) = 0 2) (a, b) such th at / ' (c ) = 0

3) (a, b) such thatb-a

= 0 4) (a, b) such that = /' (c )

In the law of mean , the value o f' 0' satis fies the condition

1) 0 > 0 2) 9 < 0 3) 0 < 1 4) 0 < 0 < 1

Which of the following statements are correct?

(a) Rolle's theo rem is a parti cula r case of Lagra nges law of mea n

(h) Lagrange s law of mean is a part icul ar case of gener alised law of mea n (Cauchy)

(c) Lagranges law of mean is a particular case of Rolle's theorem

(d) Generalised law of mea n is a particu lar case of Lagran ges law of mea n (Cauchy)

l)(b) . lc)

Book back Questions

2) (c), (d) 3) (a), (b) 4) (a), (d)

Unit 6 Diffe ren tial Calculas and its App lica tion s II

45.

46.

47.

48.

49.

50.

duIf u = xy th en is equal to

dx1)yxy-^ 2 ) u l o g x 3) u logy

(X + y'1\

^T^yl) an 0 and a > b

1) an asymptote x =a 2) an asymptote x = b 3) an asy mpt ote y =a 4) no asy mpt ote32. The cur vey 2= (x - a) (x - b)2 ha sa, b >0 and a> b

1) a loop between x - aand x =b 2) two loops bet ween x =ciand x=ib

3) two loops between x = 0 and x =a 4) no loo p

33 The cur ve y 2 ( l + x) = x 2(l - x) is defined for

1) - 1 < x < I 2) - 1


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61. The val ue of

62.

64.

65.

66.

67.

60.

69.

70.

71.

72.

tt/2sin x-cosx

1+ sin x cosx

2) 0

d.x is

1

The value ol Jx( 1 x)4dx is

0

63. The valu e of

t t /2

- t t / 2

/ _sin_x_

V.2 + COSX

1)0

j dx is

2)2

The value of J * s / r t 4 x dx is

0

1) 3rt/ 16 2) 3/ 16

7T/4

The value of J* ca v 3 2x dx is

0

*>1

3) --1 4

3)' 24

3) log 2

3)0

3)0

3 ) t t / 4

The value of Jsin2x cos3x dx is

0

1) N 2) TT/2

The area bounded by the liney = x ,the x-axis, the o rdi nat es x =f

l)f 2)fThe area of the regio n bo un de d by the grap h of y = sin

1)^2 + 1

4) 71

4) ' 20

4) log 4

4) 3TT/8

=cos x be tw ee n x = 0 and x = - is4

2 ) V 2 - 1 3 ) 2 \ /2 - 2

The are a betwe en the ellipse + ~ = 1 an d jt s auxiliary circle is

1) nb(a - b) 2) 27ro(o - b) 3) na(a - b)

The area bounded by the parabolay 2 =x and its latus rectum is

4) 2\f2 + 2

4) 2ttb[a - b)

x2 y

2

revolv ing + = 1 ab ou t the min or axis isThe volume of the solid obtaij

1) 48/r 2) 64 tt 3) 32/r 4) 128t t

The volume, when tjie curve y = V3 + x2 fr om x = 0 to x = 4 is ro ta te d a bo ut x-ax is

73.

74.

75.k-

X,76.

77.

78.

The vol um | generat ed when the region boun ded by y = x, y = 1, x = 0 is rotated abou ty-a xis

TT

33)? 4) J 3

id obta ined by revolv ing the are a of the ellipse + = 1 abo ut majo r an d min or axes are in the ratioVolume

b2: a

1 2) a

2: b

2 3) a : b 4 )b : a

y? vol ume gener ate d by r otati ng the tria ngle with ve rtic es at (0, 0), (3, 0) and (3, 3) about x-axis is

1 )18 7 1 2) 277" 3) 3671 4) 9tt

The length of the arc of the curve x 2 ^ + y2/i = 4 is

1 ) 4 8 2) 24 3) 12 4 ) 9 6

Thp surfa ce are a of the solid of revol ution of the regi on bo un de d b yy = 2x, x = 0 and x = 2 about x-axis is

