UPSEE - Mathematics -2006 Unsolved Paper

UPSEE–PAST PAPERS
MATHEMATICS - UNSOLVED PAPER – 2006

SECTION

 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 z1, z2 are any two complex numbers, then :

a. | z1 z2 | | z1 | | z2 |

b. | z1 z2 | | z1 | | z2 |

c. | z1 z2 | | z1 | | z2 |

d. | z1 z2 | | z1 | | z2 |

02 Problem

If z = x + iy is a variable complex number such that arg , z 1 then :
z 1 4

a. x2 – y2 – 2x = 1
b. x2 + y2 – 2x = 1
c. x2 + y2 – 2y = 1
d. x2 + y2 + 2x = 1

03 Problem

If arithmetic mean of tow positive numbers is A, their geometric mean is G and
harmonic mean is H, then H is equal to :

a. G2 / A
b. A2 / G2
c. A / G2
d. G / A2

04 Problem

The sun of n terms of two arithmetic series are in the ratio 2n + 3 : 6n + 5, then
the ratio of their 13th terms is :

a. 53 : 155
b. 27 : 87
c. 29 : 83
d. 31 : 89

05 Problem

If , , are the roots of the equation x3 + x + 1 = 0 , then the value of
3 3 3 is :

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

06 Problem
0 0 1
If , then A-1 is :
A 0 1 0
1 0 0

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

07 Problem

6 8 5
4 2 3
If A = 9 7 1 is the sum of symmetric matrix B and skew-symmetric
matrix C, then B is :
6 6 7
6 2 5
a. 9 7 1

0 2 2
2 5 2
b.
2 2 0

6 6 7
c. 6 2 5
7 5 1

0 6 2
d. 2 0 2
2 2 0

08 Problem
1 1
If A = 1 1
, then A100 is equal to :

a. 2100 A
b. 299A
c. 100 A
d. 299A

09 Problem
1 1 1 x 0 x
If 1 2 2 y 3 , then y is equal to :
1 3 1 z 4 z

0
a. 1
1

1
b. 2
3

5
2
c.
1

1
2
d. 3

10 Problem

If y = cos2 x + sec2 x, then :

a. y 2
b. y 1
c. y 2
d. 1 < y < 2

11 Problem
1 1
If x 2 cos , then x2 is equal to :
2 x3

a. sin 3
b. 2 sin 3
c. cos 3
d. 2 cos 3

12 Problem

If sin + cosec = 2, the value of sin10 + cosec10 is :

a. 2
b. 210
c. 29
d. 10

13 Problem

The value of sin 3 5 7 is :
sin sin sin
16 16 16 16

2
a.
16

b. 1
8

1
c. 16

2
d. 32

14 Problem
tan 3 1
If 3 , then the general value of is :
tan 3 1

n
a.
3 12

b. n 7
12

c. n 7
3 36

d. n
12

15 Problem
sin B
In any triangle ABC, If cos A
2 sin C
, then :

a. a = b = c
b. c = a
c. a = b
d. b = c

16 Problem

In a triangle ABC, if b + c = 2a and A = 600 then ABC is :

a. Equilateral
b. Right angled
c. Isosceles
d. Scalene

17 Problem

The co-ordinates of the point which divides the join of the points (2, -1, 3) and
(4, 3, 1) in the ratio 3 : 4 internally are given by : .

2 20 10
a. , ,
7 7 7

10 15 2
b. , ,
7 7 7

20 5 15
, ,
c. 7 7 7

15 20 3
, ,
7 7 7
d.

