1) The document contains a past paper for UPSEE mathematics with 48 multiple choice questions covering topics like vectors, functions, probability, calculus etc. 2) The questions are single answer multiple choice with 4 options labelled a, b, c, d. 3) For each question, the relevant mathematical concept is presented along with 4 possible answers and the candidate must select the correct option.

1. UPSEE–PAST PAPER
MATHEMATICS- UNSOLVED PAPER - 2003

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
1 attains minimum value at :
x2
1 x2
a. x = 0
b. x = 4
c. x = 1
d. x = 3

4. 02 Problem
     
   a b x c b a x c
If a, b, c are the non-coplanar vectors, then the value of   
is :
  
cxa  b c a x b
a. 1
b. 2
c. 0
d. none of these

5. 03 Problem
If x –2y = 4, the minimum value of xy is :
a. - 2
b. 0
c. 0
d. -3

6. 04 Problem
If z = x + iy and 1 iz 1, the locus of z is :
z i
a. x-axis
b. y-axis
c. circle with unity radius
d. none of the above

7. 05 Problem
The vertex of an equilateral triangle is (2, -1) and the equation of its base is x + 2y
= 1, the length of its sides is :
2
a. 15
4
b.
3 3
c. 1
5
4
d. 15

8. 06 Problem
The resultant of two forces P and Q is R. If the direction of P is reversed keeping
the direction Q same, the resultant remains unaltered. The angle between P and
Q is :
a. 900
b. 600
c. 450
d. 300

9. 07 Problem
The distance s (in cm) traveled by a particle in t seconds is given by, s = t3 + 2t2 + t.
The speed of the particle after 1 s will be :
a. 2 cm/s
b. 8 cm/s
c. 6 cm/s
d. none of these

10. 08 Problem
The roots of | x – 2|2 + | x - 2| - 6 = 0 are :
a. 4, 2
b. 0, 4
c. -1, 3
d. 5, 1

11. 09 Problem
The height of a tower is 7848 cm. A particle is thrown from the top of the tower
with the horizontal velocity of 1784 cm/s. The time taken by the particle to reach
the ground is (g = 981 cm/s2) ?
a. 8 s
b. 2 s
c. 4 s
d. 8 s

12. 10 Problem
The directrix of the hyperbola is :
6
a. y
13
6
x
b. 13
9
c. y
13
9
x
d. 13

13. 2
11 Problem
5 5
The value of cos 1 cos sin 1 cos is :
3 3
10
a.
3
b. 0
c. 2
5
d. 3

14. 12 Problem
1 x 2x
If f x log , then f will be equal to :
1 x 1 x2
2f(x2)
f(x2)
2f(2x)
2f(x)

15. 13 Problem
If (1+ x – 2x2)6 = 1 + a1x + a2x2+…+ a12x12 then the value of a2 + a4 + ….+ a12, is :
a. 31
b. 32
c. 64
d. 1024

16. 14 Problem
2x3 – 6x + 5 is an increasing function, if :
a. 0 < x < 1
b. -1 < x < 1
c. x < - 1 or x > 1
1
d. -1 < x < - 2

17. 15 Problem
Two trains are 2 km apart. Their lengths are 200 m and 300 m. They are
approaching towards each other with speed of 20 m/s and 30 m/s respectively.
They will cross each other after :
a. 150 s
b. 100 s
c. 50 s
25
d. 3
s

18. 16 Problem
d 3y d 2y =1, has degree and order as :
2 1
dx 3 dx 2
a. 3, 1
b. 3, 2
c. 1, 3
d. 2, 3

19. 17 Problem
1 1
The value of I x x dx is :
0 2
a. 1
4
1
b.
2
1
c.
8
d. none of these

20. 18 Problem
4 2
If A = 3 4
, | adj A| is equal to :
a. 6
b. 16
c. 10
d. none of these

21. 19 Problem
     
ab c x a b c is equal to
  
a. [a b c ]
 
b. 3 [a b c ]
c. 0
 
d. 2 [a b c ]

22. 20 Problem
A block weighing w, is supported on an inclined surface with the help of a
horizontal force P. The same block can be supported with the help of another
force Q acting parallel to the inclined surface, then the value of is :
a. w sin
b. 1
Q
c. 1
1
d. Q2

23. 21 Problem
2
|x 1| dx is equal to :
0
a. 0
b. ½
c. 1
d. 2

24. 22 Problem
For a pack of cards two are accidently dropped. Probability that they are of
opposite shade is :
13
a.
51
1
b. 52 x 51
26
c. 51
d. none of these

25. 23 Problem
If a particle is displaced from the point A (5, -5, -7) to the point B(6, 2, -2) under
  
the influence of the forces P1 10ˆ
i j ˆ
ˆ 11k , P2 i j ˆ
4ˆ 5ˆ 6k , P3 2ˆ
i ˆ
ˆ 9k ,
j
then the work done is :
a. 87
b. 85
c. 81
d. none of these

