The document contains 30 multiple choice questions from a past UPSEE mathematics exam. The questions cover a range of topics including: [1] calculating time taken to cross a canal based on speed and direction of flow; [2] determining the derivative of an exponential function; [3] finding the velocity of a particle with given acceleration over time.] The full document provides the questions and multiple choice answers but no solutions.

1. UPSEE–PAST PAPERS
MATHEMATICS - UNSOLVED PAPER – 2001

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
A man swims at a speed of 5 km/h. He wants to cross a canal of 120 m
wide, in a direction perpendicular to the direction of flow. If the canal flows
at 4 km/h, the direction and the time taken by the man to cross the canal
are :
1 1
tan , 24 min
2
a. 1 3
tan ,144 s
4
b. 1 1
tan ,100 s
2
c.
d. none of these

4. 02 Problem
y ey ey .....
If x ey e , then
dy
dx
is equal to :
1
a.
x
1 x
b. x
c. x
1 x
d. None of these

5. 03 Problem
The acceleration of a particle moving in a straight line, a time t is (2t + 1) m/s2. If
4 m/s is the initial velocity of the particle, then its velocity after 2 s is :
a. 4 m/s
b. 8 m/s
c. 10 m/s
d. none of these

6. 04 Problem
a b c 2a 2a
If a + b + c = 0, then determinant 2b b c a 2b is equal to :
2c 2c c a b
a. 0
b. 1
c. 2
d. 3

7. 05 Problem
A body of weight 40 kg rests on a rough horizontal plane, whose coefficient of
friction is 0.25. The least force which is acting horizontally would move the
body of :
a. 40 kg wt
b. 20 kg wt
c. 35 kg wt
d. 10 kg wt

8. 06 Problem
If log a log log c then aabbcc is equal to :
b c c a a b
a. -1
b. 1
c. 2
d. none of these

9. 07 Problem
The co-ordinates of a point on the parabola y2 = 8x whose focal distance is
4, is :
a. (2, 4)
b. (4, 2)
c. (4, -2)
d. (2, 4)

10. 08 Problem
Three letters are written to different persons and addresses on three envelopes
are also written. Without looking at the addresses, the probability that the
letters to into the right envelope is :
2
a.
3
1
b.
28
1
c. 27
1
d. 9

11. 09 Problem
The subtangent, ordinate and subnormal to the parabola y2 = 4ax at a point
different from the origin are in :
a. GP
b. AP
c. HP
d. None of these

12. 10 Problem
    
If a 3ˆ
i j ˆ
ˆ 2k and b 2ˆ
i ˆ
j ˆ
k , then a x (a b) is equal to

a. 3 a
b. 0
c. 3 14
d. none of these

13. 11 Problem
tan x
Evaluate dx :
sin x cos x
a. cot x c
b. 2 cot x c
c. tan x c
d. 2 tan x c

14. 12 Problem
Two cars start off to race with velocities u, u’ and move with uniform acceleration
f, f’; the result being a dead heat. The time taken by cars is :
5 f f'
a. u ' f uf '
u' u
b. 2
f f '
5 f f'
c.
u'f uf '
u' u
d. 2
f f'

15. 13 Problem
The difference between the greatest and least values of the function
x
x t 1 dt on [2, 3] is :
0
a. 3
b. 2
7
c. 2
11
d. 2

16. 14 Problem
If the circle x2 + y2 = a2 intersects the hyperbola xy = c2 in four points
P(x1, y1), Q(x2, y2), R(x3, y3), S(x4, y4) then :
a. x1 + x2 + x3 + x4 = 1
b. y1 + y2 + y3 + y4 = 0
c. x1x2x3x4 = c3
d. y1y2y3y4 = c3

17. 15 Problem
The vector moment about the point ˆ
i 2ˆ
j ˆ
3k of the resultant of the
force ˆ
i 2ˆ
j ˆ
5k and 3ˆ
j ˆ
4k acting at the point ˆ is :
2ˆ
i 3ˆ
j k
5ˆ
i ˆ
j ˆ
4k
a.
b. 3ˆ
i ˆ
j ˆ
4k
c. 5ˆ
i ˆ
j ˆ
4k
d. none of these

18. 16 Problem
x dx is equal to :
1 x4
a. tan-1 x2 + c
b. log (1 + x4) + c
1
c. 2
tan-1 x2 + c
d. none of these