18 Problem

The area of the triangle ABC, in which a = 1, b = 2, C = 600, is :

a. 4 sq unit
1
b. 2 sq unit
c. 3 sq unit
2
d. 3 sq unit

19 Problem

In a triangle ABC, b = , c = 1 and = 300 , then the largest angle of the triangle
is :

a. 600
b. 1350
c. 900
d. 1200

20 Problem

If A + B + C = , then sin 2A + sin 2B + sin2C is equal to :

a. 4 sin A sin B sic C
b. 4 cos A cos B cos C
c. 2 cos A cos B cos C
d. 2 sin A sin B sin C

21 Problem

A flag is standing vertically on a tower of height b. On a point at a distance a from
the foot of the tower, the flag and the tower subtend equal angles. The height of
the flag is :

a2 b2
a.
a2 b2
a2 b2
b.
a2 b2

c. a2 b2
a2 b2
a2 b2
d. a2 b2

22 Problem

If are the roots of the equation 6x2 – 5x + 1 = 0, then the value of tan 1 tan 1

is

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

23 Problem

The three stainght lines ax + by = c, b x + cy =a and c x + ay =b are collinear, if:

a. b+ c=a
b. c + a=b
c. a+ b + c=0
d. a + b =c

24 Problem

The length of perpendicular from the point (a cos , a sin ) upon the straight
line y = x tan + c, c > 0, is :

a. c
b. c sin2
c. c cos2
d. c sec2

25 Problem

The equation of the circumcircle of the triangle formed by the lines
x = 0, y = 0, 2x + 3y = 5 is :

a. 6(x2 + y2) + 5 (3x – 2y) = 0
b. x2 + y2 – 2x –3y + 5 = 0
c. x2 + y2 + 2x –3y - 5 = 0
d. 6(x2 + y2) - 5 (3x + 2y) = 0

26 Problem

The differential equation of system of concentric circles with centre (1, 2) is :

a. (x - 2) + (y -1) dy = 0
dx
dy
b. (x - 1) + (y -2) dx
=0
dy
c. (x + 1) dx + (y - 2) = 0
dy
d. (x + 2) dx
+ (y -1) = 0

27 Problem

The equation of pair of lines joining origin to the points of intersection of x2 +
y2 = 9 and x + y = 3 is :

a. x2 + (3 - x)2 = 9
b. xy = 0
c. (3 + y)2 + y2 = 9
d. (x - y)2 = 9

28 Problem

The value of , for which the circle x2 + y2 + 2 x + 6y + 1 = 0 intersects the circle
x2 + y2 + 4x + 2y = 0 intersects the circle x2 + y2 + 4x + 2y = 0 orthogonally, is :

a. 11/8
b. -1
c. -5/4
d. 5/2

29 Problem

The value of m, for which the line y = mx + 25 3 is a normal to the conic
x2 y2
3
1, is
16 9

2
a. 3
b. 3

c. 3
2
d. none of these

30 Problem

The value of c, for which the line y = 2x + c is a tangent to the circle x2 + y2 =
16, is :

a. -16 5

b. 4 5
c. 16 5
d. 20

31 Problem

The value of , for which the equation x2 – y2 – x + y – 2 = 0 represents a pair of
straight lines, are :

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

32 Problem

The focus of the parabola x2 + 2y + 6x = 0 is :

a. (-3, 4)
b. (3, 4)
c. (3, -4)
d. (-3, -4)

33 Problem

The value of m, for which the line y = mx + 2 becomes a tangent to the conic
4x2 – 9y2 = 36, are ;

a. 2
3

b. 2 2
3

8
c.
9

4 2
d. 3

34 Problem

The eccentricity of the conic 4x2 + 16y2 – 24x – 32y = 1 is :

a. 1
2

b. 3

3
c. 2

3
d. 4

35 Problem

The number of maximum normals which can be drawn from a point to ellipse
is :

a. 4
b. 2
c. 1
d. 3

36 Problem

The equation of line of intersection of planes 4x + 4y – 5z = 12, 8x + 12y – 31z =
32 can be written as :

a. x 1 y 2 z
2 3 4

b. x 1 y 2 z
2 3 4

x y 1 z 2
c. 2 3 4

x y z 2
d.
2 3 4

37 Problem

If a line makes angle , , , with four diagonals of a cube, then the value of
sin2 sin2 sin2 sin2 is :

a. 4/3
b. 8/3
c. 7/3
d. 1

38 Problem

The equation of the plane, which makes with co-ordinate axes, a triangle with
its centroid , , , is :

a. x y z 3

b. x y z 1
x y z
c. =3
d. x y z =1

39 Problem

If the points (1, 1), (-1, -1), (- 3, 3 ) are the vertices of a triangle, then
this triangle is :

a. Right-angled
b. Isosceles
c. Equilateral
d. None of these

40 Problem

A variable plane moves so that sum of the reciprocals of its intercepts on the
co-ordinate axes is ½. Then the plane passes through :

