EDUCATIONAL TRUST’S

Junior Science College

ANDHERI / BORIVALI / DADAR / NERUL / POWAI / THANE

MATHS PRACTICE QUESTIONS

SECTION - I

1 If A 3, 4,6,8 , determine the truth value of each of the following:

(a) x A, such that x 4 7 (b) x A, x 4 10 (c) x A, x 5 13

(d) x A, such that x is odd (e) x A, such that x 3 N

2. Without using truth table, show that p q p q p q (March 2013) 3. Write the negation of the following statements : (i) 6 is an even number or 36 is a perfect square. (ii) If diagonals of a parallelogram are perpendicular then it is a rhombus. (iii) If 10 5 and 5 8 then 8 7 . (iv) A person is rich if and only if he is a software engineer. (v) Mangoes are delicious but expensive. (vi) It is false that the sky is not blue. (vii) If the weather is fine then my friends are coming and we go for a picnic. 4. Write the converse, the inverse and the contra positive of the following conditional statements :

(i) If an angle is a right angle then its measure is 90 . (ii) If two triangles are congruent then their areas are equal. (iii) If f 2 0 then f x is divisible by x 2

5. Consider the following statements : (i) If a person is a social, then he is happy. (ii) If a person is not social, then he is not happy. (iii) If a person is unhappy, then he is not social. (iv) If a person is happy, then he is social. Identify the pairs of statements having same meaning

6. 2 24 3

Find inverse by adjoint method. (March 2013)

7. 1 0 03 3 05 2 1

Find inverse by adjoint method.

8. Find the inverse of the following matrices by using transformation method: 1 22 1

9. 2 0 15 1 00 1 3

Find inverse by using transformation method.

10. Find the inverse of cos sin 0

A sin cos 00 0 1

by

(i) elementary row transformations. (ii) elementary column transformations. 11. Show with the usual notation that for any matrix ij 3 3

A a

(i) 11 21 12 22 13 23a A a A a A 0 (ii) 11 11 12 12 13 13a A a A a A A

12. If 1 0 1

A 0 2 31 2 1

and 1 2 3

B 1 1 52 4 7

, then find a matrix X such that XA = B.

13. If three numbers are added, their sum is 2. If 2 times the second number is subtracted from the sum of first and third number, we get 8 and if three times the first number is added to the sum of second and third number, we get 4. Find the numbers using matrices. 14. The general solution of sin sin implies nn 1 , n Z .

15. If and are not multiple of 2 , then the general solution of tan tan is n .

16. The general solution of (i) 2 2sin sin is n ,n Z (ii) 2 2cos cos is n a,n Z (iii) 2 2tan tan is n ,n Z . 17. The general solution of a cos bsin c is 2n ,n Z where a, b, c

R a 0,b 0,c 0, 2 2 2 2

a bcos ,sina b a b

and 2 2

ccosa b

.

18. Find the general solutions of each of the following questions:

(i) sec x 2 (ii) cos x 1 (iii) 3sin x2

19. Find the general solution of 4sin x cos x 2sin x 2cos x 1 0 20. Find the general solution of 3tan x 3tan x 0 21. Find the general solution of cos x sin x 1 (March 2013) 22. Find the general solution of 3 cos x sin x 1

23. In ABC, prove that

24. With the usual notations, prove that a b csin A sin B sin C

25. (a) In ABC, prove that (i) 2 2 2a b c 2bx cos A (ii) 2 2 2b c a 2ca cos B (iii) 2 2 2c a b 2abcos c (b) In ABC prove

(i) a ccos B b cosc (ii) b a cosc c cos A (iii) c a cos B b cos A

26. In any ABC, if 2 2 2a , b ,c are in AP then prove that cot A,cot B,cot C are in A. P. 27. In any ABC, if 2s a b c , then

(i) s b s cAsin2 bc

(ii) s c s aBsin2 ca

(iii) s a s bCsin2 ab

28. In any ABC, if 2s a b c , then

(i) s s aAcos2 bc

(ii) s s bBcos2 ac

(iii) s s cCcos2 ab

29. (a) In any ABC , if 2s a b c , then

(i)

s b s cAtan2 s s a

(ii)

s a s cBan2 s s b

(iii)

s a s bCtan2 s s c

(b) prove that area of 1 1ABC absin c bcsin A

2 2

1 acsin B2

30. If a, b, c are the lengths of the sides BC, AC and AB of ABC and a b c 2s , then the area of ABC is given by s s a s b s c .