1) 8V5 TT 2) 2V5 7r 3 ) V 5tt 4)4V5TT

The curved sur face area of a sphe re of radius 5, interc epted betw een tw o parallel planes of distance 2 and 4 fro m the

centre is

1) 20TT 21 4077" 31 IOTT 4) 30TT

6Vrt6anb 6V6S)fi 6TLL(A.S> Sj.6V)&u(h}rh)e>6ri, Q6V6S)6rt ^,6V/T9U (ou/rmrrevicb /tf/ti&ert /tarn. [ 25


26/35

www.tnschools.co.in

Created Questions

1. If /n - Jsin nxdx thenIn =

1 . 1

1) - - sin ,,

-1x cosx +

11 n-2

' 71 71o- , 1 . , n - 1 ,.1) - - S i n " - 1 X CO S X / n-2

n n2a a

2. I*f[x) tlx - 2 JJ[x) dx if

0 0

l ) / ( 2 a -x ) = / (x ) 2)f[a-x)=J{x)2n

3. J / ( x ) dx = 0 if

0

7.

8.

2) - sin n_ 1x cos x + / n-2n n

1 n - 14) - - sin ll"1x cos x + In

3 ) / [ * ) = ~f{x)

2 ) / ( 2 a - x) =- f[x)a.

l ) / (2a - x) =f[x)

4. If/(x) is an odd functi on the n J /(x) dx is

1) 2 / Ax) df

0a a.

5. J /(x) c/x +j f[2a - x) dx =

na

2) f M dx

3 ) / [x )= - / (x )

3)0

4 ) / l -x ) = ; [x )

4 ) / l -x ) = / (x )

i ) / / w /x 2) 2jf /[*) dx

6. If'/(x) is even the n J f[x) dx is

D O

aJ f[x) dx is

0

1) J f[x-a)dx

f m dx is

a

f f\a

0

2 a 2 a

3) f / (x) dx , ^ V ) f f[ ~ *) dx

3) J /[2a - x) dx

0

4 ) - 2f f{x) dx0

4)j Ax~ 2a ) dx0

1) 2f fix) dx

0

9. Ifn is a positive integer then J xn e- ax dx =

V Q

3)J Ab ~ x) dx 4) f f\a + b -x)dx

2) 3)ln+1

4 ) - ^ -J n+1

is odd then J cos "x dx is

0, _ n _ n - 2 n - 4 7r

n - 1 " n - 3 ' n - 5 "

7 i - l n - 3 n - 5 1 tz

n/2

11. Ifn is even then J sin "xdx is

n n - 2 n - 4 2 23 ) Hz l 2 i

n - i " n - 3 " n - 5 " 2

0

, ?i n - 2 n - 4 tcn - 1 ' n - 3 ' n - 5 ' " " 2

2 ) " ~ 3 " ~ 5 1 7T . n n - 2 n - 4 3 ^

n ' n - 2 " n - 4 2 ' 2 n - 1 " n - 3 ' n - 5 " 2 '

. , n - 1 n - 3 n - 5 2

' n " n - 2 ' n - 4 3 "

4 . n - 1 n - 3 n - 5 2 ^n ' n - 2 ' n - 4 " " 3 '

&rt&)6G)6Ur G)S-LL)6U) $ID) 6l)rt(j)6l?GQ (Iftdd'OUCDfTm Q6VTgD 6T6STnQ) 6W(f?)^j


27/35

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T T / 2

12 Ifn is even then f cos "xdx is

11 11-2 11-4 K n1 773 n - 5 1 tt n ii2 n - 4

^ rt-1' n - 3 ' nS " ' 2 ^ n ' n - 2 ' n - 4 2 2 ^ n1 ' n - 3 ' 71S 2T T / 2

13. Ifn is odd then j s\n"xdxis

0n n-2 n - 4 TT n - l n - 3 n - 5 1 n n n-2 n- 4 3

14.

