1. UPSEE–PAST PAPERS
MATHEMATICS - UNSOLVED PAPER – 2005

2. SECTION -I
 Single Correct Answer Type
 There are five parts in this question. Four choices are given for each part and one of them is
correct. Indicate you choice of the correct answer for each part in your answer-book by
writing the letter (a), (b), (c) or (d) whichever is appropriate

3. 01 Problem
Of a total of 600 bolts, 20% are too large and 10% are too small. The
remainder are considered to be suitable. If a bolt is selected at random, the
probability that it will be suitable is :
1
a.
5
7
b. 10
1
c. 10
3
d. 10

4. 02 Problem
The area enclosed within the curve |x| + |y| = 1 is :
a. 1 sq unit
b. 2 2 sq unit
c. 2 sq unit
d. 2 sq unit

5. 03 Problem
3 1 1
If P(B) , P( A B C) and P(A B C) , then P(B c) is :
4 3 3
a. 1/12
b. 1/6
c. 1/15
d. 1/9

6. 04 Problem
Two masses are projected with equal velocity u at angle 300 and 600
respectively. If the ranges covered by the masses be R1 and R2, then :
a. R1 > R2
b. R1 = R2
c. R1 = 4R2
d. R2 > R1

7. 05 Problem
1 1
The value of sin sin 1
3
sec 1
3 cos tan 1
2
tan 1
2 is :
a. 1
b. 2
c. 3
d. 4

8. 06 Problem
It is given that f’(a) exists, then x f (a) af (x) is equal to :
lim
x a (x a)
a. f(a) – af’(a)
b. f’(a)
c. -f’a
d. f(a) + af’(a)

9. 07 Problem
/2 cot x is equal to :
dx
0
cot x tan x
a. 1
b. -1
c. 2
d.
4

10. 08 Problem
Area bounded by the curve y = log2 x, x = 0, y 0 and x – axis is :
a. 1 sq unit
b. sq unit
c. 2 sq unit
d. none of these

11. 09 Problem
     
If | a x b |2 | a b |2 144 and | a| 4, then| b | is equal to :
a. 12
b. 3
c. 8
d. 4

12. 10 Problem
     
Given that | a| 3, | b | 4,| a x b | 10, then| a b |2 equals :
a. 88
b. 44
c. 22
d. none of these

13. 11 Problem
lim x log sin x is equal to :
x 0
a. zero
b.
c. 1
d. cannot be determined

14. 12 Problem
If x = 1 + a + a2 + …… to infinity and y = 1 + b + b2 + ……….. to infinity, where a, b
are proper fractions, then 1 + ab + a2b2 + … to infinity is equal to :
xy
a.
x y 1
xy
b.
x y 1
xy
c.
x y 1
xy
d. x y 1

15. 13 Problem
cos4 sin4 is equal to :
1 2 sin2
a. 2
b. 2 cos -1
c. 1 2 sin2
2
d. 1 + 2 cos2

16. 14 Problem
x 2
If y f (x ) , then :
x 1
a. x = f(y)
b. f(1) = 3
c. y increases with x for x < 1
d. f is a rational function of x

17. 15 Problem
If two like parallel forces of P Q have a resultant 2N, then :
N and N
Q P
a. P = Q
b. 2P = Q
c. P2 = Q
d. P = 2Q

18. 16 Problem
A person standing on the bank of a river observes that the angle subtended by
a tree on the opposite bank is 600. When he retreats 20 ft from the bank, he
finds the angle to be 300. The breadth of the river in feet is :
a. 15
b. 15 3
c. 10 3
d. 10

19. 17 Problem
If are the cube roots of a positive number p, then for any real x, y, z the
expression x y z equals :
x y z
a. 1 3i
2
1 3i
b.
2
1 3i
c. 2
1 3i
d. 2

20. 18 Problem
m 1
If tan and tan , then is equal to :
m 1 2m 1
a.
3
b.
4
c. zero
d. 2

21. 19 Problem
If f(x) = x , [ x x 1] then
a. f(x) is continuous but not differentiable at x = 0
b. f(x) is not differentiable at x = 0
c. f(x) is differentiable at x = 0
d. none of the above

22. 20 Problem
Tan + 2 tan 2 + 4 tan 4 + 8 cot 8 is equal to :
a. tan 16
b. 0
c. cot
d. none of these