31. In any ABC ,

(i) B C b c Atan cot2 b c 2

(ii) C A c a Btan cot2 c a 2

(iii) A B a b Ctan cot2 a b 2

32. Show that

1 1 1 11 1 1 1tan tan tan tan5 7 3 8 4

33. (a) Solve the equation

1 11 x 1tan tan x , for x 0.1 x 2

(b) Prove the joint equation of two lines passing through origin is second degree of homogenous

equation in x and y.

34. (a) Prove that every homogeneous equation of second degree in x and y, i.e., 2 2ax 2hxy by 0

represents a pair of lines through the origin, if 2h ab 0 .

(b) Prove 22 h abtan and a b 0

a b

when 1 2l l MOST IMP

2h ab 0 when 1 2l ||l . 35. Find k, if (i) the slope of one of the lines give by 2 2kx 4xy y 0 exceeds the slope of the other by 8. 36. Find the joint equation of the pair of lines through the origin each of which making an angle of o30

with the line 3x 2y 11 0 . 37. Find the value of k, if the following equations represent a pair of lines : (i) kxy 10x 6y 4 0 (ii) 2 2x 3xy 2y x y k 0 . 38. OAB is formed by the lines 2 2x 4xy y 0 and the line AB. The equation of line AB is

2x 3y 1 0 . Find the equation of the median of the triangle drawn from the origin. 39. Find the separate equation of the lines represented by the following equations : 2 26x 5xy 6y 0 40. Find the joint equation of the pair of lines which bisect angles between the lines given by 2 2x 3xy 2y 0 .

41. If the lines represented by 2 2ax 2hxy by 0 make angles of equal measure with the coordinate axes, then show that a b . (or)

Show that, one of the lines represented by 2 2ax 2hxy by 0 will make an angle of the same measure with the X-axis as the other makes with the Y-axis, if a b .

42. If the slope of one of the lines given by 2 2ax 2hxy by 0 is square of the slope of the other line,

show that 2 2 3a b ab 8h 6abh . 43. Show that the difference between the slopes of the lines given by

2 2 2 2 2tan cos x 2xy tan sin y 0 is two. 44. Two non-zero vectors a and b are collinear, if and only if there exists non-zero scalar m and n, such

that ma nb 0 . 45. If a and b any two non-zero, non-collinear vectors lying in the same plane, then prove that any

vector r coplanar with them can be uniquely expressed as a linear combination of a and b . MOST IMP (March 2013) 46. There non-zero vectors a, b, c are coplanar if and only if there exist scalars x, y, z, not all zero

simultaneously such that xa yb zc 0 . 47. (a) If a, b, and c are three non-zero, non-coplanar vectors, prove that any vector r in space can be

uniquely expressed as a linear combination xa yb zc, where x, y, z are scalars. (b) Find volume of parellopited and tetra hedron. 48. State and prove section formula. (Internal / External Division.) 49. Let PQRS be a quadrilateral. If M and N are the midpoints of the sides PQ and RS respectively, then

prove that PS QR 2MN . 50. Find the volume of a tetrahedron whose vertices are A 1, 2,3 , B 3, 2,1 , C 2,1,3 and

D 1, 2,4 . 51. If four points A a ,B b ,C c and D d are coplanar, then show that

a b d b c d c a d a b c 52. Prove that the medians of a triangle are concurrent. MOST IMP 53. Prove that the bisectors of the triangle are concurrent. 54. Prove that a quadrilateral is a parallelogram if and only if its diagonals bisect each other. MOST IMP 55. Prove that the median of a trapezium is parallel to the parallel sides of the trapezium and its length is

half of the sum of the lengths of the parallel sides. 56. Prove by vector method, that the angle subtended by a diameter of a circle at any point on the

semicircle is a right angle. MOST IMP

57. Prove that the altitudes of a triangle are concurrent. 58. Using vector method, prove that the centroid of the triangle formed by joining the midpoints of the

sides of a given triangle coincides with the centroid of the given triangle. 59. Using vector method, prove that a quadrilateral is a rhombus if and only if diagonals bisect each

other at right angles. 60. Using vector method, show that the segments joining he midpoints of the consecutive sides of a

quadrilateral form of a parallelogram. 61. If the diagonals of a parallelogram are of equal length, prove that the parallelogram is a rectangle. 62. Using vector method, prove that sin sin cos cos sin 63. Prove that in an isosceles triangle, the median passing through the vertex common to equal sides of

the triangle is perpendicular to the base.

