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NDA MATH TEST-15

1. For the equation x + 2y + 3z = 1, 2x+y+3z=2, 5x 9. If the value of a third order determinant is 11,
+ 5y + 9z=4, then the value of the square of the determinant
(a) There is only one solution formed by the cofactors will be
(b) There exist infinitely many solutions (a) 11 (b) 121
(c) There is no solution (c) 1331 (d) 14641.
(d) None of these. 10. If a,b,c are in A.P. then the value of -

LM 0 a b OP LM x + 1 x+2 x+a OP
2. If  M -a 0
P
c , then  is equal to MM xx ++ 23 x+3 PP
x + b is
NM-b -c 0QP N x+4 x+c Q
(a) 3 (b) -3
(a)  =0 (b)  =2abc
(c) 0 (d) None
(c)  =-abc (d)  =a2+b2+c2. 11. If f(x) = logx(log x), then f’ (x) at x=e is
3. If A+B+C =  then (a) 0 (b) e
(c) 2e (d) 1/e.
LMsin(A + B + C) sin B cos C OP
MM-sin B 0 tan A PP FG 1  cos x IJ , then dy is
Ncos(A + B) - tan A 0 Q
12. If y = tan-1 H 1  cos x K dx
(a) 2 (b) 1 (a) 1/4 (b) 1/2
(c) zero (d) None (c) 1 (d) 2.
4. If every element of a third order det. of value F I
1+ x 2  1
If y = tan-1 G JK
 is multiplied by 5 then the value of the new
det. is -
13.
H x , they y’ (0)=

(a) 1 (b) ½
(a)  (b) 5  (c) 0 (d) None
(c) 25  (d) 125  . 14. The derivative of sin-1x w.r.t. cos-1
5. For any 2 x 2 matrix A, if A (Adj. A) = 1 -x 2 is

LM10 0 OP, then |A| equals (a) 1/ 1- x 2 (b) 1

N0 10Q (c) cos-1 x (d) tan-1(x 1- x 2 ).

(a) 0 (b) 10 FG 1 IJ w.r.t
(c) 20 (d) 100.
15. The derivative of sec-1 H 2x - 1K
2 1- x 2 at
6. If A is a 3 x 3 matrix, such that det A= 5, then det x = 1/2 is
(adj. A) = (a) 2 (b) 4
(a) 5 (b) 25 (c) 1 (d) -2.
(c) 125 (d) None 16. If x = a cos  , y = b sin  then dy/dx=
3 3

LM1+ a 1 1 OP (a) tan  (b) cot 

7. The value of the determinant M 1 1+ a 1 PP a tan  -b tan 
MN 1 1 1+ a Q (c)
b
(d)
a
.
is 17. If the tangent to the curve x=a(  + sin  ), y=a
(a) a3(1-2/a) (b) a3(1+3/a)
(c) a3(1-3/a) (d) a3(1+2/a). (1+cos  )at  =  /3 makes an angle  with x-
axis, then  =
LMa 0 0 OP (a)  /3 (b) 2  /3
8. If A = M 0 PP
a 0 , then the value of |A|| Adj. A|
(c)  /6 (d) 5  /6.
MN0 0 a Q 18. Two number x & y such that x + y = 60 and xy3
is is maximum are
(a) a3 (b) a6 (c) a9 (d) a27. (a) 15, 45 (b) 10, 25
(c) 20, 35 (d) None of these.
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19. The absolute maximum of y = x3-3x+2 in 0  x 30. The general solution of cos 2x = cos 3x is
 2 is (a)
2n 2n
(b)  1
(a) 4 (b) 6 5
(c) 2 (d) 0. 2n  2n
20. The maximum value of f(x) = 2x3-21x2+ 36x + (c) (d)
23 32
20, in the interval 0  x  2 is sin 7x  sin 3x
(a) 37 (b) 44 31. is equal to
cos 7x  cos 3x
(c) 32 (d) 30. (a) tan 7x (b) tan 3x
21. Let A = {x : x  N and x is a multiple of 2}; (c) tan 5x (d) cot 5x
B = {x : x  N and x is a multiple of 5}' 32. sin 51  cos 81 is equal to
C = {x : x  N and x is a multiple of 10}, then (a) sin132 (b) sin 60
A  B  C  is (c) cos 90 (d) cos 21
(a) A (b) B sin A  sin 3 A  sin 5 A  sin 7 A
(c) C (d)  33. is equal to
cos A  cos 3 A  cos 5 A  cos 7 A
22. If A = {4, 5, 8, 12}, B = {1, 4, 6, 9} and C = {1, (a) sin 4A (b) cos 4A
2, 3, 4}, then A – (B – A) is (c) tan 4A (d) cot 4A
(a) A (b) B 34. cos C  cos D is equal to
(c) 2A (d)  C  D  C  D 
23. If A = {4, 5, 8, 12}, B = {1, 4, 6, 9} and C = {1, (a) 2 cos 
 2  cos 
  2 