19. 17 Problem
If the intercept made on the line y = mx by lines y = 2 and y = 6 is less than by
5, then the range of the value of m is :
4 4
,
a. 3 3
4 4
b. , ,
3 3
c. 3 3
,
4 4
d. none of the above

20. 18 Problem
The equation of the tangent to the curve y = e-|x| at the point where the curve
cuts the line x = 1 is :
a. e(x + y) = 1
b. y + 2x = 1
c. y + x = e
d. none of these

21. 19 Problem
Let z = 1 – t + i t2 t 2, where t is a real parameter. The locus of z in the
argand plane is :
a. an ellipse
b. a hyperbola
c. a straight line
d. none of these

22. 20 Problem
Let z = 1- t + i t2 t 2, where t is a real parameter. The locus of z in the
argand plane is :
a. An ellipse
b. A hyperbola
c. A straight line
d. None of these

23. 21 Problem
Differential equation for y A cos x B sin x where A and B are arbitrary
constants, is :
a. d2y 2
y 0
dx 2
b. d2y
y 0
dx 2
d2y
c. y 0
dx 2
d2y 2
d. y 0
dx 2

24. 22 Problem
A billiard ball collides directly with another ball of same mass having in rest. If
the coefficient of restitution is e, then ratio of their velocities will be :
a. 2 – e : 2 + e
b. 1 – e : 1 + e
c. 1 – e2 : 1 + e2
e e
d. :
1 e 1 e

25. 23 Problem
From the gun cartridge of mass M, a fire arm of mass m with velocity u relative
to gun cartridge is fired. The real velocities of fire arms and gun cartridge will
be respectively :
Mm mu
,
a. M m M m
M m M m
b. ,
Mu Mu
c. u M m u M m
,
M M
d. none of the above

26. 24 Problem
Is the equation (ab + ca + bc) sin = 2(a2 + b2 + c2) possible for real values of a, b,
c?
a. Possible
b. Not possible
c. Insufficient data
d. None of these

27. 25 Problem
The equation 3 sin2 x + 10 cos x – 6 = 0 is satisfied, if :
1 1
a. x n cos
3
1 1
b. x 2n cos
3
1 1
c. x n cos
3
1 1
d. x 2n cos
6

28. 26 Problem
A train whose mass is 6 metric tons, moves at the rate of 72 km/h. After
applying brakes at stops at a distance of 500 m. What is the force exerted by
brakes, obtaining it to be uniform ?
a. 800 N
b. 1600 N
c. 3200 N
d. 6400 N

29. 27 Problem
Six girls are entering in a dance room with 10 boys to form a circle so that every
girl is in between two boys, then the probability of doing so, such that two
specified boy remains together, is :
a. 4
15
b. 7
15
c. 2
15
d. none of these

30. 28 Problem
2 n 1
If 1, , 2
,......, n 1
are the n roots of unity, then : 1 1 ..... 1
equals :
a. 0
b. 1
c. n
d. n2

31. 29 Problem
The number of common tangent to the circles (x + 1)2 + (y + 4)2 = 40 and (x - 2)2
+ (y- 5)2 = 10 are :
a. 1
b. 2
c. 3
d. 4

32. 30 Problem
If f(x) = xx, then f(x) is decreasing in interval :
a. ] 0, e[
1
b. ]0, [
e
c. ]0, 1[
d. none of these

33. 31 Problem
The angle between the vectors 2ˆ
i 3ˆ
j ˆ
k and 2ˆ
i ˆ
j ˆ
k is :
a. 2
b.
4
c.
3
d. 0

34. 32 Problem
A man falls vertically under gravity with a box of mass m on his head then the
reaction force is :
a. mg
b. 2 mg
c. zero
d. 1.5 mg

35. 33 Problem
100
The value of [ x ] dx is equal to : (where [.] is the greatest integer)
0
a. 400
b. 600
c. 415
d. 615

36. 34 Problem
5 9 13 .....n terms 17
If , then n is equal to :
7 9 11 .....(n 1)terms 16
a. 7
b. 12
c. 8
d. none of these

37. 35 Problem
1 a a2
The value of the determinant cos n 1 x cos nx cos n 1 x
is zero, if :
sin n 1 x sin nx sin n 1 x
a. sin x = 0
b. cos x = 0
c. a = 0
1 a2
d. cos x
2a

38. 36 Problem
If in a triangle ABC, sin A, sin B, sin C are in AP, then :
a. The altitudes are in AP
b. The altitudes are in HP
c. The altitudes are in GP
d. None of the above

