1. UPSEE–PAST PAPERS
MATHEMATICS - UNSOLVED PAPER – 2007

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
    
If is perpendicular to b and c,| a | 2,| b | 3,| c | 4 and the angle between
  2  
b and c, is , then [a bc] is equal to
3
a. 4 3
b. 6 3
c. 12 3
3
d. 18

4. 02 Problem
The general solution y2dx + (x2 – xy + y2)dy = 0 is
x
a. tan-1 y + log y + c = 0
x
b. 2 tan-1 + log x + c = 0
y
c. log (y + x2 y 2 ) + log y + c = 0
x
d. sinh-1 y + log y + c = 0

5. 03 Problem
The vector ˆ
i ˆ
xj ˆ
3k is rotated through an angle and doubled in
magnitude, then itˆ (4x 2ˆ)
ˆ
4ibecomes j ˆ
2k. The value of x is
2
a. ,2
3
1
b. 3
,2
2
c. 3
,0
d. {2, 7}

6. 04 Problem
a ball of mass 3 kg moving with a velocity of 3 m/s collides with another ball of
mass 1 kg moving with velocity u in the opposite direction. If the first ball
2
comes to rest after the impact and e = , 7 then u is in m/s, is
a. 13
3
b. 17
3
19
c. 3
23
d.
3

7. 05 Problem
Three forces of magnitude 30, 60 and P acting at a point are in equilibrium. If
the angle between the first two is 600, the value of P is
a. 30 7
b. 30 3
c. 20
6
d. 25 2

8.  
|a b|
06 Problem
Let be two equal vectors inclined at an angle , then a sin is equal to
2
 
|a b|
a.
2
 
b. |a b|
2
 
c. |a b|
 
d. |a b|

9. 07 Problem
2
The solution of the equation d y e 2x is
2
dx
1 2x cx
a. y e d
4 2
b. y = e-2x + cx + d
c. y = e-2x + cx2 + d
d. y = e-2x + cx3 + d

10. 08 Problem
dx is equal to
2
x 4x 13
a. log (x2 + 4x + 13) + c
1 1 x 2
b. tan c
3 3
c. log (2x + 4) + c
2x 4
d. 2
c
(x 4 x 13)2

11. 09 Problem
The value of 3 x 1 is
dx
2 x 2 ( x 1)
16 1
a. log 9 6
16 1
b. log
9 6
1
c. 2 log 2- 6
d. log 4 1
3 6

12. 10 Problem
/4 5 /4 /4
(cos x sin x)dx (sin x cos x)dx (cos x sin x)dx is equal to
0 /4 2
a. 2 -2
b. 2 2 - 2
c. 3 2 - 2
d. 4 2 - 2

13. 11 Problem
The length of the chord of the parabola x2 = 4y passing through the vertex
and having slope cot is
a. 4 cos cosec2
b. 4 tan sec
c. 4 sin sec2
d. none of these

14. 12 Problem
x2 y2
Equation of tangents to the ellipse 1 , which are perpendicular to the
9 4
line 3x + 4y = 7, are
a. 4x 3y 20
b. 4x 3y 12
c. 4x 3y 2
d. 4x 3y 1

15. 13 Problem
From the point P(16,7) tangents PQ and PR are drawn to the circle
x2 + y2 – 2x – 4y – 20 = 0. If c be the centre of the circle, then area of quadrilateral
PQCR is
a. 450 sq unit
b. 15 sq unit
c. 50 sq unit
d. 75 sq unit

16. 14 Problem
The distance between the lines 3x + 4y = 9 and 6x + 8y = 15 is
3
a. 2
3
b. 10
c. 6
d. none of these

17. 15 Problem
b
If tan x = , then the value a cos 2x + b sin 2x is
a
a. a
b. a - b
c. a + b
d. b

18. 16 Problem
The maximum value of is
a. 3
b. 4
c. 5
d. none of these

19. 17 Problem
In a triangle ABC, right angled at C, the value of cot A + cot B is
a. c2
ab
b. a + b
a2
c. bc
b2
d. ac