23. 21 Problem
A book contains 1000 pages numbered consecutively. The probability that the
sum of the digits of the number of a page is 9, is :
a. Zero
55
b. 1000
33
c.
1000
44
d.
1000

24. 22 Problem
     
The value of [a bb cc a] is :
  
a. 2 [a b c]
  
b. [a b c]
c. 1
d. none of these

25. 23 Problem
A number is chosen at random among the first 120 natural numbers. The
probability of the number chosen being a multiple of 5 or 15 is :
1
a.
8
1
b.
5
1
c. 24
1
d. 6

26. 24 Problem
Let A, B and C be n x n matrices. Which one of the following is a correct
statement ?
a. If AB = AC, then B = C
b. If A3 + 2A2 + 3A + 5I = 0, then A is invertible
c. If A2 = 0, then A = 0
d. None of the above

27. 25 Problem
   
a 2ˆ
i ˆ
j ˆ
k, b ˆ
i 2ˆ
j ˆ 
k, c ˆ
i ˆ
j ˆ 
k , then a x (b x c )
If equals :
a. 5ˆ
i 7ˆ
j ˆ
3k
b. 5ˆ
i 7ˆ
j ˆ
3k
c. 5ˆ
i 7ˆ
j ˆ
3k
d. zero

28. 26 Problem
 
If AB x AC 2ˆ
i 4ˆ
j ˆ
4k , then the area of ABC is :
a. 3 sq unit
b. 4 sq unit
c. 16 sq unit
d. 9 sq unit

29. 27 Problem
10
The coefficient of x4 in the expansion of x 3 is :
2 x2
504
a. 259
450
b.
263
c. 405
256
d. none of these

30. 28 Problem
Equation of the ellipse whose foci are (2, 2) and (4, 2) and the major axis is of
length 10, is :
2 2
x 3 y 2
a. 1
24 25
2 2
x 3 y 2
b. 1
24 25
2 2
x 3 y 2
c. 1
25 24
2 2
x 3 y 2
d. 1
25 24

31. 29 Problem
The volume of the solid generated by the revolution of the curve y a3
a2 x2
about x-axis is :
1 3
a2
a. 2
b. 3 a2
1
c. 2
a3
2
d. 2 a3

32. 30 Problem
The radius of the circle z i =5 is given by :
z i
13
a. 12
5
b. 12
c. 5
d. 625

33. 31 Problem
     
If a (1, p,1), b (q,2,2), a b r and a x b = (0, -3, -3), then p, q, r are in that
order :
a. 1, 5, 9
b. 9, 5, 1
c. 5, 1, 9
d. none of these

34. 32 Problem
The circle passes through the point (a, b) and cuts the circle x2 + y2 = k2
orthogonally, then the locus of its centre s given by :
a. 2ax + 2by – (a2 + b2 + k2) = 0
b. 2ax + 2by + (a2 + b2 - k2) = 0
c. 2ax + 2by + (a2 + b2 + k2) = 0
d. none of the above

35. 33 Problem
The foci of an ellipse are (0 4) and the equations for the directrices are y = 9.
the equation for the ellipse is :
a. 5x2 + 9y2 = 4
b. 2x2 - 6y2 = 28
c. 6x2 + 3y2 = 45
d. 9x2 + 5y2 = 180

36. 34 Problem
The straight lines x + y = 0, 3x + y – 4 = 0 and x +3y –4 = 0 from a triangle
which is :
a. Right angled
b. Equilateral
c. Isosceles
d. None of these

37. 35 Problem
The eccentricity of the hyperbola 9x2 – 16y2 – 18x – 64y – 199 = 0 is :
16
a. 9
5
b.
4
c. 25
16
d. zero

38. 36 Problem
A four-digit number is formed by the digits 1, 2, 3, 4 with no repetition. The
probability that the number is odd, is ;
a. Zero
1
b. 3
1
c.
4
d. none of these

39. 37 Problem
The coefficient of xn in the expansion of (a bx) is :
ex
( 1)n
a. (a bn)
n!
b. ( 1)n
(b an)
n!
c. ( 1)n 1
(a bn)
n!
d. none of these

40. 38 Problem
A and B are two independent events. The probability that both A and B occur is
1/6 and the probability that neither of them occurs is 1/3. The probability of
occurrence of A is ;
5
a.
6
b. 1
6
c. 1
2
d. none of these

41. 39 Problem
1 41
The value of cot 9 cosec-1 is given by :
4
a. 0
b. 4
c. tan-1 2
d. 2