26. 24 Problem
1
If sin x + cos x = , then tan 2x is :
5
a. 25
17
24
b. 7
c. 7
25
25
d. 7

27. 25 Problem
In a ABC, B and C . If D divides BC internally in ratio 1 : 3, then the
3 4
sin BAD
value of sin CAD
is :
1
a.
3
1
b. 6
2
c. 3
1
d. 3

28. 26 Problem
   
If | a x b | | a b | , then the angle between is :
a. π
b.2 π/3
c. π/4
d. π/2

29. 27 Problem
Let A, B and C are the angles of a triangle and A 1 B 2 C
tan , tan . Then, tan
2 3 2 3 2
is equal to :
1
a.
3
2
b.
3
2
c.
9
7
d. 9

30. 28 Problem
The value of : lim 1 x tan x
x 1 2
3
a.
4
2
b.
3
2
c.
d. 4

31. 29 Problem
x
1
If f(x) = , then the maximum value of f(x) is :
x
a. e
b. (e)1/e
e
1
c. e
d. none of these

32. 30 Problem
The volume of the solid formed by rotating the area enclosed between the
curve y = x2 and the line y = 1 about y =1 is (in cubic unit) :
9
a.
5
4
b. 3
8
c.
3
7
d.
5

33. 31 Problem
15 dx is equal to :
8
x 3 x 1
1 5
a. log
2 3
1 5
b. log
3 3
1 3
c. log
5 5
1 3
d. log
2 5

34. 32 Problem
Area of the square formed by |x| + |y| = 1 (in square unit) is :
a. 0
b. 1
c. 2
d. 4

35. 33 Problem
if x = 3 + i, then x3 – 3x2 – 8x + 15 is equal to :
a. 45
b. -15
c. 10
d. 6

36. 34 Problem
The function f(x) = log (x + x2 1 ) is :
a. Even function
b. Odd function
c. Neither even nor odd
d. Periodic function

37. 35 Problem
The perpendicular PL, PM are drawn from any point P on the rectangular
hyperbola xy = 25 to the asymptotes. The locus of the mid point of OP is curve
with eccentricity :
a. An ellipse with e = 2
b. Hyperbola with e = 2
1
c. parabola with e =
2
d. none of the above

38. 36 Problem
     
| a| | b| |c | 1 and a b c  
0, then the value of a b
   
If b c c a is :
a. 0
b. -1
3
c. 2
d. 3

39. 37 Problem
If x = logb a, y = logc b, z = loga c, then xyz is :
a. 0
b. 1
c. 3
d. none of these

40. 38 Problem
1 cos cos
The value of the determinant is :
cos 1 cos
cos cos 1
a. 0
b. 1
2 2
c.
2 2
d.

41. 39 Problem
If P(A) = P(B) = x and , then x is equal to :
1
a. 2
1
b. 4
1
c. 3
1
d.
6

42. 40 Problem
If p and q are the roots of the equation x2 + px + q = 0, then :
a. p = 1 or 0
b. p = -2 or 0
c. p = -2
d. p = 1

43. 41 Problem
If a dice is thrown twice, the probability of occurrence of 4 at least once is :
11
a. 36
35
b. 36
7
c. 26
d. none of these

44. 42 Problem
8
The value of |x 5 |dx is
0
a. 9
b. 12
c. 17
d. 18

45. 43 Problem
The value of | sin3 |d is :
0
a. 0
b. π
c. 4/3
d. 3/8

46. 44 Problem
A ball weighting 2 kg and speed 6 m/s collides with another ball of 4 kg moving
in opposite direction with speed of 3 m/s. They combine after the collision. The
speed of this combined mass (in m/s) is :
a. 4
b. 2
c. 0
d. 3

47. 45 Problem
If , , are the roots of the equation x3 + 4x +1 = 0, then
1 1 1
is equal to :
a. 2
b. 3
c. 4
d. 5

48. 46 Problem
If cosθ + cos 2 θ + cos 3 θ = 0, the general value of is :
a. 2m
4
n 2
b. m 1
3
n
c. m 1
3
d. 2m
3

49. 47 Problem
Three like parallel forces P, Q and R are acting on the vertices of a ABC whose
resultant passed through its centroid, then :
P Q R
a. b a c
P Q R
b. tan A tan B tan C
c. P = Q = R
d. None of the above

50. 48 Problem
A person observes the angle of elevation of a building as 300. The person
proceeds towards the building with a speed of 25( 3 -1) m/h. After two hours, he
observes the angel of elevation as 450. the height of the building (in m) is :
a. 50 ( 3 - 1)
b. 50( 3 + 1)
c. 50
d. 100

51. 49 Problem
x 2
The value of lim x 3 is :
x x 1
a. 0
b. 1
c. e2
d. e4

52. 50 Problem
If A + B + C =π , then cos 2A + cos 2B + cos 2C + 4 sin A sin B sin C is equal to :
a. 0
b. 1
c. 2
d. 3

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