UPSEE - Mathematics -2004 Unsolved Paper

UPSEE–PAST PAPER
MATHEMATICS- UNSOLVED PAPER - 2004

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

01 Problem
200
j
The coefficient of x100 in the expansion of 1 x is :
j 0

200
a. 100

201
b. 102

200
c. 101

201
d. 100

02 Problem

The circles x2 + y2 – 10x + 16 = 0 and x2 + y2 = r2 intersect each other at two
distinct points, if :

a. r < 2
b. r > 8
c. 2 < r < 8
d. 2 r 8

03 Problem

Three numbers are in AP such that their sum is 18 and sum of their equares is
158. The greatest number among them is :

a. 10
b. 11
c. 12
d. none of these

04 Problem

Let    
be three vectors. Then, scalar triple product [a b c] is equal to :
a, b and c


a. [ b a c]


b. [a c b]

c. 
[ c b a]


d. [b c a]

05 Problem

The roots of the equation x4 – 2x3 + x = 380 are :

a. 5, - 4,1 5 3
2

1 5 3
b. -5, 4, - 2

1 5 3
c. 5, 4, 2

1 5 3
d. -5, -4, 2

06 Problem

Let y x .... dy is equal to :
xx , then
dx

a. yxy-1

y2
b. x 1 y log x

y
c.
x 1 y log x

d. none of these

07 Problem
3
1
tan x is equal to :
dx
1 x2

a. 3 (tan-1 x)2 + c

4
b. tan 1 x
c
4

c. (tan-1 x)4 + c
d. none of these

08 Problem

‘X’ speaks truth in 60% and ‘Y’ in 50% of the cases. The probability that they
contradict each other narrating the same incident is :

a. ¼
b. 1/3
c. ½
d. 2/3

09 Problem

A set contains 2n + 1 elements. The number of subsets of this set containing
more than n elements is equal to :

a. 2n-1
b. 2n
c. 2n+1
d. 22n

1
3

10 Problem

The area between the parabola y = x2 and the line y = x is :

1
a. 6
sq unit
1
b. sq unit
3
1
c. 2
sq unit
d. none of these

11 Problem

The eccentricity of the hyperbola 5x2 – 4y2 + 20 x + 8y = 4 is :

a. 2
3
b.
2
c. 2
d. 3

12 Problem
e x esin x is equal to :
lim
x 0 x sin x

a. -1
b. 0
c. 1
d. none of these

13 Problem
x dx is equal to :
0 1 sin x

a. - π
b. π/2
c. π
d. none of these

14 Problem

A man of mass 80 kg is traveling in a lift. The reaction between the floor of the lift
and the man when the lift is accelerating upwards at 4 m/s2 and the acceleration
due to gravity g = 9.81 m/s2, is equal to :

a. 884.8 N
b. 784.8 N
c. 464 N
d. 1104.8 N

15 Problem

The argument of 1 i 3 / 1 i 3 is :

a. 600
b. 1200
c. 2100
d. 2400

16 Problem

The points z1, z2, z3, z4 in a complex plane are vertices of a parallelogram taken in
order, then :

a. z1 + z4 = z2 + z3
b. z1 + z3 = z2 + z4
c. z1 + z2 = z3 + z4
d. none of these

17 Problem
2
The harmonic mean between two numbers is 14 5 and the geometric mean is
24. The greater number between them is :

a. 72
b. 54
c. 36
d. none of these

18 Problem

The angle between two forces each equal to P when their resultant is also equal
to P is :

a. 2π/3
b. π/3
c. π
d. π/2

19 Problem

The solution of the differential equation sec2 x and y dx + sec2y tan x dy = 0 is :

a. tan y tan x = c
tan y
b. tan x c

tan2 x
c. c
tan y
d. none of these

20 Problem

The real roots of the equation x2/3 + x1/3 – 2 = 0 are :

a. 1, 8
b. - 1, -8
c. - 1, 8
d. 1, -8

21 Problem

Let f (x) = g(x) = ex. Then, (gof)’(0) is :

a. 1
b. -1
c. 0
d. none of these

22 Problem

Cosine of the angle between two diagonals of a cube is equal to :

2
a. 6

b. 1
3

1
c.
2

d. none of these

23 Problem

In a certain population 10% of the people are rich, 5% are famous and 3% are rich
and famous. The probability that a person picked at random from the population
is either famous or rich but not both, is equal to :

a. 0.7
b. 0.08
c. 0.09
d. 0.12

24 Problem

Three numbers are in GP such that their sum is 38 and their product is 1728. The
greatest number among them is :

a. 18
b. 16
c. 14
d. none of these

25 Problem

The equation of the circle touching x = 0, y = 0 and x = 4 is :

a. x2 + y2 – 4x – 4y + 16 = 0
b. x2 + y2 – 8x – 8y + 16 = 0
c. x2 + y2 + 4x + 4y - 4 = 0
d. x2 + y2 – 4x – 4y + 4 = 0