1 1 1
, ,
a. 2 2 2

b. (-1, 1, 1)
c. (2, 2, 2)
d. (0, 0, 0)

41 Problem

The direction cosines l, m, n of two lines are connected by the relations l +
m + n = 0, lm = 0, then the angle between them is ;

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

42 Problem
     
The value of [a b ca b c] is :

  
a. [a b c ]

b. 0
  
c. 2 [a b c ]

  
d. a x (b x c )

43 Problem

The area of the triangle having vertices as ˆ
i 2ˆ
j ˆ i
3k , ˆ 3ˆ
j ˆ i
3k , 4ˆ 7ˆ
j ˆ
7k , is
:

a. 36 sq unit
b. 0 sq unit
c. 39 sq unit
d. 11 sq unit

44 Problem

The figure formed by the four points ˆ
i ˆ
j ˆ i
k , 2ˆ 3ˆ,5ˆ
j j ˆ ˆ
2k and k ˆ
j is :

a. Trapezium
b. Rectangle
c. Parallelogram
d. None of the above

45 Problem
  
The equation of the plane passing through three non-collinear points a, b, c is :

      
a. r (b x c cxa a x b) 0

         
b. r (b x c cxa a x b) [a b c]

     
r (a x (b x c )) [a b c ]
c.

   
d. r (a b c) 0

46 Problem

The unit vector perpendicular to and coplanar with is :

2ˆ
i 5ˆ
j
a. 29

b. 2ˆ 5 ˆ
i j

1
c. 2 (ˆ
i ˆ
j )

d. ˆ
i ˆ
j

47 Problem
  2   2
(a x b) (a b) is equal to :

 
a. a2 b2
 
b. a2 b2

c. 1
 
d. 2 a2 b2

48 Problem

The domain of the function f(x) = exp ( 5x 3 2x2 ) is :

a. [3/2, )
b. [1, 3/2]
c. (- , 1]
d. (1, 3/2)

49 Problem
sin x
lim is equal to :
x x

a.
b. 1
c. 0
d. does not exist

50 Problem

For the function which of the following is correct :

a. lim f ( x ) does not exist
x 0

b. lim f ( x) =1
x 0

c. lim f (x) exists but f(x) is not continuous at x = 0
x 0

d. f(x) is continuous at x = 0

51 Problem
x (fo fo...of )(x)
If f ( x) , then is equal to :
x 1 19 times

x
a. x 1
19
b. x
x 1
c. 19 x
x 1

d. x

52 Problem

A function f is defined by f(x) = 2 + (x - 1)2/3 in [0, 2]. What of the following is not
correct ?

a. f is not derivable in (0, 2)
b. f is continuous in [0, 2]
c. f(0) = f(2)
d. Rolle’s theoren is ture in [0, 2]

53 Problem
2x 1
If f(x) = x 5
(x 5) , then f-1 (x) is equal to :

x 5 1
a. ,x
2x 1 2

b. 5x 1
,x 2
2 x

c. 5x 1
,x 2
2 x

x 5 1
d. ,x
2x 1 2

54 Problem
d 1 1 x2 1 is equal to :
tan
dx x

1
a.
1 x2

x2
b.
2 1 x 2 ( 1 x 2 1)

c. 2
1 x2

1
d. 2 1 x2

55 Problem
x is equal to :
1 cos
d 1 2
tan
dx x
1 cos
2

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

56 Problem

The maximum value of x1/x is :

a. 1/ee
b. e
c. e1/e
d. 1/e

57 Problem

The function f defined by f(x) = 4x4 – 2x + 1 is increasing for :

a. x < 1
b. x > 0
c. x < 1/2
d. x > 1/2

58 Problem

A particle moves in a straight line so that s = t , then its acceleration is
proportional to :

a. (veloctiy)3
b. velocity
c. (velocity)2
d. (velocity)3/2

59 Problem

32 x 2 (log x )2 dx is equal to :

a. 8x4(log x)2 + c
b. x4{8(log x)2 – 4(log x} + c
c. x4{8(log x)2 – 4 log x} + c
d. x3 {(log x)2 – 2 log x} + c

