OURS ACADEMY

NDA MATH TEST-16

1. Equation of the parabola whose vertex is the 11. If A, B are symmetric matrices of the same order
origin, axis along the x axis and which passes then AB-BA is
through the point (-2, 4) is (a) symmetric matrix (b) skew symmetric matrix
(a) y 2  8 x (b) y 2  12 x (c) Diagonal matrix (d) identity matrix
12. A 3x3 is a non - singular matrix  A2 (Adj A) =
(c) y 2  8 x (d) x 2  y
2
2. When the eccentricity tends to zero, then the (a) A A (b) I (c) A I (d) A I
equation of ellipse becomes
 Cos Sin 
(a) a parabola (b) a circle 13. If A =  Sin 
Cos 
then A . A =

(c) an ellipse (d) a hyperbola
(a) A   (b) A (c) A  A (d)I
3. Equation of the ellipse with vertices (5,0) foci
(4,0) is a 0 0
0 a 0
x2 y2 x2 y2 14. If A =   then An =
(a)  1 (b)  1  0 0 a 
25 9 32 16
(a) an. A (b) an-1.A
x2 y2 x2 y2
(c)  7 (d)  1 (c) an+1.A (d) a3nI
25 7 25 12

4. The eccentricity of the ellipse

x2 y 2
 1 is
2 x  3 x  2
9 16  3 2 1 
5 15. If A=  is a symmetric matrix
7 7 7  4 1 5 
(a) (b) 4(c) (d)
16 4 2
5. Equation of the hyperbola with vertex (4,0) and then x=
focus (6,0) is (a) 0 (b) 3 (c) 6 (d) 8
x 2 y2 x 2 y2  o pq p  r
(a)  1 (b)  1 
16 20 20 16 q  r 
16. Det  q  p o
=
x 2
y 2
x y2 2  r  p r q o 
(c)  1 (d)  1
16 36 36 16 (a) (p-q) (q-r) (r-p) (b) 0
6. The vertices of the hyperbola are 2, 0 and (c) pqr (d) 4 pqr
the foci are 3, 0 then its equation is  0 c  b
2 2  c 0 a 
x 2 y2 x y If A =   then (a2+b2+c2) A =
(a)  1 (b)  1 17.
5 4 4 5  b  a 0 
x 2 y2 x 2 y2 (a) abc (b) a + b + c
(c)  1 (d)  1
5 2 2 5 (c) (a3 + b3+c3) (d) 0
7. cos[Sin-1 (2 cos2  - 1) + Cos-1 (1 - 2sin2  )] =
1
 18. The real part of is
(a) 0 (b) 1 (c) -1 (d) 1 – cos   i sin 
2
tan  1 1
3 (a) 2 (b) (c) (d)
8. Sin-1( ) - Tan-1(- 3) = 2 2 1 – cos 
2
19. The number of solutions of the equations
(a) -  /3 (b) 2  /3 (c)  /6 (d) 0
2x - 3y = 5, x + 2y=7 is ....
 (a) 1 (b) 2 (c) 4 (d) 0
9. If Sin-1 x - Cos-1 x = then x =
6
1  sin x  sin 2 x  1
20. L im =
3 3 x 0 x
(a) 1/2 (b) (c) - 1/2 (d) -
2 2 (a) 0 (b) 1 (c) 2 (d) 1/2

10.
  1  1 
cos  cos 1     sin 1      21. lim[ x  [ x]] 
  7  7  x 1

1 1 4 (a) 1 (b) -1 (c) 0 (d) does not exist

(a)  (b) 0 (c) (d)
3 3 9
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sin 1 ax  sin 1 bx d
22. L im  31. { xtanx } =
x 0 x dx
(a) 0 (b) 1 (c) a  b (d) a  b  tan x
x tan x 