42. 40 Problem
Let a, b, c be distinct non-negative numbers. If the vectors
ˆ
ai ˆ
aj ˆ
ck , ˆ
i ˆ ˆ
k and ci ˆ
cj ˆ
bk lie in a plane, then :
a. c2 = ab
b. a2 = bc
c. b2 = bc
d. none of these

43. 41 Problem
The greatest coefficient in the expansion of (1 + x)2n is :
a. 2nC
n
b. 2nC
n+1
c. 2nC
n-1
d. 2nC
2n-1

44. 42 Problem
ex log(1 x) (1 x) 2
The value of lim is equal to :
x 0 x2
a. 0
b. -3
c. -1
d. infinity

45. 43 Problem
The values of k for which the equations x2 – k x- 21 = 0 and x2 – 3k x + 35 = 0
will have a common roots are :
a. k = 4
b. k = 1
c. k = 3
d. k = 0

46. 44 Problem
   
are two non-zero vectors, then (a b) (a b) is equal to :
a. a + b
b. (a - b)2
c. (a + b)2
d. (a2- b2)

47. 45 Problem
If sin x + sin2 x = 1, then cos6 x + cos12x + 3 cos10 x + 3 cos8 x is equal to :
a. 1
b. cos3 x sin3 x
c. 0
d.

48. 46 Problem
The integrating factor of the differential equation is :
a. x
b. ln x
c. 0
d.

49. 47 Problem
/2
x sin2 x cos2 x dx is equal to :
0
2
a. 32
2
b.
16
c.
32
d. none of these

50. 48 Problem
H H
If H is harmonic mean between P and Q, then the value of P Q
is ;
a. 2
PQ
b. (P Q)
(P Q)
c.
PQ
d. none of these

51. 49 Problem
The value of ‘p’ for which the equation x2 + pxy + y2 – 5x – 7y + 6 = 0
represents a pair of straight lines is :
5
a.
2
b. 5
c. 2
2
d. 5

52. 50 Problem
Angle between the vectors       is :
3(a x b) and b (a b)a
a. 2
b. 0
c. 4
d.
3

53. 51 Problem
The equation of the circle passing through (4,5) having the centre (2, 2) is :
a. x2 + y2 + 4x + 4y – 5 = 0
b. x2 + y2 - 4x - 4y – 5 = 0
c. x2 + y2 - 4x = 13
d. x2 + y2 - 4x - 4y + 5 = 0

54. 52 Problem
n
the smallest positive integer n for which 1 i is :
1
1 i
a. n = 8
b. n = 12
c. n = 16
d. none of these

55. 53 Problem
The equation of tangents drawn from the origin to the circle x2 + y2 – 2rx –
2hy + h2 = 0 are :
a. x = 0, y = 0
b. x = 1, y = 0
c. (h2 - r2) x – 2 rhy = 0, y = 0
d. (h2 – r2) x – 2 rhy = 0, x = 0

56. 54 Problem
The value of 91/3 x 91/9 x 91/27 x ….. is :
a. 9
b. 1
c. 3
d. none of these

57. 55 Problem
  
If a, b, c are any three coplanar unit vectors then :
  
a. a (b x c ) =1
  
b. a (b x c ) =3
  
c. (a x b) c 0
  
d. c x a) b 1

58. 56 Problem
Let 0 < P(A)<1, 0 < P(B) < 1 and P(A B) = P(A) + P(B) – P(A) P(B), then
a. P(B/A) = P(B) – P(A)
b. P(A’ B’) = P(A’) + P(B’)
c. P(A B) = P(A’) P(B’)
d. None of the above

59. 57 Problem
The probability that in the toss of two dice we obtain the sum 7 or 11, is :
1
a.
6
1
b.
18
2
c. 9
23
d. 108

60. 58 Problem
dy
If 2x + 2y = 2x + y, then dx
is equal to :
(2x 2y )
a. (2x 2y )
(2x 2y )
b. (1 2x y )
x y 2y 1
c. 2
1 2x
2x y
2x
d.
2y

61. 59 Problem
If the probability of A to fail in an examination is 0.2 and that for B is 0.3, then
probability that either A or B is fail, is :
a. 0.5
b. 0.44
c. 0.8
d. 0.25

62. 60 Problem
If the line ax + by + c = 0 is normal to the curve xy = 1, then :
a. a > 0, b > 0
b. a > 0, b < 0
c. a < 0, b < 0
d. data is unsufficient