^ n - l * n-3 ' n -5 ""2 ^ n 'n - 2 ' n - 4 " " 2 ' 2 ^ n - l ' n - 3 ' n - 5 " " 2 ' 1

h

j Ax) dx =

n - l n- 3 n- 5 2 ^

?i ' n -2 "n -4 3"

n - l n-3 n- 5 2

N

1) - / / W dx 2 ) - J / ( x ) r / x

0

3 ) - / / W c/x

0

4) 2 J /(x) f/x4

xc/x

rt 0 0 ()

: 5. The area bounde d by the curve x - ,g(yj to the right ofy- axi s and the two li ne sy = c an dy =d is given by

d a d d

2) jxdy 3) Jydy A) jxdy

c c c c16. The area boun ded by the cur ve x =J(y), y-axis and the lines/ =candy =d is rotated abouty-axis. Then the volume of

the solid is

d d d

1) n j x2 dy 2 ) n jx2 dx 3) n j y

1 dx 4)rr jy

2dy

c c c ""* " V I " c1". The area bou nde d by the curve x =f[y), to the left of y-axis between the lijisy =c andy =d is

d d d ( \ / d

1) jxdy 2 ) - J x d y 4) - j y dx

c c18. The arc length of the cur ve y =f[x) from x =a to x =b is

vsf+W~x

a c ' \ j

19. The surface area obtai ned by revolvi ng the area boii hded by the cu rv ey =/( x), the two ordin ates x=a,x = band

x-axis, about x-axis is

W1 +

(2)2rfAr 2n

/ yj1 +(%)

2dx

3 ) 2 7 i f y j l + ( ) ' d x 4 ) 2t i f y j l + ( ^ ) 2 d x

3 ) 1J J 4 s

3 )7 7 7 1 + 1

74 )S

22. j x6e~x/2dx =

I6

11 J 2

7

23. If/n = Jcosnxdxthen /n=

2)1 , n _ 1 .

1J - - COS n _ 1 XSin X + / n-2n n

o-v 1

a . n - l ,3 J - cos

n _1x sin x / n-2J

n n

3 ) 2 6 [ 6 4 ) 2 ' [ 6

21 cosn _ 1x sin x + 1n-2n

. ^ 1 , " - 1 .4 ) - COS n _ 1 X S i n X + I n - 2

n n

&6rf)6vr s^tjibuu q m o f .


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w w w l n s c h o o l s . r o . i n

Unit 8 Differen tial Equations

79. The integ rat ing fac tor of ~ + 2 ^ = p4x is

1) log x 2)x2 3 )e x 4) xdy

80. If cos x is an integrating factor of the differential equation + Py = Q,then P =

1) coLx 2) cotx 3) tan x 4) - tanx81. The integr ating factor ofdx + xdy = e~>' sec2ydyis

1) ex 2 je-x 3)ey 4)


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97. The different ial equa t ion obtained by el iminat ing aand b f ro m y = ae3x + be is

(I V2 ) g - 9 y = 0

f > . _ q d yJ dx* 'dx

d2v

A) i + Qy = 0J d.x2

98. The different ial equat ion form ed by el iminat ing A and B from the relat iony =cx [Acos x +B sin x) is

l ) y 2 + y i = 0 2)y/ - y i = 0

99. I f ^ = ^ t h e ndx x+y

1) 2xy +y2 + x

2 = c 2) x

2 + y

2 - x +y = c

100. If/ ' (x) = \ [x and/(1J = 2 then/(x) is

1 ) - - ( x V x + 2 ) 2) ~ [x\[x +2)

3) y2 - 2yi + 2y = 0

3) x2 +y 2 - 2xy = c

3)-(x^+2)

4)yi - 2yi - 2y - 0

4) x2 -y

2 - 2 xy=c

4 ) - x ( V x + 2 )

101. On puttingy =vx, the homogenous different ial equat ion x2dy + y(x + y)dx =0 become s

1) xdv + (2v + v2)dx = 0 2) vdx +(2x +x2)dv = 0 3) v2dx - (x +x2)dv = 0 4) vdv + (2x +x2)dx =0

dy102. The integ rati ng fact or of the d iffer ential equ atio n - y tan x = cos x is

3)e< 4) cot xl ) s e c x 2 ) c o s x

103. The F.l. of (3D 2 +D -U)y = 13e2x is

1) 26x e2x 2) 13 x e2x 3 ) x e 2 * 4) x2 / 2 e2x \

104. The partic ular integr al of the differ entia l equ ati on f[D)y = eax where/[D) = (D - a) g(DJ,

X1v

1) meax

Created Questions

^ g(a)3 )g(a)e"