2, 3, 4}, then A – (C – B) is
C  D  C  D 
(a) (A – C) + B (b) A (b)  2 s in   s in  
 2   2 
(c) B (d) C
 C D C  D 
24. If A and B are finite sets, then n  A  B  is (c) 2 sin   sin  2 
 2   
(a) n  A   n  B  (d) none of these
(b) n  A   n  B   n  A  B  35. cos10  cos 50 is equal to
(a) cos 20° (b) sin 20°
(c) n  A   n  B   n  B  A  (c) sin 40° (d) sin 50°
(d) n  A   n  B   n  A  B  36. The roots of x 2  7ix  12  0 are
25. If set A has n elements, then power set P(A) (a) 2i, 3i (b) –3i, 4i
has (c) 3i, 4i (d) none of these
(a) 2n elements (b) 2n elements 37. The roots of x 2   7  i  x  18  i   0 are
(c) (2n + 1) elements (d) 22n elements
26. Which of the following functions is an odd (a) 4  3i ,3  2i (b) 4  3i ,3  2i
function? (c) 4  3i ,3  2i (d) 4  3i , 3  2i
(a) x6 + 2x4 + 3x2 + 1 (b) 2x3 + 5x a  ib
(c) |x + 7| (d) 2x2 + 3 38. If  x  iy , then x 2  y 2 is equal to
c  id
27. f (x) = 3x4 + 9x2 + 3 is a
(a) odd function (b) even function a b a 2  b2
(a) (b)
(c) absolute function (d) linear function a b c 2  d2

3x  5 a 2  b2 a 2  b2
28. f x   is a (c) (d)
2x 2  8 x  9 a 2  b2 c 2  d2
(a) even function (b) odd function 39. The square root of 2  2 3i is
(c) rational function (d) linear function
29. The general solution of cosec x = –2 is 
(a)  1  3i  
(b)  1  3i 
  (c) 1  3i (d) 1  3i
(a) (b)
6 6
40. The modulus of 2  3 is equal to
 
(c) n    1 (d) n    1
n n

6 6 (a) 7 (b) 7 (c) 7i (d) –7

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52. If A1,A2 be two A.M’s and G1,G2 be two G..M.’s
1  i   2  i  between a and b, then (A1+A2)/G1G2 is equal to
41. The modulus of
 3  i  is equal to
ab ab
(a) 0 (b) 1 (c) 2 (d) 1/2 (a) (b)
ab ab
z1
42. If z1  1  2i and z 2  6  5i , then z is equal 2ab ab
2 (c) (d)
ab 2ab
to
53. The value of 1 + 3 C1 + 4 C 2 is
5 5 5
(a) (b) (c) (d) None (a) 5 C1 (b) 5 C2
61 6 61
(c) 5 C3 (d) none
43. If z is a complex number, then z  z is
54. The number of lines that can be drawn through
(a) 2 Im (z) (b) 2Re (z) n points on a circle, is
(c) 2i Im (z) (d) 2i Re (z) (a) n (b) (n – 1)
1i n n  1
44. The conjugate of complex number is (c) n (n – 1) (d)
1 i 2
(a) –i (b) i 55. The number of chords that can be drawn through
(c) i2 (d) 1 21 points on a circle, is
45. If z is a complex number and z is conjugate of (a) 210 (b) 211
z, then z  z is (c) 209 (d) none
(a) |z| (b) |z|2 (c) 2|z| (d) 0 56. The ratio of coefficient of xn in 1  x 
2n
and
46. Which of the following is a linear inequation of
coefficient of x n in 1  x 
2n 1
two variables is
(a) a  bx  0 (b) ax + by (a) 1 : 1 (b) 2 : 1
(c) ax  by  0 (d) all of these (c) 1 : 2 (d) none
47. Which of the following points lies in the graph of 57. The sum of coefficient of two middle terms in
the expansion of 1  x 
2n 1
the inequation 2x  y  8  0 is
(a) (4, 3) (b) 1, 3) (a) 2n 1
Cn (b) 2n 1
Cn 1
(c) (0, 7) (d) none
(c) Cn2n
(d) Cn 1 2n
48. Which of the points lies in the half-plane
58. If the 17th and 18th terms in the expansion of
x y
 1  0
 2  a  are equal, then a is equal to
50
2 3
(a) (0, 1) (b) (1, 5) (a) 1 (b) 0
(c) (–3, –4) (d) none (c) 2 (d) 3
49. The solution set of the inequality y  0 is In the expansion of 1  x  , coefficient of x r is
n
59.
(a) half plane below the x-axis excluding the
(b)  1 n Cr
r
(a) n Cr
points on x-axis
(b) half plane below the x-axis including the points (c) n 1
Cr (d) none of these
on x-axis 9
(c) half plane above the x-axis  3x 2 1 
60. The term independent of x in    is
(d) none of these  2 3x 
50. Region represented by the inequalities
7 18
x  0, y  0 lies in (a) (b)
18 7
(a) first quadrant (b) second quadrant (c) 7 (d) 18
(c) third quadrant (d) fourth quadrant
61. The value of  e x cos e x dx is
51. 2n
C 3 : n C 2  12 : 1 , then n is
(a) sin e x  e x  c
(a) 10 (b) 5
(c) 15 (d) none (b) sin e x  c
(c) e x sin e x  cos e x  c
(d) e x sin x  sin e x  c