63. 61 Problem
If f(x) = cos(log x), then f (x )f (y ) 1 x has the value :
f f (xy )
2 y
a. -1
1
b. 2
c. -2
d. zero

64. 62 Problem
If y = 3x-1 + 3- x –1 (x real), then the least value of y is :
a. 2
b. 6
c. 2/3
d. none of these

65. 63 Problem
the value of lying between 0 and and satisfying the equation
2
1 sin2 cos2 4 sin 4
sin2 1 cos2 4 sin 4 0
are :
sin2 cos2 1 4 sin 4
a. 7
24
b. 5
24
11
c.
2
d. 24

66. 64 Problem
100 100
1 3 1 3 is equal to :
2 2
a. 2
b. zero
c. - 1
d. 1

67. 65 Problem
If , be the two roots of the equation x2 + x + 1 = 0, then the equation
whose roots are and is :
a. x2 + x + 1 = 0
b. x2 - x + 1 = 0
c. x2 - x - 1 = 0
d. x2 + x - 1 = 0

68. 66 Problem
In a binomial distribution, the mean is 4 and variance is 3. Then, its mode is :
a. 5
b. 6
c. 4
d. none of these

69. 67 Problem

If a force F 3ˆ
i 2ˆ
j ˆ
4k is acting at the point P(1, -1,2), then the

moment of F about the point Q(2, -1, 3) is :
a. 57
b. 39
c. 12
d. 17

70. 68 Problem
the equation of a line passing through (-2, -4) and perpendicular to the line 3x –
y + 5 = 0 is ;
a. 3y + x – 8 = 0
b. 3x + y + 6 = 0
c. x + 3y + 14 = 0
d. none of these

71. 69 Problem
(cosec x)1/logx is equal to :
a. 0
b. 1
c. 1/e
d. none of these

72. 70 Problem
the minimum value of f(x) = sin4 x + cos4 x, 0 x is :
2
1
a.
2 2
1
b. 4
1
c. 2
1
d.
2

73. 71 Problem
Which of the following is a true statement ?
a. {a} {a, b, c}
b. {a} {a, b, c}
c. {a, b, c}
d. none of these

74. 72 Problem
A vector of magnitude of 5 and perpendicular to (ˆ
i 2ˆ
j ˆ
k ) and (2ˆ
i ˆ
j ˆ
3k )
5 3 ˆ ˆ ˆ
a. (i j k)
3
b. 5 3 ˆ ˆ ˆ
(i j k)
3
5 3 ˆ ˆ ˆ
c. (i j k)
3
5 3 ˆ
d. ( ˆ
i ˆ
j k)
3

75. 73 Problem
/3 x sin x is :
dx
/3 cos2 x
a. 1
(4 1)
3
b. 4 5
2log tan
3 12
4 5
c. log tan
3 12
d. none of these

76. 74 Problem
m
n r
Cn is equal to :
r 0
a. n + m + 1C
n+1
b. n + m + 2C
n
c. n + m + 3C
n-1
d. none of these

77. 75 Problem
The angle between the lines 2x = 3y = - z and 6x = -y = -4z is :
a. 900
b. 00
c. 300
d. 450

78. 76 Problem
A particle is projected vertically upwards at a height h after t1 seconds and
again after t2 seconds from the start. Then h is equal to :
a. 1 g(t – t2)
1
2
1
b. g(t1 + t2)
2
c. 1 Gt1t2
2
d. None of these

79. 77 Problem
If sin + cosec =2, then sin2 + cosec2 is equal to :
a. 1
b. 4
c. 2
d. none of these

80. 78 Problem
/2 sin x
The value of dx , is :
0
sin x cos x
a.
2
b.
4
c.
8
d. 6

81. 79 Problem
sin2 y 1 cos y sin y
The value of expression 1 is equal to :
1 cos y sin y 1 cos y
a. 0
b. 1
c. - sin y
d. cos y

82. 80 Problem
a 1 0
If f(x) = ax a 1 , then f(2x) – f(x) equal to :
ax 2 ax a
a. a (2a + 3x)
b. ax (2x + 3a)
c. ax (2a + 3x)
d. x (2a + 3x)

83. 81 Problem
2 2
1 2 3 2 3 3
If is a non-real cube root of unity, then 2 2 is equal
2 3 3 3 2
to :
a. -2
b. 2
c. -
d. 0