The order and degree

1) 3,1


1) 2,1

The order and degree of the differential

of the differential

2 )1 , 3

of the differential

2 ) 1 , 2

1 ) 2 , 1


1) 2,1


1) 1,1


1) 2,1


1) 2, 2

2) 1,2

of the differential

2) 1,2

of the differential

4

equat ion are (1 +y' ) 2 = y ' 2

f i e *a r e

^+

>/= *

2

3) 2,1

j a t ion a r e y ' +y2 =x

3) 1,0

equat ion are y" + 3y ' 2 + y 3 = 0

3) 1,2

8. The ord er an d de

1 ) 2 , 1

9. The or de r and de gr ee of the

i ) 2 y y *

10. The or de r and deg re e of the

1) 2, 3


\ h ) i , I

12. The or de r and degr ee of the

1) 2,2


1) 2, 2

14. The or de r and degr ee of the

1) 1,1

2 ) 1 ,

of the differe

2 ) i f ! )

a&'al

^ different ial equat ion are + x = Jy + d2y

2) 1,2

differential

2) 3 ,3

differential

2) 3 ,3

differential

2) 1,2

differential

2) 2, 1

dx2

13) 2, -

2

dy

. d2y (dy , d?y\ ^^

c a t i o n a r e ^ - y + ^ + ^ j

3) 3,2

equat ion ar ey " = (y-y'3)2/3

3) 3,2

equat ion arey' + (y")2 = (a +y") 2

3) 2,1

equat ion arey' + (y")2 =x[x +y") 2

3) 1 ,2different ial equat ion are j +x = ^ + x2

2) 2 , 1 3) 1,2

differential are sin x (dx + dy) =co s x (dx - dy)

2 ) 0 , 0 3 ) 1 , 2

4) 0,1

4) 0,1

4) 3, 1

4 ) 2 , 2

= 0

4) 2,2

4) 2,2

4) 2,2

4) 1 ,1

4) 1 ,3

4) 2 ,1


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15. The differential equat ion corre spo ndi ng to xy = czw h e r e c is an arbi trar y const ant, is

l)xy" + x = 0 2)y" = 0 3) a t/ + y = 0 4) xy" - x = 0

16. In f indi ngtho di f ferentia l equat ion correspo nding t oy = cmx w h e r e m is the arbitr ary const ant, then m is

1) 2) ~ J ) / 4)y

17. The soluti on of a line ar diff eren tial equ ati on ^ +Px = Qw h e r e Pa nd Qare funct ions of y, is

1) yV-F) = J (//=) Qd x+C 2) x(I.F) = J (IF.) Qdy + c3) y(I-F) =J * {I F) Qdy + c 4) x(I.F) =J (I .F .) Qdx + c

dy

18. The soluti on of a lin ear diff eren tial equ ati on +Py = Qw h e r e Pa nd Qare fu ncti ons of x, is

1) yV-F) = J V-F.) Qdx +c 2) x(I.F) =J (I.F.) Qdy + c

3) y ( I . F )= J (I.F.) Qdy + c 4) x(I.F) = J (I.F.) Qdx + c

19. Identify the incorr ect sta tem ent .

1) The ord er of a differential equatio n is the ord er of the highest or der de rivati ve occur ing in i t

2) The degree o ft he differ ential equat ion is the degr ee of the highes t orde r derivative wh ich occur s in i t (the

derivatives are free from radicals and fractions)dy (x , y )

3) = ^ ^ is the first ord er first degr ee ho mog ene ous differential equa tion

dy4) +xy = e

x is a l inear differential equation in x

Unit 9 Dicret e Math emat ics

Book back Questions

105. Which of th e fol lowing are s ta tem ents? ; 1 " \ ^

(i) May God ble ss yo u (ii) Ros e is a fl ow er (iii) Milk is (iv) 1 is a pr im e nu mb er

1) (i), (ii), (iii) 2) (i), (ii), (iv) 3) (i), ( i i i ^ t i v j ^ ^ > 4) (ii), (iii), (iv )

106. If a comp ound s tat eme nt is mad e up of thr ee simple st at em en ts ^& ei ft he nu mb er of row s in the truth table is

1)8 2)6 4 ) 2

107. Ifp isTand q is /', the n whic h of th e following have the tr dih ^a lUe 7?

(i)pvq (ii) ~p\Jq (iii ) p V ~ q (ty%X~q

1) (0/ 00 C0 2) (i), (ii), (iv) ()(>v) 4) (ii), (iii), (iv)