20. 18 Problem
The records of a hospital show that 10% of the cases of a certain disease are fatal.
If 6 patients are suffering from the disease, then the probability that only three
will die, is
a. 8748 x 10-5
b. 1458 x 10-5
c. 1458 x 10-6
d. 41 x 10-6

21. 19 Problem
Out of 40 consecutive natural numbers, two are chosen at random, Probability
the that sum of the number is odd, is
14
a. 29
20
b. 39
1
c.
2
d. none of these

22. 20 Problem
If arg(z) = , then arg is equal to (z )
a.
b.
c.
d. -

23. 21 Problem
z 1
If z is a complex number such that z 1
is purely imaginary, then |z| is equal to
a. 0
b. 1
c. 2
d. none of these

24. 22 Problem
If , are the roots of the equation tx2 + mx + n = 0, then the equation 1whose 1
tan tan
3 3
roots are and , is
a. l4x2 – nl (m2 – 2nl)x + n4 = 0
b. l4x2 + nl (m2 – 2nl)x + n4 = 0
c. l4x2 + nl (m2 – 2nl)x - n4 = 0
d. l4x2 – nl (m2 + 2nl)x + n4 = 0

25. 23 Problem
1 1
If the 7th term of a HP is and the 12th term is , then the 20th term is
10 25
a. 1
41
1
b. 45
1
c. 49
1
d. 37

26. 24 Problem
The value of 21/4. 41/8. 81/16 . 161/32
3
a. 2
5
b. 2
c. 2
d. 1

27. 25 Problem
Out of 6 boys and 4 girls, a group of 7 is to be formed .In how many ways can
this be done, if the group is to have a majority of boys ?
a. 120
b. 80
c. 90
d. 100

28. 26 Problem
1
1
[2 1 -1] is equal to
2
2
1
a. 2
2 1 1
2 1 1
b. 4 2 2
c. [-1]
d. not defined

29. 27 Problem
The domain of the function 1 x2 is
sin log2
2
a. [-1, 2] – {0}
b. [-2, 2] – (-1, 1)
c. [-2, 2] – {0}
d. [1, 2]

30. 28 Problem
(2x 3)(3x 4) is equal to
lim
x (4x 5)(5x 6)
1
a. 10
b. 0
1
c. 5
3
d. 10

31. 29 Problem
x 1, x 2
function is f (x) a continuous function
2 x 3, x 2
a. for x = 2 only
b. for all real values of x such that x 2
c. for all real value of x
d. for all integral values of x only

32. 30 Problem
Differential coefficient of sec x is
1
a. sec x sin x
4 x
1
b. (sec x )3 / 2.sin x
4 x
1
c. x sec x sin x
2
1
d. x (sec x )3 / 2 sin x
2

33. 31 Problem
The function x5 – 5x4 + 5x3 - 1 is
a. Neither maximum nor minimum at x = 0
b. Maximum at x = 0
c. Maximum at x = 1 and minimum at x = 3
d. Minimum at x = 0

34. 32 Problem
dy
If 1 y2 , then dx
is equal to
a. X
1 y2
b. 1 2y 2
c. 1 y2
1 2y 2
d. 0

35. 33 Problem
If the planes x + 2y + kz = 0 and 2x + y – 2z = 0, are at right angles, then the
value of k is
a. 2
b. -2
1
c. 2
1
d. - 2

36. 34 Problem
The ratio in which the line joining (2, 4, 5), (3, 5, -4) is divided by the yz-plane is
a. 2 : 3
b. 3 : 2
c. - 2 : 3
d. 4 : - 3

37. 35 Problem
The radical centre of the circles
x2 + y2 – 16x + 60 = 0,
x2 + y2 – 12x + 27 = 0 and x2 + y2 – 12y + 8 = 0 is
33
13,
a. 4
33
b. , 13
4
c. 33
,13
4
d. none of these

38. 36 Problem
If the lines 3x + 4y + 1 = 0, 5x + y + 3 = 0 and 2x + y –1 = 0 are concurrent, then
is equal to
a. - 8
b. 8
c. 4
d. -4