39. 37 Problem
A variable chord is drawn through the origin to the circle x2 + y2 –2ax = 0. The
locus of the centre of the circle drawn on this chordas diameter is :
a. x2 + y2 + ax = 0
b. x2 + y2 + ay = 0
c. x2 + y2 - ax = 0
d. x2 + y2 - ay = 0

40. 38 Problem
The straight lines x + y = 0, 3x + y – 4 = 0 and x + 3y – 4 = 0 from a triangle
which is :
a. Isosceles
b. Right angled
c. Equilateral
d. None of these

41. 39 Problem
  
For three vectors u, v, w which of the following expression is not equal to
any of the remaining three ?
  
a. u (v x w)
  
b. (v x w) u
c.   
v (u x w)
 
d. (u x v) u

42. 40 Problem
[f ( x )g ''( x ) f ''( x )g( x )]dx is equal to :
f (x)
a. g '(x)
b. f’(x) g(x) –f(x)g’(x)
c. f(x)g’(x) – f’(x)g(x)
d. f(x)g’(x) + f(x)g(x)

43. 41 Problem
xe x is equal to :
2
dx
1 x
ex
a. c
x 1
b. ex(x + 1) + c
ex
c
c. (x 1)2
ex
d. c
1 x2

44. 42 Problem
The top of a hill observed from the top and bottom of a building of height h is at
angles of elevation p and q respectively. The height of the hill is :
h cos q
a. cot q cot p
h cot p
b. cot p cot q
h tan p
c. tan p tan q
d. none of these

45. 43 Problem
The probability that in a random arrangement of the letters of the word
‘UNIVERSITY’, the two I’s do not come together is :
4
a.
5
b. 1
5
1
c.
10
9
d. 10

46. 44 Problem
     
If a and b are two vectors such that , a x b a b 0 then :
a. A is equal to zero
b. B is equal to zero
c. Either a or b is zero
d. Both a and b are necessarily zero

47. 45 Problem
2 2
If (1 + i) (1 - 2i) (1 - 3i)…. (1 - ni) = i then equals :
a. 1 . 2 . 3 ……..n
b. 12. 22 . 32 …. n2
c. 12 + 22 + 32 + … n2
d. 2 .5 . 10 ….. (n2 + 1)

48. 46 Problem
If f(x) = ax + b and g(x) = cx + d, then f[g(x)] = f[f(x)] is equivalent to :
a. f(a) = g(c)
b. f(b) = g(b)
c. f(d) = g (b)
d. f(c) = g (a)

49. 47 Problem
A rough plane is inclined at an angle to the horizon. A body is just to slide due
to its own weight. The angle of friction would be :
a. tan-1
b.
c. tan
d. 2

50. 48 Problem
If a , then f(x) has maximum value at x = 3, then :
f (x) x2
x
a. a < -27
b. a > -27
c. a > 27
d. a < 27

51. 49 Problem
If a + b + c = 0, |a| = 3, |b| = 5, |c| = 7 then the angle between a and b is
equal to :
a.
6
2
b. 3
5
c. 3
d. 3

52. 50 Problem
A bag contains 4 red, 6 white and 5 black balls. 2 balls are drawn at random.
Find the probability of getting one red and one white ball is :
2
a. 3
b. 4
35
c. 15
10
8
d.
35

53. 51 Problem
5 5 3
In two events P( A B)
6
P( A)
6
,P B
2
then A and B are :
a. Independent
b. Mutually exclusive
c. Mutually exhaustive
d. Dependent

54. 52 Problem
If P(not A) = 0.7, P(B) = 0.7 and P(B/A) = 0.5, then P(A/B) equals :
3
a.
13
b. 3
14
c. 1
12
d. none of these

55. 53 Problem
p q y r z p q r
If = 0, then the value of x y z
is :
p x q r z
p x q y r
a. 0
b. 1
c. 2
d. 4pqr

56. 54 Problem
x m
Let f :R R be a function defined by f x , where m n, then :
x n
a. f is one-one onto
b. f is one-one into
c. f is many-one onto
d. f is many-one into

57. 55 Problem
1.3 1.3.5
The sum of the series 1 ... is :
6 6.8
a. 1
b. 0
c.
d. 4

58. 56 Problem
The locus of the pole of normal chords of an ellipse is given by :
a6 b6 2
a. a2 b2
x2 y2
a3 b3 2
a2 b2
b. x2 y2
a6 b6 2
c. a2 b2
x2 y2
a3 b3 2
a2 b2
d. x2 y2