 sec2 x. log x 
(a)
 x 
Lim
2n  1  tan x 
23. = (b) x tan x   sec 2 x. log x 
n 
2n  1  x 
 x 
1 (c) x tan x   sec 2 x.log x 
(a) 1 (b) (c)-1 (d) 0  tan x 
2  x 
(d) x tan x   sec 2 x. log x 
f  x  3  tan x 
If f  9   9, f  9   4, L im 
1
24.
x 9 x 3 32.    is increasing in
1 1 1 (a) (-5, 0) (b) (0, 5)
(a) 4 (b) (c) (d) 
4 2 2
(c)   ,  5  5,   (d) (-5, 5)
25. A lin e passes through (2,2) and is
perpendicular to the line 3x + y = 3 its y 33. If z 1 ,z 2 ,z 3 represent the vertices of a n
intercept is equilateral triangle such that |z1| = |z2| = |z3|,
then
(a) 4/3 (b) 1
(a) z1+z2 = z3 (b) z1+z2+z3 = 0
(c) 2/3 (d) 1/3
(c) z1z2 = 1/z3 (d) z1–z2 = z3–z2
d  3x  5  34. The slope of the tangent at (1, 6) on the curve
26.  =
dx  2 x  3 
2x 2  3y 2  5 is
19 14
(a) 2 (b) 1 1
( 2 x  3) ( 2 x  3) 2 (a) –9 (b) (c) (d) 9
9 9
 19 1 35. If the displacement in time t of a particle is
(c) 2 (d) 2 given by s = ae t + be-t, then the acceleration
( 2 x  3) ( 2 x  3)
is equal to
1    dy (a) velocity (b) displacement
27. If y= Tan  cot 2  x   then =
   dx (c) initial velocity (d) aet
1  x 2 x3 
(a) 1 (b) -1 (c) 0 (d)
2 36.  
 1  x    .........dx 
2! 3! 
dy (a) ex +c (b) -ex + c
28. If x = at2, y = 2at then 
dx ex ex
(a) 1/t (b) t (c) t 2
(d) 1 (c) c (d)  c
2 2
d
29 [log{log(logx)}]= 1
dx 37.  Cos 2 x  sin 2
x
dx 
1 1
(a) x log x log(log x) (b) x log x log(log x) (a) tan x + c (b) cot x + c
2
tan x cot 2 x
x 1 (c) c (d) c
(c) log x log(log x) (d) log x log(log x) 2 2

0 cos
3
38. x dx 
 5 
d  3 log 3 ( 2 x 1) 
30. 3 = (a) –1 (b) 0 (c) 1 (d) 
dx  
 1
 10 10
39.  (1  e x
)(1  e  x )
dx 

(a) 8 (b) 8
1
3( 2 x  1) 3
3( 2 x  1) 3
(a)
1
c (b)  c
1 ex 1  ex
1 1
(c) (d) 1 1
(2 x  1)
8
3
(2 x  1)
8
3 (c)  c (d) c
1 ex 1  ex

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49. If k is the diameter of a circle and A is the
x2 1
 x  12 dx = area of a sector of the circle whose vertical
x
40. e
angle is q then dA/dt =
k 2  d   k 2   d 
 x 1 
x  x 1
x (a)   (b)   
(a) e  + c (b) e   +c 8  dt   4   dt 
 x 1  x 1  d  d 
(c) (d) k 
x x 1 x 1 dt  dt 
(c) e c (d) e
x
+c
(x 1)2 ( x  1) 2 50. A line makes an angle  ,  ,  with the X,Y,Z
axes Then sin 2   sin 2   sin 2  =
tan 1 x
1

41. 0 1  x 2 dx = (a) 1 (b) 2 (c) 3/2 (d) 4

2
–2 ( a x
3
2 2 2 2 51. The value of  bx  c )dx 
(a) (b) (c) (d) 1
4 18 32 8 dpends on
 (a) b (b) c
2 (c) a (d) a and b
1
42.  1  Cotx dx = 52. If a = 2i+j+2k and b = 5i-3j+k, then the projection
0 of b on a is
(a) 1 (b) 2 (c) 3 (d) 4

(a)  (b)  2 (c)  4 (d) 53. The angle between any two diagonals of a cube
6 is equal to

  x  2  x  5  dx (a) Cos 1 (1 / 2) (b) Cos 1 (1 / 4)

