UPSEE - Mathematics -1998 Unsolved Paper

UPSEE–PAST PAPERS
MATHEMATICS - UNSOLVED PAPER - 1998

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

If the probability that A and B will die within a year are p and q
respectively, then the probability that only one of them will be alive at the
end of they year, is :

a. p+q
b. p + q – 2pq
c. p + q – pq
d. p + q + pq

02 Problem

Ten different letters of alphabet are given. Words with five letters are formed
from these given letters. Then, the number of words which have atleast one
letter repeated, is :

a. 69760
b. 30240
c. 99748
d. none of these

03 Problem

If (1 + x + x2)n = C0 + C1x + C2x2 + … then the value of C0C1- C1C2 + C2C3 –

a. 0
b. 3n
c. (-1)n
d. 2n

04 Problem

Given positive integers r > 1, n > 2 and that the coefficient of (3r)th and (r +
2)th term in the binomial expansion of (1 + x)2n are equal, then :

a. n = 2r
b. n = 3r
c. n = 2r + 1
d. none of these

05 Problem

A ball of mass 1 kg moving with velocity 7 m/s, overtakes and collides with a
ball of mass 2 kg moving with velocity 1 m/s in the same direction. If 2 =
¾, the velocity of lighter ball after impact is :

a. 6 m/s
3
b. 2
m/s
c. 1 m/s
d. 0 m/s

06 Problem

A bullet of 0.05 kg moving with a speed of 120 m/s penetrates deeply into a fixed
target and is brought to rest in 0.01 s. The distance through which it penetrates is
:

a. 3 cm
b. 6 cm
c. 30 cm
d. 60 cm

07 Problem

a, b, c are real a b, the roots of the equation (a - b)x2 –5 (a + b) x – 2 (a - b) =
0 are :

a. real and equal
b. complex
c. real and unequal
d. none of these

08 Problem
 /2
The value of 
0
| sin x  cos x | dx is equal to :

a. 0
b. 2( 2 -1)
c. 2 2

d. 2( 2 + 1)

09 Problem
1
The value of 
1
x | x | dx is equal to

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

10 Problem
x sin1 x
The value of  1  x2
dx is equal to :

a. (1  x 2 ) sin-1 x + c
b. x sin–1 x + c
c. x - (1  x 2 ) sin-1 x + c
d. (sin-1 x)2 + c

11 Problem

 cos3 2n  1 xdx has the value
2
For any integer n, the integral esin x
0

a. -1
b. 0
c. 1
d. 

12 Problem

The cube roots of unity when represented argand diagram from the vertices of :

a. an equilateral triangle
b. an isosceles triangle
c. a right angled triangle
d. none of the above

x
a

13 Problem

 (a
x
The value of / x )dx is equal to :

x
a. a loge a + c
b. 2 a x log10 a + c
x
c. 2 a loga e + c
x
d. 2 a loge a + c

14 Problem

If +b then

1
a. a = 3

2
b. a =
3
1
c. a = - 3

2
d. a = - 3

15 Problem
a
If f(a - x) = f(x), then x f(x)d

0
x is equal to :

a a
2 0
a. f(x) dx
a
b. a 
0
f(x)dx
a2 a
c.
2 
0
f(x)dx

d. none of these

16 Problem
 1 
If z   5  i  Then z lies
  
   , where |  | 1.
z

a. a circle
b. a parabola
c. an ellipse
d. none of these

17 Problem

If log 2, log (2x - 1), log (2x + 3) are in AP Then x is equal to :

a. 5/2
b. log2 5
c. log3 2
d. log5 3

18 Problem

Points D, E are taken on the side BC of a triangle such that BD = DE = EC. If
BAD   , DAE   , EAC   , then the value of sin     sin      is
sin  sin 
equal to :

a. 1
b. 2
c. 4
d. none of these

19 Problem

The value of cos (2 cos-1 x + sin-1 x) at x = 1/5 is equal to :

2 6
a.
5
2 5
b.
6
2 6
c. -
5
2 5
d. - 6

20 Problem
 /2
If f(x) is an odd function of x, then 
 / 2
f (cos x )dx is equal to :

a. 0
 /2
b.  0
f (cos x )dx

 /2

c. 
0
f (cos x )dx

d.  f cos x  dx

21 Problem

y = cos-1 [sin (1  x) / 2] + xx, then dy/dx at x = 1 is equal to :

a. 3/4
b. 0
c. 1/2
d. -1/2

22 Problem

f(x) = (sin x + cos 2x), (x > 0) has minimum value for x is equal to :

n
a. 2
3
b.  n  1 
2
1
c. 2n  1 
2

d. none of these

23 Problem

The point P on curve y2 = 2x3 such that the tangent at P is perpendicular to the
line 4x – 3y + 2 = 0 is given by :

a. (2, 4)

b. (1, 2)