108. The nu mb er of row s in the truth table of ~ [ p % ^ a f ] n s

1) 2 2 ) 4 * 3) 6 4) 8

109. The conditional statement p> qis equ ivaf lt | t fep?

1 ) p V q 2) 3) ~ p v q 4) p Aq

1 10 . W h ic h o f t h e f ol lo wi ng i s t a u t o l o s ^ N r ^

1) p v ? 3) (p*q) V ( q * p ) 4) ( p ^ q ) A (q->p)

113. Wh ich of the. follow ing is not a binar y ope ra tio n on R?

1) a * b= 2)a* b = a - b 3) a* b = Vab 4)a* b =V a 2 + b2

114. A mon oid beco nfe s a gr ou p if it also sati sfie s the

1) cl0sl^e,,axiom 2) asso ciati ve axiom 3) iden tity axiom 4) inverse ax iom

115. Which "of the following is not a group?

4} (2 *, +n ) . 2)( Z,+ ) 3) (Z , . ) 4) ( /? ,+)

Inithe set of integ ers with o pera tion * define d by a * b = a + b - ab, the value of 3*(4*5) is is

- 1) 2 5 2) 15 3) 10 4) 5 *

117. The order of [7] in (Zy, +9) is

1) 9 2) 6 3 ) 3 4) 1

118. In the mulplicative group of cube root of unity, the order of wz is

1 ) 4 2 ) 3 3 ) 2 4 )1

119 . The val ue of [3] +n ( [5] +11 [6]) is

1) [0] 2) 11] 3) [2] 4) [3 ]

(D60T$6V tflbus,*6)0>rt6BBTUrT6V #rt$6B)6Sr 6V&(D(T(g,Cb.30


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www.tnschools.co.in

120. In the set of real numbers R,an op era tio n * is defin ed by a * b =Va2 + b

2. Th en t he v alu e of (3 * 4) * 5 is

1) 5 2) 5V 2 3) 25 4) 50

121. Which of the following is correct

1J An elem ent of a gr ou p can have m or e than on e inver se.

2) If eve ry el em en t of a gr ou p is its own in vers e, th en the grou p is abe lia n.

3) The set of a ll 2 x 2 real matr ices for ms a grou p und er matrix mult ipl icat ion4) (a *b) 1 = cr1 * f r1 for al l a, heG

122. The orde r o f - / in the mulpl icativ e gro up of 4th

r oot s of unity is

1) 4 2) 3 3) 2 4 ) 1

123. In the mulplicative group of n th r oo ts of unity, th e in ve rs e of a)* is (k < n)

l j t o 1 /* 2) co-' 3) oi "- k 4) co"/"

124. In the set of int ege rs und er th e oper ati on * defin ed by a * b = a + b - 1,the identi ty elem en t is

1)0

Created Questions

l .

2 ) 1 3) a 4)b

2.

3.

5.

7.

(ii) The earth is a planet.(iv) Fvery tr ian gle i


32/35

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13

14.

15.

16.

17.

1 8 .

19.

20.

21 .

22.

23

24.

25.

26.

27.

28.

29.

30

31.

32.

33

34.

35.

36.

38.

39.

(c) a * b = a (d )u * b =b

3) (b), (c) an d (d) 4) (c), (d)

Which oft he following is not tru e?

1) Negation of a negation of a statement is the statement itself

2) If the last column of its trut h tab le cont ain s only T then it is tautology

3) If the last, col umn of its tr ut h ta ble co nta ins o nly Fthen it is contrad iction

4) Ifp an d q are any two statements then p qis a taut olo gy

Which ofthe following are binary operations on R?

(a ) a* b - min {a, b} (b) a* b =m ax{a, b)

1) all 2) (a), (b) and (c)' + ' is not a binary operation on

1)N 2)1

' - ' is a binary operation on

1)N 2)CM0)

' - r' is a bina ry ope rat ion on

1) TV 2) R 3) Z

In congruence modulo 5, (xe z / x =5k + 2 ,k e z} rep res ent s

1) [0] 2) [5] 3) [7]

15J.12[11] is .