84. 82 Problem
a b
If in a ABC ,
cos A cos B '
then :
a. sin2 A + sin2 B = sin2 C
b. 2 sin A cos B = sin C
c. 2 sin A sin B sin C = 1
d. none of the above

85. 83 Problem
The graph of the function y = f(x) has a unique tangent at the point (a, 0)
loge {1 6f (x)}
through which the graph passes, Then lim is :
x a 3f (x)
a. 0
b. 1
c. 2
d. none of these

86. 84 Problem
n
a is equal to :
lim 1 sin
n n
a. ea
b. e
c. e2a
d. 0

87. 85 Problem
3c
If the equation ax2 + 2bx – 3c = 0 has no real roots and 4
< a + b, then :
a. c < 0
b. c > 0
c. c 0
d. c = 0

88. 86 Problem
The line 3x – 4y = touches the circle x2 + y2 – 4x – 8y – 5 = 0 if the value of is :
a. - 35
b. 5
c. 20
d. 31

89. 87 Problem
     
If OA ˆ
i 2ˆ
j 3k, OB 3ˆ
i ˆ
j ˆ
2k, OC 2ˆ
i 3ˆ
j ˆ
k. Then AB AC is equal to :
a. 0
b. 17
c. 15
d. none of these

90. 88 Problem
The value of tan2 (sec-1 2) + cot2 (cosec-1 3) is
a. 15
b. 13
c. 11
d. 10

91. 89 Problem
The sum of all proper divisor of 9900 is :
a. 29351
b. 23951
c. 33851
d. none of these

92. 90 Problem
The combined equation of the pair of lines through the point (1, 0) and parallel
to the lines represented 2x2 – xy – y2 = 0 is :
a. 2x2 – xy – y2 – 4x – y = 0
b. 2x2 – xy – y2 – 4x + y + 2 = 0
c. 2x2 + xy + y2 –2x + y = 0
d. none of the above

93. 91 Problem
a 1 2
If a, b, c are in AP, then , , are in :
bc c b
a. AP
b. GP
c. HP
d. None of these

94. 92 Problem
A particle is in equilibrium when the forces ,
  u  u
F1 ˆ
10k, F2 (4ˆ
i 12ˆ
j ˆ
3k), F2 (4ˆ
i 12ˆ
j ˆ
3k)
13 13
 v 
F3 ( 4i j ˆ
ˆ 12ˆ 3k) and F4 (cos ˆ sin ˆ) act on it, then :
i j
13
65
v 65 cot
a. 3
b. u = 65 (1 – 3 cot )
c. w = 65 cosec
d. none of the above

95. 93 Problem
There are 10 points in a plane out of these 6 are collinear. The number of
triangles formed by joining these point is :
a. 100
b. 120
c. 150
d. none of these

96. 94 Problem
 
If x and y are two unit vectors and is the angle between them, then
1  
|x y| is equal to :
2
a. 0
b. 2
sin
c. 2
cos
d. 2

97. 95 Problem
      
   a b c a b x a c
If a, b and c are three non-coplanar vectors, then is
equal to :
a. 0
  
b. [a b c ]
  
c. 2 [a b c ]
  
d. - [a b c ]

98. 96 Problem
The coefficient of x5 in the expansion of (1 + x2)5 (1+ x)4 is :
a. 30
b. 60
c. 40
d. none of these

99. 97 Problem
The function f(x) = x3 – 3x is :
a. Increasing on (- , -1) (1, ) and decreasing on (-1, 1)
b. Decreasing on (- , -1) (1, ) and increasing on (-1, 1)
c. Increasing on (0, ) and decreasing on (- , 0)
d. decreasing on (0, ) and increasing on (- , 0)

100. 98 Problem
A man in a balloon rising vertically with an acceleration of 4.9 m/s2, releases a
ball 2 s after the balloon is let go from the ground. The greatest height above the
ground reached by the ball, is :
a. 19.6 m
b. 14.7 m
c. 9.8 m
d. 24.5 m

101. 99 Problem
A bag contain n + 1 coins. It is known that one of these coins shows heads on
both sides, whereas the other coins are fair. One coin is selected at random and
7
tossed. If the probability that toss results in heads is , then the value of n is :
12
a. 3
b. 4
c. 5
d. none of these

102. 100 Problem
x
If (x) sin t 2dt , then ' (1) is equal to :
1/ x
a. sin 1
b. 2 sin 1
3
c. 2
sin 1
d. none of these

103. FOR SOLUTIONS VISIT WWW.VASISTA.NET