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1
 sin tan x 
dx is
 69.

The value of 
sec 2 x
dx is
62. The value of   tan2 x  4
 1  x2

  c   c (a) log tan x  tan x  4  c

2
cos tan1 x cos tan1 x
(a) (b) 
1  2x 1  x2 (b) log sec x  tan x  c
(c)  cos  tan x   c (d) none of these
1
(c) log 1  tan x  4  c
2

 1 (d) none of these

63. The value of  x log x 2 dx is
   70. If f  2a  x   f  x  , then the value of
2a
1 1
(a)  log x  c (b) 
x
c  f  x  dx
0
is

log x (a) 2a (b) 0

(c)  log x  c (d) c a
x
(c) 2 f  x  dx (d) none of these
 1  tan x dx
2 0

64. The value of  is a

 1  tan x 1
x dx 
2
71. If 3
, then the value of a is
(a) sec2 x  c (b) log 1  tan x  c
2 0

(a) 1 (b) 2 (c) 1/2 (d) none

(c) log sec x  tan x  c (d) log 1  tan x  c 3
dy  dy 
65. The value of  sec x  tan x dx is
3 72. If y   1   , then the order and
dx  dx 
sec3 x degree of given differential equation is
(a) sec x  tan x  c (b) c respectively
3
(a) 1 and 3/2 (b) 1 and 3
sec2 x (c) 2 and 1 (d) 2 and 3
(c) c (d) none
2 73. The differential equation for the family of curves
 dx y = mx, where m is arbitrary constant is given by
66. The value of  is dy dy
 a  x2
2
(a) y 0 (b) x y 0
dx dx
1 ax
(a) 2a log a  x  c (b) log x  a  x  c
2 2
dy dy x
(c) x 1 (d) dx  y  0
dx
1 x 1 x
(c) sin c (d) sin 1  c 74. If y  cx  c 2  c 3 , where c is an arbitrary
a a a
constant, then the differential equation for the
 dx given curve is
67. The value of  is
 x 2  16 dy
(a) y  x
1 x 4 1 x 4 dx
(a) 4 log x  4  c (b) 8 log x  4  c 2
dy  dy 
(b) y  x 
1 x 2 dx  dx 
(c) 4 log x  2  c (d) none of these 2 3
dy  dy   dy 
 dx (c) y  x    
dx  dx   dx 
68. The value of  is
 9  16x 2 (d) none of these
4x dy
(a) sin
1
c 75. The solution of differential equation x  2 is
3 dx
given by
1 1 4 x
(b) sin c 1
4 3 (a) y  c (b) y  2 log x  c
x
(c) log x  9 16x  c
2
x
(c) y  log 2  c (d) none of these
1
(d) log x  9  16x 2  c
4