1) 15 5] 2) 1121 3) 17J

[3] +g [7] is

1J [10 ] 2) [8] 3) [51

In the group (G, .) ,G={1, - 1, /, - /}, ord er of - 1 isi ) - i z j 1 3 ) 2

In the group (G,.),G={1, - 1, /, - /'}, o rd er o f - / is

1) 2 2) 0 3 )4

In the group (G,.) G={1, co, oj2), a) is cube root of unity, 0(oo 2) is

1)2 2) 1 3) 4

In the gr ou p (Z-t, +4), or de r of [0] is

1) 1 2) 00 3) can 't

In the grou p (Z4, +4),0([ 3j) is

1) 4 2) 3 3) 2

In (5, o),xoy = x,x,ye s then '0' is

1) only associative 2) only connifyUttiv e

3) associative and commutative 4) nei tt j ^r^r oci ati ve nor commutative

In [N,*), x *y = max {x,y), x,ye N then (/V, *) is . V

1)only closed 2) only semi gr oi i ' ^ i f \ %3) only monoid

The set of posit ive even integers , wit h usyal multip lica tion f orm s

1) a finite group 2) onl y a s ^ i g r p t i p 3) only a mon oid

The set of posit ive even int eger s wjt h uStj al add iti on fo rms

1) a finite group 2) only a seni i gro up 3) only a mono id

In the g ro up (Zs - {[01), .5), 0([3])^f

! ) 5 3 ) 4

In the group [G,.), G={1, -% J SJ J^ or de r of 1 is

1) 2 G f

3 )4

In th e gr ou p ((7, ~ 0 or der of /' is

1 ) 2 - C j L V 2 ) 0 3 ) 4

In the grou p (G,.) G&i%o, a)2), a) is cube root of unity, 0(o>) is

1)2 f "

r

2 ) 1 3 ) 4In the group (?if%^={l, to, o>2), a> is cub e r oo t of unity , 0( 1) is

1) 2 % V - 2 ) 1

In th&J!>!^'(Z4, +4), 0([ 1]) is

1) 1 V 2) 00

th gr ou p (Z4 +4), 0([ 2]) is

2) 2

In the group (Zs - {[0]}, .s), 0([2j) is

fl ) 5 2) 3

In the group (Zs - {[01), .5), 0([4]) is

1) 5 2) 3

In the group (Zs - {[0]}, .5), 0([1]) is

1) 1 2) 2

3) 4

3) can't be determined 4) 4

3) can 't be det erm in ed 4) 0

3 ) 4 4 ) 2

3) 4 4) 2

3) 3 4) 4

4) a group

4) an infinite group

4) an infinite group

4) 2

4)1

4)3

4 ) 3

4) 3

(o 6 JJ6 3 3 f@W 6T6tif/Y) Qsutf)1 6?{puCL^fT6V &>fT6ST 6)5>((?)W Sj6V)t-(L) ( 0 Q ^ ( > .

32


33/35

vwmtnschools .co . in

Book back Questions

nr I c ft \ \kx2, 0 < X ? 4) ' 12

126. If f i x ) = - ~-T, - oo < x < oo is a p.d.f. of a con ti nu ou s r an do m v ar ia bl e^ , t hen the v alu e ofA is

' v y

n 16+x2 1

1) 16 2 ) 8 3 ) 4 4) 1

127. A ran dom variable X h as the following proba bility distr ibut ion

X 0 1 2 3 4 5

P(X = x) 1 /4 2 a 3a 4a 5a 1/ 4

then P(l< x < 4) is

^ 2 1

128. A random variable X has the following probability mass function as follows

ofX - 2 3 1

P(X = x)A /I A

P(X = x)6 4 12

the n the value of A is

1) 1 2) 2 3) 3

129.X is a discrete random variable which takes the values 0,1, 2 and

P[X= 0) = 77^,P[X= 1) = th en the val ue ofP[X = 2) is169 169

i) J 169

2) J 169

130. A rand om variable X has the following probability distrib ution

X 0 1 2 3 4 5 6

P(X = x) 0 k 2k 2k . 3k k2 2 \M +k

4) ' 169

The v alu e of /c is

131. Given E[X+ c) = 8 and E(X - c) = 12 then the vgmj

1 ) -2 2 ) 4 G *

132. Xis a random variable which takes th

1)5 2) 7

133. Variance of random variable X

1)2

134. [12= 20, \i2 = 276 for a di s

1) 16

4) -1 or } 10

4 )2

l lue^ 3, 4 and 12 with probab ilitie s - and Then E(X) is

4 )33)6

ean is 2. Then E[X2) is

3) 6 4) 8

ndom variable X. Then the mean of the random variable X is

) 5 3) 2 4) 1

135. Var (4X+3;ys

1) 7 2) 16 Var (A) 3 ) 1 9 4) 0136. In 5 th ro ws of a die, getting 1 or 2 is a success. The mean number of success is

3)"} 9

137. TJie melfh of a binomial distribution is 5 and its standard deviation is 2. Then the value ofn and p are2H

25.f)

3)(i

25) .