26 Problem
1, when x is rational
Let f(x) = f x then lim f (x ) is :
0, when x is irraitonal ' x 0

a. 0
b. 1
c. 1/2
d. none of these

27 Problem
  
   | a | 4,| b | 4,|c | 2 and 
a, b and c are three vectors with magnitude such that a
      
is perpendicular a b c ,b is perpendicular to (c a) and c is perpendicular to
    
(a b) . It follows that |a b c | is equal to :

a. 9
b. 6
c. 5
d. 4

28 Problem

Let z1 and z2 be complex numbers, then |z1 + z2|2 + |z1 – z2|2 is equal to :

a. |z1|2 + |z2|2
b. 2 (|z1|2 + |z2|2)
c. 2(z12 + z22)
d. 4z1z2

29 Problem
2
If tan tan
3
tan
3
3, then :

tan 2θ = 1
tan 3 θ = 1
tan2 θ = 1
tan3 θ = 1

30 Problem
d2 y
Let y = t10 + 1 and x = t8 + 1, then 2 is equal to :
dx

5
a. 2 t
b. 20t8
5
c.
16t 6
d. none of these

31 Problem
 
The vectors AB i j ˆ
3ˆ 5ˆ 4k and AC i j ˆ
5ˆ 5ˆ 2k are the side of a triangle ABC.
The length of the median through A is :

a. 13 unit
b. 2 5 unit

c. 5 unit
d. 10 unit

32 Problem
         
If a, b, c are three non-coplanar vectors, then (a b c).[(a b) x(a c)] is

a. 0

b. 2 [a b c]

c. – [a b c]

d. [a b c]

33 Problem
dx is equal to :
x( x 5 1)

1 5 5
a. 5 log x (x 1) c

1 x5 1
log c
b. 5 x5

1 x5
c. log 5 c
5 x 1

d. none of the above

34 Problem

A function f on R into itself is continuous at a point a in R, iff for each > 0, there
exists, > 0 such that :

a. | f (x) f (a) | |x a|
b. | f (x) f (a) | |x a|
c. | x a| | f (x) f (a) |
d. | x a| | f (x) f (a) |

35 Problem

x sin x is equal to :
dx
1 cos x

x
a. x tan +c
2
x
b. x sec2 2 + c
x
c. log cos
2
d. none of these

36 Problem

A straight line through the point (1, 1) meets the x-axis at ‘A’ and the y-axis at ‘B’.
The locus of the mid point of AB is :

a. 2xy + x + y = 0
b. x + y – 2xy = 0
c. x + y + 2 = 0
d. x + y – 2 = 0

37 Problem
2 4 5
If A 4 8 10 , then rank of A is equal to :
6 12 15

a. 0
b. 1
c. 2
d. 3

38 Problem

A bag contains 8 red and 7 black balls. Two balls are drawn at random. The
probability that both the balls are of the same colour, is :

14
a.
15

11
b. 15

c. 7
15

4
d.
15

39 Problem
2 x2 y2 1
If sin , then x must be :
2x

a. - 3
b. - 2
c. 1
d. none of these

40 Problem

The solution of equation cos2 θ + sin θ + 1 = 0 lies in the interval :

a. ,
4 4

b. 3
,
4 4

c. 3 , 5
4 4

5 7
d. ,
4 4

41 Problem

Coefficient of x19 in the polynomial (x –1) (x - 2) ……..(x - 20) is equal to :

a. 210
b. - 210
c. 20!
d. None of these

42 Problem

Two pillars of equal height stand on either side of a road way which is 60 m wide.
The a point in the road way between the pillars, the elevation of the top of pillars
are 600 and 300. The height of the pillars is :

a. 15 3m
15
b. 3 m
c. 15 m
d. 20 m

43 Problem

A light string passing over a light smooth pulley carries masses of 3 kg and 5kg at
its ends. If the string is allowed to move from the rest, the acceleration of the
motion is equal to :

a. (g/2)m/s2
b. (g/4)m/s2
c. 2g m/s2
d. 4g m/s2

44 Problem

The equation of the directrix of the parabola x2 + 8y – 2x = 7 is :

a. y = 3
b. y = -3
c. y = 2
d. y = 0

45 Problem

If iz4 + 1 = 0, the z can take the value :

1 i
a.
2

b. cos i sin
8 8

1
c. 4i

d. i

46 Problem
  
If a i j ˆ
ˆ ˆ k, b i j ˆ
2ˆ 3ˆ k, and c ˆ
i ˆ
j are coplanar vectors, the value of a is :

a. - 4
3

3
b. 4

4
c. 3

d. 2

47 Problem

The equation of the tangent parallel to y – x + 5 = 0 drawn to

a. x – y – 1= 0
b. x – y + 2 = 0
c. x + y – 1 = 0
d. x + y + 2 = 0

48 Problem

The equation y2 – x2 + 2x – 1 = 0 represents :

a. A hyperbola
b. An ellipse
c. A pair of straight lines
d. A rectangular hyperbola