1 1
c. 2 , 2
 

1 1 
d. , 
 8 16
 

24 Problem

Let I1 = 2 dx 2 dx then :

1
1  x2
and I2  
1 x
,

a. I1 > I2
b. I2 > I1
c. I1 = I2
d. I1 > 2I2

25 Problem

The value of the integral is equal to :

a. 0
1
b. 2

c. 1
2
d. none of these

26 Problem

In  ABC, the value of cosec A (sin B cos C + cos B sin C) is equal to :

c
a. a

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

27 Problem

The value of tan 90 – tan 270 – cot 270 + cot 90 is :

a. 2
b. 3
c. 4
d. none of these

28 Problem

In  ABC, 3 sin A = 6 sin B = 2 3 sin C, then the value of A is equal to :

a. 00
b. 450
c. 600
d. 900

29 Problem

If the side of the triangle are 5k, 6k, 5k and radius of incircle is 6, then the
value of ‘k’ is

a. 4
b. 5
c. 6
d. 7

30 Problem

The angle of depression of the top and the foot of the chimney as seen from the
top of second chimney which is 150 m high and standing on the same level as the
first are  and  respectively. The distance between their tops when tan
4 5 is equal to :
  and tan  
3 2

50 m
100 m
150 m
none of these

31 Problem

The medians AD and BE of the triangle with vertices A (0, b), B (0, 0) and C
(a, 0) are mutually perpendicular, if :

a. b = 2 a
b. a = b 2

c. b = - 2 a
d. a = - 2b

32 Problem

The distance between the chords of contact of the tangent to the circle x2 + y2
+ 2gx + 2fy + c = 0 from the origin and the point (g, f) is :

a. g2 + f2

b. (g2 + f2 + c)
g2  f 2  c
c.
2 g2  f 2

d. g2  f 2  c
2 g2  f 2

33 Problem

A line is drawn through a fixed point p (h, k) to cut the circle x2 + y2 = a2 at Q
and R. Then PQ . PR is equal to :

a. (h + k)2 – a2
b. h2 + k2 – a2
c. (h – k)2 + a2
d. h2 + k2 + a2

34 Problem

The locus of the mid point of a focal chord of a parabola is :

a. Circle
b. Parabola
c. Ellipse
d. Hyperbola

35 Problem
2 2
The straight line x + y = c will be tangent to the ellipse x  y  1 then c is equal
9 16
to :

a. 8
b.  5
c.  10
d.  6

36 Problem

The length of the subnormal of the curve y2 = 2ax is equal to

a. a
b. 2a
c. a/2
d. -a

37 Problem
2 2
If is the angle between the asymptotes of the hyperbola x  y  1 with
 a2 b2
eccentricity e, then sec 2 is equal to :

a. 0
b. e
c. e2
e
d. 2

38 Problem

The latus rectum of the hyperbola 9x2 – 16y2 + 72x –32y –16 = 0 is :

9
a. 2

b. - 9
2

c. 32
3

d. - 32
3

39 Problem
 x2  1 
The value of p and q from lim   px  q   0 are :
 x 1
x 


a. p = 0, q = 0
b. p = 1, q = -1
c. p = -1, q = 1
d. p = 2, q = - 1

40 Problem

Which of the following functions is an even function ?

ax  1
a. f  x   x x
a 1

b. f(x) = tan x
ax  a x
c. f x  x x
a 1
ax  1
d. f x  x
a 1

41 Problem
 x 2 sin1 / x, x 0
If f(x) =  , then :
 0, x 0

a. f and f’ are continuous at x = 0
b. f is derivable at x = 0
c. f is derivable at x = 0 and f’ is not continuous at x = 0
d. f is derivable at x = 0

42 Problem
p q r
Let p, q, r be positive and not all equal, then the value of the determinant q r p
r p q
is equal to :

a. Positive
b. Negative
c. 0
d. none of these

43 Problem

Suppose n  3 person are sitting in a row. Two them are selected at random.
Then probability that they are not together, is L:

a. 1- 2/n
b. 2/(n -1)
c. 1 – 1/n
d. none of these

44 Problem

The minimum value of x2 – 3x + 3 in the interval (-3, 3/2) is equal to :

a. 3/4
b. 5
c. - 15
d. - 20

45 Problem
100 100
Let tn be the nth term of the GP of positive numbers. Let t
n1
2n  x and  t2n1  y
n1
such that x  y , then the common ratio is

x
a. y

x
b.
y

c. y
x

d. none of these

46 Problem

If every element of third order determinant value of is multiplied by five, then
the value of the new determinant is :

a. 
b. 5 
c. 25 
d. 125 

47 Problem

In the expansion of (1 + x)50, the sum of the coefficients of odd powers of x is :