64. Can 2 2 1, ,3 3 3

be the direction cosines of any directed line? Justify your answer.

65. If a line makes angle , , with the coordinate axes, prove that (i) 2 2 2sin sin sin 2 (ii) cos 2 cos 2 cos 2 1 . 66. Find the vector of magnitude 9 which is equally inclined to the coordinate axes. 67. The direction cosines of the two lines are determined by the relations 5m 3n 0 l and

2 2 27 5m 3n 0 l , find them. 68. Find the measure of acute angle between the vectors whose direction ratios are (i) 3,2,6 and 2,1,2 (ii) 1,2,2 and 3,6, 2 .

69. If the angle between the vectors a and b having direction ratios 1, 2, 1 and 3,3k,1 is 4 , find k.

70. If the points A 5,5, , B 1,3, 2 and C 4,2, 2 are collinear, find the value of . 71. Prove that the line joining the points A and B having position vectors 6a 4b 4c and 4c and the

line joining the points C and D having position vectors a 2b 3c and a 2b 5c intersect. Find the position vector of their point of intersection.

72. A line passes through 3, 1,2 and perpendicular to the lines ˆ ˆ ˆ ˆ ˆ ˆr i j k 2i 2 j k and

ˆ ˆ ˆ ˆ ˆ ˆr 2i j 3k i 2 j 2k , find its equation.

73. Find the shortest distance between the lines x 1 y 1 z 17 6 1

and x 3 y 5 z 71 2 1

.

74. By computing the shortest distance, determine whether the following lines intersect of not :

(i) x 5 y 7 z 3 x 8 y 7 z 5,4 5 5 7 1 3

.

75. Find the vector and the Cartesian equation of the plane that passes through the point 0,1, 2 and

normal to the3 plane is ˆ ˆ ˆi j k . 76. Find the equation of the plane passing through the intersection of the planes 3x 2y z 1 0 and

x y z 2 0 and the point 2,2,1 . 77. Find the vector equation of the plane passing through the intersection of the planes

ˆ ˆ ˆr. 2i 3j 4k 1 and ˆ ˆr. 2i j 4 0 and perpendicular to the plane ˆ ˆ ˆr. 2i j k 5 .

78. Minimize z 800x 640y, subject to 4x 2y 16,12x 2y 24,2x 6y 18,x 0, y 0 .

SECTION - II 79. Discuss the continuity of the following function at the points shown against them. If a function is

discontinuous, determine whether the discontinuity is removable. In this case, redefine the function,

so that it becomes continuous:

2sin x3 1f x

x log 1 x

for x 0, f x 2log 3 for x 0

80. (i) If 1 tan xf x ,1 2 sin x

for x

4

is continuous at x4

, find f4

.

(ii) If 2x

2

e cos xf xx

, for x 0 is continuous at x 0 , find f 0 .

81. Prove that every differentiable function is continuous. Using counter example, prove that the

converse is not true. MOST IMP 82. Show that the function f x defined as

1f x x cos , x 0x

0 x 0

is continuous at x = 0

83. If y is a differentiable function of u and u is a differentiable function of x, then prove that

dy dy du.dx du dx

. MOST IMP

84. If y f x is a derivable function of x such that the inverse function 1x f y is defined, then

show that

dx 1dy dy / dx

, where dy 0dx

. MOST IMP

85. Differentiate the following w.r.t. x :

1 3cos x 4sin xcos5

86. If 12

5 xy tan6x 5x 3

,

dyfinddx

87. If 2 2

1 12 2

x ycos tan ax y

, show that dy ydx x

.