5
43.
2
(a) 0 (b) 3 (c) 9/2 (d) 9 (c) Cos 1 (1 / 3) (d) Cos 1 (1 / 5)
44. The area bounded by the curve y  cos x , x- 54. The distance between the points
axis between the ordinates x  0, x  2 is (sin  , cos  ) and (cos  , -sin  ) is
(a) 1 sq. unit (b) 4 sq. units (a) 1 (b) 2 (c) 2 (d) 6
2 55. If the extremities of a diagonal of a square are
(c) sq, units (d) 2 sq. units
3 (1, -2, 3) and (2, -3, 5) then the length of its
45. The area bounded by the two parabolas side is
y 2  8 x and x 2  8 y is (a) 6 (b) 3 (c) 5 (d) 7
64 56. A card is drawn at random from a pack of 52
(a) 64 sq. units (b) sq, units cards. What is the probability that the card drawn
3 is red or queen?
32 1 (a) 1/13 (b) 1/26 (c) 7/13 (d) 1/2.
(c) sq. units (d) sq. units 57. What is the probability that on ordinary year has
3 3
53 sunday?
46. The type of D.E 1  y 2  dx   tan 1 y  x  dy is (a) 53/365 (b) 1/7
(a) Linear in x (b) Bernoulli (c) 2/7 (d) 48/53.
(c) Linear in y (d) Homogenious 58. Average of six numbers is 4 but when we divides
every number by two, then new mean is :
47. The D.E whose solution is y  ax 2  bx  c is (a) 2 (b) 4 (c) 6 (d) 8.
(a) y3  y (b) y3  x 59. What is average of 1 , 2 , 3 , ..., 203?
3 3 3

(a) 2215 (b) 2205 (c) 2202 (d) 2212

(c) y3  0 (d) y2  0 60. The mean of first n odd natural numbers is
dy (a) 2n - 1 (b) n
48.  e 2 y , x  5  y  0, then y  3  2 x  (c) 1/n (d) None of these.
dx
(a) e5  9 (b) e 6  9 x2  x
61. L im 
(c) e8  9 (d) e 4  9 x 1 x 1
(a) 1 (b) 2 (c) 3 (d) 4

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1  x  x2  1 72. If f  2   4 and f '  2   1, then
62. L im 
x
xf  x   2 f  x 
x 0

lim 
(a) 1 (b) 1/2 (c) 1/3 (d) 0 x 2 x2
(a) 2 (b) 1
x If x2 (c) 4 (d) 3

f  x  x  5 If 2 x3 L im f  x   d  2 2 
63.
x  7 x 3 73 log( x  a  x )  =
 If x3 dx  
1 x
(a) -2 (b) -4 (c) 0 (d) 1 (a) (b)
(x  a2  x2 ) a2  x2
tan x  sin x 1
64. L im  1
x 0 x3 (c) 2 2 (d)
x( x  a  x ) a  x2
2

1 1 1 74. If s = e t (sint - cost), then the acceleration is

(a) 1 (b) (c) (d)  (a) 2et (Cost - sint) (b) 2et (cost + sint)
2 4 3
(c) e t (cost + sint) (d) e t (cost-sint)
75. The equation of the normal to the curve y 2  4ax
log 1  ax   log 1  bx 
65. L im  at the origin is
x 0 x
(a) x = 0 (b) x = 2 (c) y = 0 (d) y = 2
If the curves ay  x 2  7 and x 3  y cut
(a) a  b (b) a  b (c)   a  b  (d) ab 76.
orthogonally at (1, 1) then a =