a. 0
b. 249
c. 250
d. 251

48 Problem

The equation sin6   cos6   a has a real solution, if :

a. 1/2  a  1
b. 1/4  a  1
c. -1  a  1
d. 0  a  1/2

49 Problem

  (1  x
2
) sin x cos2 x dx is equal to :


a. 0
b.  - 3/3
c. 2  -  3
7
d.  2 3
2

50 Problem

The derivative of f  x   x3 dt
 (x  0) is equal to :
x2 log t

a. 1 1

3log x 2log x

1
b.
3log x

3x 2
c.
3 log x

 x  1  x 
d.
log x

51 Problem
4 4 1
If  f  x  dx  4 and  [3  f (x)]dx  7 , then the value of  f  x  dx is equal
1 2 2

to :

a. 2
b. - 3
c. - 5
d. none of these

52 Problem

e [ (x)   '(x)]dx is equal to :

x

a.  ex '(x)dx

b. e x ( x )  c

c. ex '(x)  c
d. none of these

53 Problem
1
The value of  | sin2 x |
0
dx is equal to :

a. 0
1
b. - 
1
c. 
2
d.


54 Problem

A house has multi-storeys. The lowest storey is 20 ft high. A stone which is
dropped from the top of the house passes the lowest storey in 1/4 s, then
the height of the house is

a. 110.00 ft
b. 110.2 ft
c. 110.25 ft
d. none of these

55 Problem

A particle was dropped from the top of the tower and at the same time
another body is thrown vertically upwards from the bottom of the tower with
such a velocity that it can just reach the top of the tower, then they will meet
at the height of :

a. h/4
b. 3h/4
c. h
d. h/2

56 Problem

A particle starts with a velocity of 200 cm/s and moves in a straight line with
a retardation of 10 cm/s2. Then the time it takes to describe 1500 cm is :

a. 10 s, 30 s
b. 5 s, 15 s
c. 10 s
d. 30 s

57 Problem

The area bounded by the curve y = x3, the x – axis and the ordinates x = -2 and
x = 1 is :

a. -9 sq unit
b. -15/4 sq unit
c. 15/4 sq unit
d. 17/4 sq unit

58 Problem

Two balls are projected respectively from the same point in direction inclined
at 300 and 600 to the horizontal. If they attain the same height, then the ratio
of their velocities of projection is

a. 1 : 3

b. 3 :1
c. 1 : 1
d. 1 : 2

59 Problem

A particle is projected under gravity (g = 9.81 m/s2) with a velocity of 29.43
m/s at an angle of 300. The time of flight in seconds to a height of 9.81 m are
:

a. 5, 1.5
b. 1, 2
c. 1.5, 2
d. 2, 3

60 Problem

The path of a projectile in vacuum is a :

a. A straight line
b. Circle
c. Ellipse
d. Parabola

61 Problem

A particle is projected with initial velocity u making an angle with the
horizontal, its time of flight will be given by :

a. 2u sin 
g

b. 2u 2 sin 
g

u sin 
c.
g

u2 sin 
d. g

62 Problem

If x2 + px + 1 is a factor of x2 + bx + c, then :

a. a2 + c2 = - ab
b. a2 – c2 = - ab
c. a2 – c2 = ab

63 Problem

The probabilities of occurrence of two events E and F are 0.25 and 0.50
respectively. The probability of their simultaneous occurrence is 0.14. The
probability that neither E occurs nor F occurs :

a. 0.39
b. 0.25
c. 0.11
d. none of these

64 Problem

A sphere impinges directly one an equal sphere which is at rest. Then the
original kinetic energy lost is equal to :

1  e2
a. 2
times the initial KE
1  e2
b.
2
1  e2
c. times the initial KE
2
d. none of these

65 Problem

A hockey stick ball is at rest for 0.01s with an average force of 5 N. If the ball
weight 0.2 kg, then the velocity of the ball after being pushed is equal to :

a. 2.5 m/s
b. 2 m/s
c. 3.0 m/s
d. 5 m/s

66 Problem

A given force is resovled into components P and Q is equally inclined to it, then :

a. P = 2Q
b. P = Q
c. 2P = Q
d. none of these

67 Problem

If the forces of 12,5 and 13 unit weight balance at a point, two of them are
inclined at :

a. 300
b. 450
c. 900
d. 600

68 Problem

ABC is a triangle. Forces P, Q, R act along the lines OA, OB and OC and are in
equilibrium, if O is incentre of ABC, then :

P Q R
 
A B C
a. cos cos cos
2 2 2

P Q R
b.  
OA OB OC
P Q R
c.  
A B C
sin sin sin
2 2 2

69 Problem

If two equal perfectly elastic balls impinges directly, after impact :

a. Their velocities are not effected
b. They interchange their velocities
c. Their velocities changes their direction
d. Their velocities get doubled