88. If y log x log x log x ... , find dydx

.

89. If x and y are differentiable functions of t, then dy dy / dt dx, if 0dx dx / dt dt

MOST IMP

90. If 2

2 2

2bt 1 tu , v a1 t 1 t

, show that 2

2

du b vdv a u

.

91. If 2 2ax 2hxy by 0 , show that 2

2

d y 0dx

.

92. If 1y sin msin x , show that 2

2 22

d y dy1 x x m y 0dx dx

.

93. A man of height 180 cm is moving away from a lamp post at the rate of 1.2 metres per sec. if the

height of the lamp post is 4.5 metres. find the rates at which (i) his shadow is lengthening. (ii) the tip of his shadow is moving.

94. Verify Rolle’s theorem for the function 2f x x 5x 9, x 1,4 . 95. Find the values of x such that 3 2f x 2x 15x 84x 7 is a decreasing function. 96. If x t is a differentiable function of t then 1f x dx f t . t dt .

97.

1 dxcos(x a).cos x b

98. 1 dxcos cos x

99. If u and v are functions of x, then duuvdx u vdx . vdx dxdx

.

100. Prove that :

(i) 2 2

1 1 a xdn log ca x 2a a x

(ii) 2 2

1 1 x adn log cx a 2a x a

MOST IMP

(iii) 12 2

1 1 xdn tan cx a a a

.

(iv) 2 2

2 2

1 dx log x x ax a

(v) 1

2 2

1 xdx sin caa x

(vi) 2 2

2 2

1 dx log x x a cx a

101. ne f (x) f '(x) dx 102. Evaluate the following:

(i) /2

2 3

0

sin x cos x dx

(ii) /4

4 40

sin 2x dxsin x cos x

.

103. Prove the following properties of definite integrals :

(1) (i) a a

0 0

f x dx f a x dx (ii) b b

a a

f x dx f a b x dx

(2)

2a a a

0 0 0

f x dx f x dx f 2a x dx

(3) a a

a 0

f x dx 2 f x dx

, if f is an even function

= 0 if f is an odd function. 104. Evaluate the following :

(1) 0

x tan x dxsec x cos x

(2) /2

0

log sin x dx

.

105. Find the area of the region bonded by the curve 2y 2x x and X-axis. 106. Find the area (1) enclosed between the circle 2 2x y 1 and the line x y 1 lying in the first quadrant (2) bounded by the curve 2 2y 4a x 3 and the lines x = 3, y = 4a.

107. x 2xy Ae Be is a solution of the 2

2

d y dyD.E. 2y 0dx dx

.

108. Solve the following differential equations with the help of the substitutions shown against them :

(1) 2 2dyx y a , x y udx

(2) / y x/ y x x1 e dx e 1 dy 0, uy y

109. In a certain culture of bacteria, the rate of increase is proportional to the number present. If it is found that the number doubles in 4 hours, find the number of times the bacteria are increased in 12 hours. 110. The rate of decay of certain substance is directly proportional to the amount present at that instant. Initially, these are 27 gm of certain substance and three hours later it is found that 8 gm are left. Find the amount left after one more hour.

111. Let X B n, p . If E X 5 and Var X 2.5 , find n and p. 112. Suppose that 80% of all families own a television set. If 10 families are interviewed at random, find the probability that

(i) Seven families own a television set. (ii) At most three

113. Find the c.d.f. F x associated with the following p.d.f. f x :

23 1 2x , 0 x 1f x

0, otherwise

Find 1 1P X4 3

by using p.d.f. and c.d.f.

114. Calculate the expected value of the sum of two numbers obtained when two fair dice are rolled. 115. Following is the distribution function F x of a discrete r.v.X :

x 1 2 3 4 5 6 F(x) 0.2 0.37 0.48 0.62 0.85 1

(i) Find the probability distribution of X. (ii) Find P X 3 , P 2 X 5

(iii) Find P X 5 X 3 . 116. The p.m.f. of a r.v. X is as follows : 3 2P X 0 3k , P X 1 4k 10k , P X 2 5k 1, P X x 0 for any other value of x.

117. Find the solution of dy Py Qdx

where P & Q function of x or constant.