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76. A vector of magnitude 4 units which is parallel 3x  5
84. f x   is a
to the vector 3 iˆ  ˆj is given by 2x 2  8 x  9
(a) even function (b) odd function
3 ˆ 1 ˆ
(a) 3 iˆ  ˆj (b) i  j (c) rational function (d) linear function
2 2
2x  1
(c) 2 3 iˆ  2 ˆj (d) none of these 85. If f : R  R is defined by f  x   then
3
77. If P(3, –1, 7) and Q(4, –5, –1) are two given
 f 1  x  
points, then the vector PQ is given by
3x  1 x3 2x  1 x4
(a) iˆ  4 ˆj  8kˆ (b) iˆ  4 ˆj  8kˆ (a) (b) (c) (d)
2 2 3 3
(c) iˆ  4 ˆj  kˆ (d) 2iˆ  4 ˆj  8kˆ
86. If  lies in the first quadrant and 5 tan   4 ,
78. The unit vector in the direction of the vector
 5sin   3cos 

 then  sin   2 cos   
a  iˆ  ˆj  2kˆ is given by  
1
(a)
2
6
iˆ  ˆj  2kˆ  (a)
5
14
(b)
3
14
(c)
14
(d) 0

1 ˆ 1 ˆ 1 ˆ 87. sin 2 50 0  cos 2 1300 

(b) i  j k
6 6 6 (a) 0 (b) 1 (c) –1 (d) –2
1 ˆ 1 ˆ 2 ˆ 88. cos 24  cos 55  cos125  cos 2040 
0 0 0

(c) i  j k (a) –1 (b) 0 (c) 1 (d) 2

6 6 6
(d) none of these 89. cos 72 - sin2540 =
2 0


79. If angle between any two vectors a and b is 0, 5  3  5 5
 (a) (b) (c) (d)
then the value of a  b is 2 4 4 8
(a) 1 (b) ab (c) 0 (d) –ab 3
  90. If 90    180 , sin  
0 0
, then sin 3 =
80. If a and b are any two vectors with magnitude 5
 
3 and 2 respectively such that a  b  3 , then 117 117 125 125

angle between a and b is (a) (b) (c) (d)
125 125 117 117
(a)  / 6 (b)  / 3 91. cos 255 0  sin 165 0 =
(c)  / 2 (d) 0
81. Which of the following intervals is a set-builder 3 1 3 1 2 1
(a) 0 (b) (c) (d)
form of (6, 12]: 2 2 2 2
(a) x : x  R,6  x  12 92. If z1  z2  z3 and z1  z2  z3  0 then
(b) x : x  R,6  x  12 z1 , z2 , z3 are vertices of
(c) x : x  R, x is multiple of 6 (a) equilateral triangle (b) issosceles triangel
(c) Right angled triangle (d) scalene triangle
(d) x : x  R,6  x  12
1 1
82. If A  B    , then which of the following 93.  is
i 1 i  1
statement is true: (a) Positive rational number
(a) A  B (b) B  A (b) Purely imaginary
(c) A  B  A (d) A  B  B (c) Positive Integer
83. If A = {1, –1}, then A × A × A is (d) Negative integer
(a) {(1, 1, 1), (–1, –1, –1)}
(b) {(1, –1, 1), (1, –1, –1)} 94. The modulus of 1  i  3  4i  
(c) {(1, 1, 1),(1, 1, –1),(1, –1, 1),(1, –1, –1),(–1, 1,
(a) 50 (b) 25 (c) 25 (d) 5 / 2
1)
(d) {(1, 1, 1), (1, 1, –1), (1, –1, 1), (1, –1, –1), The integral solution of 1  i   2 x is
x
95.
(–1, 1, 1), (–1, 1, –1), (–1, –1, 1), (–1, –1, –1)}
(a) x = 1 (b) x = -1 (c) x = 2 (d) x = 0