39. 37 Problem
ax / 2 dx is equal to
x
a ax
1
a. sin 1 ax c
log a
1
b. tan 1 ax c
log a
c. 2 a x
ax c
d. log (ax - 1) + c

40. 38 Problem
The value of 1 x4 1 is
dx
0 x2 1
1
a. 6 (3 - 4 )
1
b. 6 (3 + 4)
1
c. 6 (3 + 4 )
1
d. 6 (3 - 4)

41. 39 Problem
dy
The solution of the differential equation dx
= y tan x – 2 sin x, is
a. y sin x = c + sin 2x
1
b. y cos x = c + 2
sin 2x
c. y cos x = c – sin 2x
1
d. y cos x = c + cos 2x
2

42. 40 Problem
 
If a 1 1 and b
, ( 2, m) are two collinear vectors, then m is equal to
a. 2
b. 4
c. 3
d. 0

43. 41 Problem
A ball falls from rest from top of a tower. If the ball reaches the foot of the
tower in 3 s, then height of tower is (take g = 10 m/s2)
a. 45 m
b. 50 m
c. 40 m
d. none of these

44. 42 Problem
A particle is thrown vertically upwards with a velocity of 490 cm/s. It will return
to this position after
a. 1 s
b. 0.5 s
c. 2 s
d. none of these

45. 43 Problem
The resultant of two forces P and 2P . 3 If 1st forces is doubled and reversed. The
resultant of forces is
a. 2P 3
b. P 3
c. 4P
d. none of these

46. 44 Problem
2
The value of 1 – log 2 + (log2) (log2)3 + …is
2! 3!
a. log 3
b. log 2
c. 1/2
d. None of these

47. 45 Problem
x
The maximum f ( x) on [-1, 1] is
4 x x2
a. - 1
3
1
b. - 4
1
c. 5
d. 1
6

48. 46 Problem
ex
dx is equal to
(2 e x )(e x 1)
ex 1
a. log +c
ex 2
ex 2
log
b. ex 1 +c
ex 1
c. ex 2 +c
ex 2
d. ex 1
+c

49. 47 Problem
If the radius of a circle be increasing at a uniform rate of 2 cm/s. The area of
increasing of area of circle, at the instant when the radius is 20 cm, is
a. 70 cm2/s
b. 70 cm2/s
c. 80 cm2/s
d. 80 cm2/s

50. 48 Problem
5 boys and 5 girls are sitting in a row randomly. The probability that boys and
girls sit alternatively, is
a. 5
126
1
b. 42
4
c. 126
1
d. 126

51. 49 Problem
1
If P(A) = P(B) = x and P(A B) = P(A’ B’) = , then x is equal to
3
1
a.
2
1
b. 3
1
c.
4
1
d. 6

52. 50 Problem
The focus of parabola y2 – x – 2y + 2 = 0 is
1
a. ,0
4
b. (1, 2)
5
,1
c. 4
3 5
,
d. 4 2

53. 51 Problem
the equation of normal at the point (0, 3) of the ellipse 9x2 + 5y2 = 45 is
a. x-axis
b. y-axis
c. y + 3 = 0
d. y – 3 = 0

54. 52 Problem
x2 y2
The equation of the tangent parallel to y – x +5 = 0 drawn to 1 is
3 2
a. x – y + 1 = 0
b. x – y + 2 = 0
c. x + y - 1 = 0
d. x + y + 2 = 0

55. 53 Problem
Let the functions f, g, h are defined from the set of real numbers R to R such
that f(x) = x2 – 1, g (x) = (x2 1) and
0, if x 0
h(x) = x, if x 0
, then ho(fog)(x) is defined by
a. x
b. x2
c. 0
d. none of these

56. 54 Problem
The angle of elevation of the sun, if the length of the shadow of a tower is
times the height of the pole, is
a. 1500
b. 300
c. 600
d. 450