59. 57 Problem
A body is projected through an angle from vertical so that its range is half of
maximum range. Value of is :
a. 600
b. 750
c. 300
d. 22.40

60. 58 Problem
The sun of the magnitudes of two forces acting at a point is 18 and
magnitudes of their resultant is 12. If the resultant is at 900 with the force of
smaller magnitude, then their magnitudes are :
a. 3, 15
b. 4, 14
c. 5, 13
d. 6, 12

61. 59 Problem
To be semigroup the elements of a subset of a group must obey the axioms
of :
a. Associativity and commutativity
b. Closure and identity
c. Closure and associativity
d. Closure and inverse

62. 60 Problem
Let A and B be two events such that, 5 1 1 then :
P A B ,P A B and P A
6 3 2
a. P(B) P (A)
b. P(A) = P(B)
c. A and B are independent
d. A and B are mutually exclusive

63. 61 Problem
2
If z = z 3 i 5 , then the locus of z is a :
a. Circle
b. Hyperbola
c. Parabola
d. None of these

64. 62 Problem
Let a > 0, b > 0 and c > 0. Then both the roots of the equation ax2 + bx + c = 0
a. Are real and negative
b. Have negative real parts
c. Are rational numbers
d. None of the above

65. 63 Problem
If 1 1 1 + …. upto n terms, then
y tan 1 tan 1 tan 1
1 x x2 x 2
3x 3 x 2
5x 7
y’(0) is equal to :
1
a. 1 n2
n2
b. 1 n2
n
c. 1 n2
d. none of these

66. 64 Problem
Equation of the tangent to the hyperbola 2x2 – 3y2 = 6 which is parallel to the
line y = 3x + 4 is :
a. y = 3x + 5
b. y = 3x – 5
c. y = 3x + 5 and y = 3x – 5
d. none of the above

67. 65 Problem
1 2x 1 2x
Differential coefficient of tan
1 x2
with respect to sin
1 x2
will be :
a. 1
b. -1
c. -1/2
d. x

68. 66 Problem
A particle is moving in a straight line with constant acceleration a. If x is the
space described in t seconds and x’ is the space described during next t’
seconds, then a is equal to :
2 x x'
a. t t' t t'
2 x' x
b. t t' t' t
2 x' x
c. t t' t' t
2 x' x
d. t t' t' t

69. 67 Problem
The numbers P, Q and R for which the function f(x) = Pe2x + Qex + Rx satisfies the
conditions f(0) = -1, f’ (log 2) = 31 and log 4 [f (x ) Rx]dx 39 are given by :
0 2
a. P = 2, Q = -3, R = 4
b. P = -5, Q = 2, R = 3
c. P = 5, Q = -2, R = 3
d. P = 5, Q = -6, R = 3

70. 68 Problem
2
ex cos x is equal to :
lim
x 0 x2
3
a. 2
b. 1
2
c. 2
3
d. none of these

71. 69 Problem
(x 2 x 6)2
lim is :
x 2 (x 2)2
a. 6
b. 25
c. 9
d. 16

72. 70 Problem
If in a triangle ABC, B 600 , then :
a. (a - b)2 = c2 – ab
b. (b - c)2 = a2 – bc
c. (c - a)2 = b2 – ac
d. a2 + b2 + c2 = 2b2 – ac

73. 71 Problem
10
The coefficient of the term independent of x in the expansion of x 3 is
3 2x 2
:
5
a. 4
7
b. 4
c. 9
4
d. none of these

74. 72 Problem
A set contains (2n +1) elements. The number of subsets of the set which
contain at most n element, is :
a. 2n
b. 2n+1
c. 2n-1
d. 22n

75. 73 Problem
A unit vector perpendicular to the vector 4ˆ
i ˆ
j ˆ
3k and 2ˆ
i ˆ
j ˆ
2k is :
1 ˆ ˆ
a. (i 2ˆ
j 2k )
3
1 ˆ ˆ
b. ( i 2ˆ
j 2k )
3
1 ˆ ˆ
c. (2i 2ˆ
j 2k)
3
1 ˆ ˆ
(2i 2ˆ
j 2k )
d. 3

76. 74 Problem
The radius of the incircle of a triangle whose sides are 18, 24 and 30 cms, is :
a. 2 cm
b. 4 cm
c. 6 cm
d. 9 cm

77. 75 Problem
The area in the first quadrant bound by y = 4x2, x = 0, y = 1 and y = 4 is :
7
a. sq unit
3
4
b. sq unit
5
c. 3 sq unit
4
d. none of these

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