49 Problem

The minimum value of 3 sin θ + 4 cos θ is :

a. 5
b. 1
c. 3
d. - 5

50 Problem

A man in swimming with the uniform velocity of 6 km/h straight across a river
which is flowing at the rate of 2 km/h. If the breadth of the river is 300 m, the
distance between the point and the man is initially directed to and the point it
will reach on the opposite bank of the river is equal to :

a. 100 m
b. 200 m
c. 300 m
d. 400 m

51 Problem

A ball is thrown vertically upwards from the ground with velocity 15 m/s and
rebounds from the ground with one-third of its striking velocity. The ratio of its
greatest heights before and after striking the ground is equal to :

a. 4 : 1
b. 9 : 1
c. 5 : 1
d. 3 : 1

52 Problem

If the position vectors of the vertices A, B, C of a triangle ABC are
j ˆ i j ˆ
7ˆ 10k, ˆ 6ˆ 6k and i j ˆ
4ˆ 9ˆ 6k respectively, the triangle is :

a. Equilateral
b. Isosceles
c. Scalene
d. Right angled and isosceles also

53 Problem

The number of solutions of the equation 2 cos (ex) = 5x + 5-x, are :

a. No solution
b. One solution
c. Two solution
d. Infinitely many solutions

54 Problem

Probability of throwing 16 in one throw with three dice is ;

a. 1
36

1
b.
18

c. 1
72

1
d.
9

55 Problem

The differential equation of all straight lines passing through origin is :

dy
a. y x
dx

dy
b. dx
=y+x

dy
c. dx
=y–x

d. Nome of these

56 Problem
    
  
Ifa, b, c are three unit vectors such that a b c 0 where 0 is null
     
vector, thenb c c a
a b is :

a. - 3
b. - 2
3
c. - 2

d. 0

57 Problem

The expression equal to :

a. -1
b. 0
c. 1
d. none of these

58 Problem

is equal to :

a.
4

b.
6

c. 3

2
d.
3

59 Problem

If f(x) = (a - xn)1/n, where a > 0 and n N, then f0f(x) is equal to :

a. a
b. x
c. xn
d. an

60 Problem

The number of reflexive relations of a set with four elements is equal to :

a. 216
b. 212
c. 28
d. 24

61 Problem

The maximum horizontal range of a ball projected with a velocity of 39.2 m/s is
(take g = 9.8 m/s2)

a. 100 m
b. 127 m
c. 157 m
d. 177 m

62 Problem

Maximum value of sin x – cos x is equal to :

a. 2

b. 1
c. 0
d. none of these

63 Problem

The equation of the bisector of the acute angles between the lines 3x – 4y + 7 = 0
and 12x + 5y -2 = 0 is :

a. 99x – 27y – 81 = 0
b. 11x -3y + 9 = 0
c. 21x + 77y – 101 = 0
d. 21 x + 77y + 101 = 0

64 Problem

To reduce the differential equation + P(x)y = Q(x).yn to linear form, the subsitution
is

a. v =
b. v =
c. v = yn
d. v = yn – 1

65 Problem

A particle possess two velocities simultaneously at an angle of tan-1 to each other.
Their resultant is 15 m/s. If one velocity is 13 m/s, then the other will be :

a. 5 m/s
b. 4 m/s
c. 12 m/s
d. 13 m/s

66 Problem

If in the expansion of (1 + x)21, the coefficients of xr and xr + 1 be equal, then r is
equal to :

a. 9
b. 10
c. 11
d. 12

67 Problem

A train is running at 5 m/s and a man jumps out of it with a velocity 10 m/s in a
direction making an angle of 600 with the direction of the train. The velocity of
the man relative to the ground is equal to :

a. 12.24 m/s
b. 11.25 m/s
c. 14.23 m/s
d. 13.23 m/s

68 Problem

A ball is projected vertically upward with a velocity 112 m/s. The time taken by it
to return to the point of projection is (g = 10 m/s2) :

a. 11 s
b. 33 s
c. 5.5 s
d. 22 s

69 Problem

If the sides of triangle are 4, 5 and 6 cm, then area of the triangle is equal to :

15
a. 4
cm2
15
b. 7 cm2
4
4
c. 7 cm2
15
d. none of these

70 Problem

The volume of a spherical cap of height h cut off from a sphere of radius a is
equal to :

a. 3 h2 (3a - h)
b. π (a - h)(2a2 – h2 - ah)
4
c. h3
3

71 Problem

The eccentricity of the hyperbola conjugate to x2 –3y2 = 2x + 8 is :

2
a.
3
b. 3

c. 2
d. none of these

72 Problem

The area of the parallelogram whose adjacent sides are is :

a. 2
b. 4
c. 17
d. 2 13

73 Problem

Integrating factor of the differential equation dy is :
P(x)y Q(x)
dx

a. P dx

b. Q dx

c. P dx
e

d. e Q dx

74 Problem

Angle of intersection of the curves r = sin + cos and r = 2 sin is equal to :

a. 2

b. 3

c.
4

d. none of these

75 Problem
1
Define f on R into itself by x sin , when x 0, then :
f (x) x
0, when x 0

a. f is continuous at 0 but not differentiable at 0
b. f is both continuous and differentiable at 0
c. f is differentiable but not continuous at 0

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UPSEE - Mathematics -2004 Unsolved Paper

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