d  log 1 cot 2 x 
1
66. e = (a) 1 (b) – 6 (c) 6 (d)
dx   6
(a) cosecx cotx (b) -cosecx.cotx
77. 42  42  42  ... 
(c) cosec2x.cotx (d) 0
x x dy
67. If y= a .e then = (a) 7 (b)-6 (c) 5 (d)4
dx
(a) y(1+loga) (b) y2(1+loga)  1 4  2  1  4 7
78. If A =   B  then  3 4 is
(c) -y(1+loga) (d) y(1-loga)  1 1  1 2   
1 x2 1 dy (a) 2A + B (b) A - B (c) AB (d) A - 2B
68. If y= Tan 1 then =
x dx 0 1
79. If A   , then A4 =
(a)
1
(b)
1
(c)
1
(d)
2 1 0
1 x2 1 x2 2(1  x 2 ) 1  x2 (a) I (b) 0 (c) A (d) 4 I
dy  cos  sin  
69. If x = at2, y = 2at then  80. If A ()    then A ( ) A () =
dx   sin  cos  
(a) 1/t (b) t
(c) t2 (d) 1
70. The derivative of ax w.r.t. sin-1x is (a) A ( )  A () (b) A ( )  A ()
(c) A (  ) (d) A (  )
a x . log e a
(a) a x . log a 1  x 2 (b) 1 1 1
e 1  x2 x y z
81. =
ax  ax x3 y3 z3
(c) (d)
1 x2 1 x2 (a) (x+y+z) (x+y) (y+z)(z+x)
dy (b) (x+y+z)(x-y)(y-z) (z-x)
71. If ax2 + 2hxy + by2 = 0 then =
dx (c) (x-y) (y - z) (z -x)
 ax  hy   ax  hy  (d) (x+y) (y+z) (z+x)
(a) -  hx  by  (b)  hx  by  82. If 15Pr = 32760, then r =
   
(c) -(ax+hy)(hx+by) (d) (ax+hy)(hx+by) (a) 4 (b) 5
(c) 6 (d) 7

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83. For a binomial distribution n = 20, q = 0.75. The  1  Sin x 
 e  1  Cos x  dx =
x
mean of the distribution is 92.
(a) 10 (b) 15 (c) 5 (d) 20
84. If a random variable x has the following probability
x x
distribution X  xi :0 1 2 3 , (a) ex Tan +c (b) - ex Cot +c
2 2
P X  xi  : 2 K 2 3K 2 5 K 2 6 K 2
x x
then the value of K is (c) ex csc2 + c (d) e-x cot + c
2 2
1 1 1 1
(a) (b) (c)  (d)
e cos x dx 
log x
4 4 4 2 93.

85.  (sin-1x + Cos-1x)2 dx =

(a) x Sinx  Cosx  c
x
Sinx  Cos2 x  c
(b)
x x 2
 2x
(a) 0 (b) +c (c) c (d) c (c) x Sin x  Cos x  c (d) x Sin x  Sin x  c
4 4 2
 x.e dx 
x
1  Sin x 94.
86.  1 Sin x dx 
. x c
(a) xe (b) e x ( x  1)  c
(a) 2 tan x - 2 Sec x - x + c
(b) 2 tan x - Secx - x + c (c) e x ( x  1)  c (d) e x ( x  2)  c
(c) tan x + 2 Sec x + x + c  x4 
 e  x  6  dx =
x
(d) tan x - 2 Sec x + x + c 95. 3

 e5 log x  e 4 log x  ex 1
87.   e3log x  e 2 log x dx = (a)
(x6)2
c (b)ex
(x4)2
c
(a) x + c (b) 3x2+c x 4
(c) ex c (d) ex +c
x3
x 3 x6 ( x  6) 2
(c) c (d) c
3 2 e x (1  x)
96.  Cos 2 ( xe x ) dx =
x2
88.  1  x6
dx =
(a) Tan (xex) + c (b) Sec 2 (ex x )  c
(c) Cos(xe ) + c x
(d) Cot(xex) + c
1 1
(a) Sin-1 (x(c) + c (b) Cos-1 (x(c) + c
 x cos
3
3 3 97. x 2 . sin x 2 dx =

1 1 1 4 2 1
(c) - Sin-1 (x(c) + c (d) Cos-1 (x(c) + c (a) sin x  c (b) cos 4 x 2  c
3 4 8 8
1 1 1 4 2
89.  (1  e x
)(1  e  x )
dx 
(c)  cos x  c
4 2
(d) - sin x  c
8 8
1 1
c (b)  c dx
(a)
1 ex 1  ex 98. 1 e x =
1 1
(c)  c (d) c (a) 1  e x  c (b) 1 / 2  (1  e x )  c
1 ex 1  e x
 1  x
1
(c) log (1  e x )  c (d) 1 / 2 log (1  e x )  c
90.  1  x 2 .e x dx 
1 1
99.  cos 7 x. cos 3x dx =
(b) 1  2  c
x
(a) e x
c Sin10 x Sin8 x
x (a)  c
4 6
1 1
(d) 1  2  c
x
(c) e x
c 1
x (b) [2 Sin10 x  5Sin 4 x ]  c
40
e (Cot x  Cot 2 x ) dx 
x
91.
1
(a) ex(cot x) + c (b) ex (1-cot2 x) + c (c) [2Cos10 x  5Cos 4 x]  c
40
(c) ex cot2 x + c (d) ex (cot x + (a) + c
(d) [ 2Cos10 x  5Cos 4 x ]  c