70 Problem

In triangle ABC (sin A + sin B + sin C) (sin A + sin B – sin C) = 3 sin A sin B then :

a. A = 600
b. B = 600
c. C = 600
d. A = 900

71 Problem

If sin A  p, cos A  q then :
sin B cos B

p q2  1
tan A 
a. q 1  p2

2
b. tan A   p q  1
2
q 1 p
q2  1
c. tan B 
1  p2

d. all are correct

72 Problem

In a triangle ABC, A   and AD is median then :
3

a. 4AD2 = b2 + bc + c2
b. AD2 = b2 + bc + c2
c. 2AD2 = b2 + bc + c2
d. 4AD2 = b2 – bc + c2

73 Problem

If the radius of the circumcircle of isoseceles triangle ABC is equal to AB =
AC, then angle A is equal to :

a. 300
b. 600
c. 900
d. 1200

74 Problem

The focus of the parabola y2 – x – 2y + 2 = 0 is :

a. (1/4, 0)
b. (1/2)
c. (3/4, 1)
d. (5/4, 1)

75 Problem

If A (3, 1), B (6, 5) and C (x , y) are three points such that the angel CAB is a right
angle and the area of CAB = 7, then the number of the point C is ;

a. 0
b. 1
c. 2
d. 4

76 Problem
dy
Solve, dx
= (4x + 3y - 1)2.

77 Problem
lim
Evaluate x 0 [sin (x + a) + sin (a - x) – 2sin a]/x sin x

78 Problem

A rod is moveable in a vertical plane about a hinge at one end, another end is
fastened to a weight equal to half the weight of the rod, this end is fastened by
a string of length l to a pint at a height c vertically over the hinge find the
tension in of the string.

79 Problem

Sun of infinity of the series
12  22 12  22  32
1   ........
2! 3!

80 Problem

Evaluate, cos x dx
 1  sin x  2  sin x 
2

81 Problem

Find the equation to the chord of the hyperbola 25x2 – 16y2 = 400 having mid
point at (6, 2).

82 Problem

i j ˆ i j ˆ i j ˆ
Find the vector moment f the three vectors ˆ  2ˆ  3k,2ˆ  3ˆ  4k, ˆ  ˆ  k acting
on a particle at point P (0, 1, 2) about the point A (1, -2, 0).

83 Problem

If the equation k(6x2+3) + rx+2x2 -1 =0 and 6k (2x2 +1 ) + px +4x 2 -2 =0, have
both the common roots find the value of (2r – p)

84 Problem

Find the equation to the common tangnt to the parabola y2=2x and x2 = 16y

85 Problem

 
x 5
Determine the value of ‘x’ in the expansion of x  xlog
10 if the third term in
the expansion is 10,00,000.

86 Problem

In parallelogram ABCD the interior bisectors of the consecutive angles B and C
intersect at P, then find . BPC

87 Problem

Find the area bounded by the curve y = 2x – x2 and the straight line y = - x.

88 Problem

b2  c 2 a2 a2
Find the value of determinant b2 c 2  a2 b2
c2 c2 a2  b2

89 Problem

The probability of getting sum more than 15 in three dice will be 5/108. Prove
it.

90 Problem

If tan 2 tan  =1, then find the value of .

91 Problem

If A = [1 2 3] and B =  5 4 0  , then find AB.
 
 0 2 1
 1 3 2 
 

92 Problem

Find the length of tangent of circle x2 + y2 + 6x – 4y – 3 = 0 from point (5, 1).

93 Problem

Find the value of cos 200 cos 400 cos 600 cos 800.

94 Problem
x 3 y  4 z 5
Find the distance from point (3, 4, 5) of that point where line 1

2

2
cut the plane x + y + z – 17 = 0.

95 Problem

Find the point on the curve 9y2 = x3, where normal to the curve makes equal
intercepts with the axes.

96 Problem

A ladder 15 m long leans against a wall 7 m high and a portion of the ladder
protrudes over the wall such that its projection along the vertical is 3 m. How fast
does the bottom start to slip away from the wall, if the ladder slides down along
the top edge of the wall at 2 m/s ?

97 Problem

Solve the equation sin [2 cos-1 {cot (2 tan-1 x) }]= 0

98 Problem

The side AB, BC, CD and DA of a quadrilateral have the equations x + 2y = 3, x =
1, x – 3y = 4, 5x + y + 12 = 0 respectively. Find the angle between the diagonals
AC and BD.

99 Problem

The angles of top of the tower from the foot and top of a building are . Find the
height of tower.

100 Problem

If lines px2 – pxy – y2 = 0, make the angle from x-axis, then find the value of tan .
   

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UPSEE - Mathematics -1998 Unsolved Paper

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