60 Problem
cos x 1 x is equal to :
e dx
sin x 1

e x cos x
a. c
1 sin x

e x sin x
c
b. 1 sin x

ex
c
c. 1 sin x

e x cos x
c
d. 1 sin x

61 Problem

If f(x) dx = g(x) + c, then f-1 (x) dx is equal to :

a. x f-1 (x) + c
b. f(g-1 (x)) + c
c. xf-1(x) – g (f-1 (x)) + c
d. g-1(x) + c

62 Problem

The value of 2 dx is :
1 x(1 x 4 )

1 17
a. log
4 32

1 32
b. log
4 17

17
c. log
2

1 17
log
d. 4 2

63 Problem

The value of the integral b x dx is :
a
x a b x

a.
1
b. 2
(b - a)
c. /2
d. b – a

64 Problem

The area bounded by y = log x, x-axis and ordinates x = 1, x = 2 is :

a. (log 2)2
b. log 2/e
c. log 4/e
d. log 4

65 Problem

The area of the segment of a circle of radius a subtending an angle of 2 at the
centre is :

1
a2 sin2
a. 2

1 2
b. a sin 2
2

1
c. a2
2
sin2

d. a2

66 Problem
dy 2yx 1
The solution of the differential equation is :
dx 1 x2 (1 x 2 )2

a. y(1 + x2) = x + tan-1 x
y
b. 1 x2 = c + tan-1 x
c. y log (1 + x2) = c + tan-1 x
d. y (1 + x2) = c + sin-1 x

67 Problem

The solution of the differential equation xdy – y dx = x2 y2 dx is :

2
a. x + x y 2 = cx2

b. y - x2 y 2 = cx

c. x - x2 y 2 = cx

d. y + x2 y 2 = cx2

68 Problem
dy
The solution of the differential equation ex y
x 2e y
is :
dx

a. y = ex-y – x2e-y + c
1
b. ey - ex = 3 x3 + c
1
c. ex + ey = x3 + c
3
d. ex – ey =1 x3 + c
3

69 Problem

If A and B are 2 x 2 matrices, then which of the following is true ?

a. (A + B)2 = A2 + B2 + 2AB
b. (A - B)2 = A2 + B2 – 2AB
c. (A - B) (A + B) = A2 + AB – BA – B2
d. (A + B) (A - B) = A2 – B2

70 Problem

If M and N are any two events. The probability. That exactly one of them
occurs, is :

a. P(M) + P(N) – P(M N)
b. P(M) + P(N) + P(M N)
c. P(M) + P(N)
d. P(M) + P(N) – 2P(M N)

71 Problem

If four dice are thrown together. Probability that the sum of the number
appearing on them is 13, is :

a. 35
324

5
b. 216

11
c. 216

11
d. 432

72 Problem

If is the angle between two regression lines with correlation coefficient
, then :

2
a. sin 1
2
b. sin 1
2
c. sin 1
2
sin 1
d.

73 Problem

The value of .0 , where 0.0 stands for the number 0.0373737 …., is :

a. 37/1000
b. 37/990
c. 1/37
d. 1/27

74 Problem
2
a b a
If is an imaginary root of unity, then the value of 2
is :
b c b
2
c a c

a. a3 + b3 + c3
b. a2b – b2c
c. 0
d. a3 + b3 + c3 – 3abc

75 Problem

If A = {x, y}, then the power set of A is ;

{xy, yx}
{ , x, y}
{ , {x}, {2y}}
{ , {x}, {y}, {x, y}}

FOR SOLUTIONS VISIT WWW.VASISTA.NET

UPSEE - Mathematics -2006 Unsolved Paper

More Related Content

What's hot (20)

Similar to UPSEE - Mathematics -2006 Unsolved Paper (16)

More from Vasista Vinuthan (20)

Recently uploaded (20)

UPSEE - Mathematics -2006 Unsolved Paper