57. 55 Problem
If sin A = n sin B, then n 1 A B is equal to
tan
n 1 2
A B
a. sin
2
A B
b. tan
2
c. cot A B
2
d. none of these

58. 56 Problem
Both the roots of the given equation (x - a) (x - b) + (x - b) (x - c) + (x - c) (x - a)
= 0 are always
a. Positive
b. Negative
c. Real
d. Imaginary

59. 57 Problem
3 tan-1 a is equal to
3a a3
a. tan-1 1 3a2
3a a3
b. tan-1 1 3a2
3a a3
c. tan-1 1 3a2
3a a3
d. tan-1 1 3a2

60. 58 Problem
1 2i
In which quadrant of the complex plane, the point lies ?
1 i
a. Fourth
b. First
c. Second
d. Third

61. 59 Problem
The argument of the complex number 13
4
5i
9i
is
a.
3
b.
4
c.
5
d. 6

62. 60 Problem
If sin and cos are the roots of the equation px2 + qx + r = 0, then
a. p2 + q2 – 2pr = 0
b. p2 - q2 + 2pr = 0
c. p2 - q2 – 2pr = 0
d. p2 + q2 + 2pr = 0

63. 61 Problem
if a, b, c are in GP, then the equations ax2 + 2bx + c = 0 and dx2 + 2ex + f = 0 have
a common root, if d e f are in
, ,
a b c
a. AP
b. GP
c. HP
d. None of these

64. 62 Problem
The sum of 24 terms of the following series 2 8 18 32 .... is
a. 300
b. 300 2
c. 200 2
d. none of these

65. 63 Problem
x = 0, the function f(x) = |x| is
a. continuous but not differentiable
b. discontinuous and differentiable
c. discontinuous and not differentiable
d. continuous and differentiable

66. 64 Problem
12
In the expansion of 2 1 , the term independent of x is
2x
x
a. 8th
b. 7th
c. 9th
d. 10th

67. 65 Problem
The general value of in the equation cos 1 is
, tan 1
2
a. 2n ,n I
6
7
b. 2n ,n I
4
c. n ( 1)n ,n I
3
d. n ( 1)n ,n I
4

68. 66 Problem
In a ABC, if r1 = 2r2 = 3r3, then
a 4
a.
b 5
b. a 5
b 4
c. a + b – 2c = 0
d. 2a = b + c

69. 67 Problem
1 2
If A = , then A-1 is equal to
3 5
5 2
a. 3 1
5 / 11 2 / 11
b. 3 / 11 1 / 11
5 / 11 2 / 11
c. 3 / 11 1 / 11
5 2
d. 3 1

70. 68 Problem
The range of f(x) = cos x – sin x is
a. [-1, 1]
b. (-1, 2)
c. ,
2 2
d. [- 2 , 2]

71. 69 Problem
the value of x2 bx 4 is
lim
x x2 ax 5
b
a. a
b. 0
c. 1
4
d. 5

72. 70 Problem
sin x
Let x 0 if f(x) is continuous at x = 0, then k is equal
f (x) 5x ,
k, x 0
to
a. 5
b. 5
c. 1
d. 0

73. 71 Problem
 
If be the angle between the vectors a 2ˆ
i 2ˆ
j ˆ
k and b 6ˆ 3ˆ
i j ˆ
2k
, then
a. 4
cos
21
3
b. cos
19
2
c. cos
19
5
d. cos
21

74. 72 Problem
  
Let a, b and c be vectors with magnitudes 3, 4 and 5 respectively and
     
a b c 0 then    
, the values of a b b c c a is
a. 47
b. 25
c. 50
d. -25

75. 73 Problem
A particle is moving in a straight line with uniform velocity. Its acceleration is
a. Positive
b. Negative
c. Zero
d. None of these

76. 74 Problem
If a particle is projected with a velocity 49 m/s making an angle 600 with the
horizontal, its time of flight is
a. 10 3
b. 5 3 s
c. 3 s
d. none of these

77. 75 Problem
If two equal components (P) of a force (F) are acting at an angle 1200, then F is
a. P
b. P 2
c. 2P
P
d.
2

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