4)(

25J)

ft he mea n and stan dar d deviati on of a binomia l distri buti on are 12 and 2 respe ctive ly. Then the value of its

paramete r p is

2 ' 33

) i 4] -J 4

139. In 16 throws of a die, getting an even number is considered a success. Then the variance of success is

1 ) 4 2) 6 3) 2 4) 256

(tp(L)rp-uL](b uuyrp&iLjQu) $(b6B)(D33


34/35

www.tnschools.co.in

i ) -J 2

2) 1 51

140. A box contains 6 red and 4 whi te balls, if 3 balls are d raw n at ran dom, the probabili ty of getting 2 white balls wit hou t

replacement, is1 , 1 8 4 ,. 3

^ 2 0 "l25 25 >10

141. If 2 cards are drawn from a well shuffled pack of 52 cards, the probability that they are of the same colours without

replacement, is

3) 4} ' 51 10 2

142. If in a Poiss on di str ibu tio n P[X= 0) =k then the variance is

l ) l o g ; 2) logk 3)e 4 ) i

143. If a rando m variable X follows Poisson dis trib ution such that H(X2) = 30 then the variance of the distribution i;

1) 6 2) 5 3) 30 4) 25

144. The distribution functionF(X) of a random variable X is

1) a decreasi ng function 2) a non-decreasing function

3) a const ant function 4) increasi ng first and then decre asin g

145. For a Poisson dis tribut ion with pa ram ete r A = 0.25 the value of the 2nd moment about the orig

1 )0 . 2 5 2 ) 0 . 3 1 2 5 3 ) 0 . 0 6 2 5 4 ) 0 . 0 2 5 ^ /

146. In a Poisson distribution ifP[X= 2) =P(X= 3) the n tji e value of its par am et er A is1) 6 2) 2 3) 3

147. If f(x) is a p.d.f. of a norm al d ist rib uti on w ith me an p t hen j f ( x ) dx is

1)1 2) 0.5

148. The random variable Xfollows normal dis tribution f[x) = ce

1 ) V 2 tt 2 ) -T== 3 ) 5 ^

149. If/ fx) is a probability d istri bution function of a nor

len the value ofc is

1

5V2n

V

!e XandX - N{\x, o2)then J fix) dx is

00

1) undefi ned 2) 1 05 40 ~ - 5

150. The marks scored by 400 stu den ts in a mat hem ati cs tes t wer e normally distri buted with mean 65. If 120 st ude nts got

more mar ks above 85, the nu mb er of stu dent s securing mar ks betw een 45 and 65 is

1) 120 3) 80 4) 160

2 .

Created Questions1. A discrete ran dom vari

1) only a finite number ofva

3) infinite number of v

A continuous ran^o % vafrjlble takes

1) only a finite nurn^erof values

3) infinite p#mber owalues

IfX is a discrefesfemdom variab le t henP(X > a) =

1) P[X


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9.

10.

n .

12.

13.

14.

15.

Which ofthe following is or arc correct regarding normal distribution curve?

(a) Symmetrical about the lineX=\i (mea n) (b) Mean = medi an = mode

(c) Unimou ai (clj Poin ts of infle ction ai t- atX= p o

1) (a)- 03) o n l y 2) (b), (d) only 3) (a), (b), (c) only 4) all

For a stan dar d normal dist ribut ion the mean and vari ance are

1 ) p , a 2 2 )p , a 3 ) 0 , 1 ( 4 )1 , 1

Th e p.d.f of thn standard normal variate Z is cp(z) =

l ) ^ = e ~ 27

'-j2no

l \ f 2 n e1 ~7? 1 ~~Z

2

4)-=e Sm2tt

\fX is a discrete ra ndo m varia ble then which o ft he following is corrcct?

1) 0 < F(x)

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12th magudam maths em onemark 123.pdf