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 x2 x3  110. In a throw of a coin, the probability of getting tail

100.  
1  x    .........dx 
2! 3! 
is :
(a) ex +c (b) -ex + c (a) 1/2 (b) 1/4 (c) 1 (d) none
111. What is the probability that on ordinary year has
ex ex
(c) c (d)  c 53 sunday?
2 2
(a) 53/365 (b) 1/7
101. If  is an acute angle between the lines
(c) 2/7 (d) 48/53.
y=2x+3, y=x+1 then the value of tan  =
112. In a simultanous throws of two dice, what is the
2 1 3 1 probability of getting a doublet?
(a) (b) (c) (d)
3 3 4 2 (a) 1/6 (b) 1/4
 2 x  3  x 1  (c) 3/4 (d) 2/3.
102. L im 113. In a simultaneous throw of two dice, what is the
x 1 2x  x  3
2
probability of getting a total of 10 or 11?
1 1 2 2 (a) 7/12 (b) 5/36
(a) (b)  (c) (d) 
10 10 5 5 (c) 1/6 (d) 1/4.
5 5
114. The probability that a leap year selected at
x8  a8
103. L im 1 1
 random will contain 53 sunday is :
xa
x a3 3
(a) 1/7 (b) 2/7
15 247 15 247 (c) 2/366 (d) 3/366.
(a) a (b) a 115. If E and F are independent events such that P(E)
8 4
= 0.7 and P(F) = 0.3, then P(E  F) is :
15 247 15  247
(c)  a (d) a (a) 0.4 (b) 1 (c) 0.21 (d) none
8 4
116. A bag contains 5 blue and 4 black balls. Three
3 5 x balls are drawn at random. What is the probability
104. L im 
x4 x4 that 2 is blue and 1 is black?
1 1 1 1 (a) 1/3 (b) 2/5 (c) 1/6 (d) none.
(a)  (b) (c) (d)  117. The mean or average number of heads when
5 6 5 6
we toss 10 unbiased coins is
sin x (a) 20 (b) 10 (c) 5 (d) 15
105. Lx 
im
0 | x|
=
118. A random variable x has the following probability
(a) 0 (b) – 1
distribution X  x : 1 2 3 4 then
(c) 1 (d) does not exist
P X  x  k 2k 3k 4k
tan x  sin x the value of K is
106. L im 
x 0 x3 1 1 1
(a) 10 (b) (c) (d)
1 1 1 10 5 15
(a) 1 (b) (c) (d)  119. If x is a random variable with the following probability
2 4 3
distribution X  x : 3 6 9 then E(x) =
107. A coin is tossed two times, then the probability
1 1 1
of getting atleast one times head. P  X  xi  :
6 2 3
(a) 3/4 (b) 1/4 (c) 1/2 (d) 2/3.
11 11 11
108. A card is drawn at random from a pack of 52 (a) 11 (b) (c) (d)
2 3 4
cards. What is the probability that the card drawn
120. If a random variable x has the following probability
is red or queen?
distribution X  xi :0 1 2 3 , then the
(a) 1/13 (b) 1/26 (c) 7/13 (d) 1/2.
P X  xi  : 2K 2 3K 2 5K 2 6K 2
109. A bag contains 5 red and 3 yellow ball, two balls value of K is
are drown at a time, the probability of both red
1 1 1 1
balls. (a) (b) (c)  (d)
4 4 4 2
(a) 5/8 (b) 3/28 (c) 1/14 (d) 5/14.

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