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96. If  ,  are the roots ax2-2bx+c=0 then 109. If in an A.P., 3rd term is 18 and 7th term is
30, then sum of its 17 terms is
3 3 + 2  3 + 3 2 = (a) 600 (b) 612 (c) 624 (d) None
c2 bc 3 110. If S 1 =a 2 +a 4 +a 6 +...... upto 100 terms and
(a) (c  2b) (b) S2=a1+a3+a5+.....upto 100 terms of a certain A.P.
a3 a3 then its common difference d is
c3 bc (a) S1-S2 (b) S2-S 1
(c) (c  2b) (d) (c) S1 - S2/2 (d) None of these.
a3 a3
111. If a,b,c are in A.P. then 3 a,3b,3c are in
97. 42  42  42  ...  (a) A.P. (b) G.P. (c) equal (d) None
(a) 7 (b)-6 (c) 5 (d)4 112. If the value of 1 + 2 + 3 + ..... + n is 55, then
98. One root of the equation the value of 1 3+23+33+....+n3 is-
 x  1  x  3  x  2  x  4   12 0 is (a) 165 (b) 835
(c) 3025 (d) None of these.
(a) 1 (b) 2 (c) 1 (d) 0
113. If the straight line y=mx+c passes through
99. If     2 and     56 then the
3 3
the points (3,-4) and (-1,2) then value of m is
quadratic equation whose roots are is 2 3 3 2
(a) x 2  2 x  16  0 (b) x 2  2 x  15  0 (a) (b) (c) (d)
3 2 2 3
(c) x 2  2 x  12  0 (d) x 2  2 x  8  0 114. Equation of the line on which the length of
the perpendicular from origin is 5 and the
100. If  ,  are the roots of x 2 -x+1=0 then angle which this perpendicular makes with
the x axis is 60O
5   5 
(a) 2 (b) 1 (c) 12 (d) 4 (a) x  3 y  12 (b) 3 x  y  10
101. Which of the following is a linear inequation of (c) x  3 y  8 (d) x  3 y  10
two variables
(a) a  bx  0 (b) ax + by 115. The distance between the parallel lines
8x+6y+5=0 and 4x+3y-25=0 is
(c) ax  by  0 (d) all of these
7 9 11 5
102. The number of permutations of the letters of the (a) (b) (c) (d)
word "ENGINEERING" is 2 2 2 4
11! 11! 11!
116. Equation of the circle with centre (1,2) and
11!
(a) 3! 2! (b) (3! 2!) 2 (c) (3!) 2 . 2! (d) 3!(2!) 2 passing through (2,1) is
(a) x2 + y2 – 6x + 10y + 18 = 0
103. If nP4 : nP5 = 1 : 2 then n = (b) x2 + y2 + 6x - 10y + 18 = 0
(a) 4 (b) 5 (c) 6 (d) 7 (c) x2 + y2 – 6x + 10y + 25 = 0
104. The number of ways in which 7 persons can be (d) x2 + y2 – 2x - 4y + 3 = 0
arranged around a circle is 117. The length of the diameter of the circle
(a) 360 (b) 720 (c) 5040 (d) 1440 x2 + y2 – 6x – 8y = 0 is
105 A polygon has 35 diagonals. The number of its (a) 100 (b) 10 (c) 20 (d) 5
sides are
118. The point on the parabola which is nearest to
(a) 8 (b) 9 (c) 10 (d) 11 directrix is
12
106. The 4th term in the expansion of  1
 x  is (a) End of latusrectum (b) Focus
 x
(c) Vertex (d) Centre.
3 3
(a) 110 x 2 (b) 220 x 2 (c) 220x2 (d) 110x2 119. The parabolas y 2  4ax (a>0) and
9

107 The term independent of x in

 3x 2 1 
   is
y 2  4ax (a>0) touches at
 2 3 x 

(a) (0, 0) (b)  0, a 
7 5 11 13
(a)
18
(b)
18
3)
18
4)
18 (c)  a, 0  (d)  a, a 
n
cr 120. The vertex and focus of a parabola are (-2, 2), (-
108. n
 6, 6). Then its length of latusrectum is
cr 1
(a) 8 2 (b) 4 2 (c) 16 2 (d) 12 2
n  r 1 n  r 1 n  r 1 n  r 1
1) 2) 3) 4)
r r r r

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 NDA MATH TEST-15 ANSWER KEY
1 A 26 B 51 B 76 C 101 C
2 A 27 B 52 B 77 B 102 B
3 C 28 C 53 B 78 C 103 C
4 D 29 D 54 D 79 B 104 B
5 B 30 C 55 A 80 A 105 C
6 B 31 C 56 B 81 A 106 B
7 B 32 D 57 C 82 A 107 A
8 C 33 C 58 A 83 D 108 B
9 B 34 A 59 B 84 C 109 D
10 C 35 B 60 A 85 A 110 D
11 D 36 C 61 B 86 A 111 B
12 B 37 A 62 C 87 B 112 C
13 B 38 D 63 A 88 B 113 B
14 B 39 A 64 D 89 C 114 D
15 B 40 B 65 B 90 A 115 C
16 D 41 B 66 C 91 A 116 D
17 C 42 C 67 B 92 A 117 B
18 A 43 B 68 B 93 B 118 C
19 A 44 B 69 A 94 A 119 A
20 A 45 B 70 C 95 D 120 C
21 C 46 C 71 A 96 A
22 A 47 A 72 B 97 A
23 B 48 B 73 B 98 A
24 D 49 B 74 C 99 D
25 B 50 A 75 B 100 B