1) The differential equation representing the family of curves y2 = 2 c ( 2 + ) , where C > 0 , is a parameter, is of order and degree as follows :

1 )   order 1, degree 3

2 )   order 2, degree 2

3 )   order 1, degree 2

4 )   order 1, degree 1

2) Area of the greatest rectangle that can be inscribed in the ellipse x2/a2 + y2/b2 = 1 is

1 )

2 )   a/b

3 )   2ab

4 )   ab

3)

1 )   tan 1

2 )   (1/2) tan 1

3 )   (1/2) sec 1

4 )   (1/2) cosec 1

4) If the cube root of unity are 1, w, w2 then roots of equation (x-1)3 + 8 = 0, are

1 )   -1 , 1 - 2w , 1 - 2w2

2 )   -1 , 1 + 2w , 1 + 2w2

3 )   -1 , -1 + 2w , -1 - 2w2

4 )   -1 , -1 , -1

5) If A2 - A + I = 0, then the inverse of A is

1 )   A - I

2 )   I - A

3 )   A + I

4 )   A

6) Let R = {(3, 3), (6, 6), (9, 9), (12, 12), (6, 12), (3, 9), (3, 12), (3, 6)} be a relation on the set A = {3, 6, 9, 12}. The relation is

1 )   an equivalence relation

2 )   reflexive and symmetric only

3 )   reflexive and transitive only

4 )   reflexive only

7) If in a frequency distribution, the mean and median are 21 and 22 respectively, then its mode is approximately

1 )   25.5

2 )   24.0

3 )   22.0

4 )   20.5

8) Let P be the point (1, 0) and Q a point on the locus y2 = 8x. The locus of mid point of PQ is

1 )   x2 + 4y + 2 = 0

2 )   x2 - 4y + 2 = 0

3 )   y2 - 4x + 2 = 0

4 )   y2 + 4x + 2 = 0

9) If C is the mid point of AB and P is any point outside AB, then

1 )

2 )

3 )

4 )

10) ABC is a triangle. Forces P, Q, R acting along IA, IB and IC respectively are in equilibrium, where I is the incentre of DABC. Then P : Q : R is

1 )   cos A/2 : cos B/2 : cos C/2

2 )   cos A : cos B : cos C

3 )   sin A : sin B : sin C

4 )   sin A/2 : sin B/2 : sin C/2

11) In a triangle PQR, ∠R = π/2 . If tan (P/2) and tan(Q/2) are the roots of ax2 + bx + c = 0 , a ≠ 0 , then

1 )   b = c

2 )   b = a + c

3 )   a = b + c

4 )   c = a + b

12) If the coefficient of rth, (r + 1)th and (r + 1)th and (r + 2)th terms in the binomial expansion of (1 + y)m are in A.P., then m and r satisfy the equation

1 )   m2 - m(4r + 1) + 4r2 - 2 = 0

2 )   m2 - m(4r - 1) + 4r2 + 2 = 0

3 )   m2 - m(4r - 1) + 4r2 - 2 = 0

4 )   m2 - m(4r + 1) + 4r2 + 2 = 0

13) Let f : (-1, 1) → B, be a function defined by f(x) = tan-1( 2x / (1 - x2) ) , then f is both one-one and onto when B is interval

1 )   [ - π/2 , π/2 ]

2 )   ( - π/2 , π/2 )

3 )   ( 0 , π/2 )

4 )   [ 0 , π/2 )

14) If the coefficient of x7 in equals the coefficient of x -7 in then a and b satisfy the relation

1 )   a / b = 1

2 )   ab = 1

3 )   a - b = 1

4 )   a + b = 1

15) If and |w| = 1, then z lies on

1 )   a straight line

2 )   a parabola

3 )   an ellipse

4 )   a circle

16) If a2 + b2 + c2 = -2 and Then f(x) is a polynomial of degree

1 )   3

2 )   2

3 )   1

4 )

17) If z1 and z2 are two non-zero complex numbers such that |z1 + z2| = |z1| + |z2|, then arg z1 - arg z2 is equal to

1 )

2 )   -π/2

3 )   π/2

4 )   π

18) The value of a for which the sum of the squares of the roots of the equation x2 - (a - 2)x - a - 1 = 0 assume the least value is

1 )   3

2 )   2

3 )   1

4 )    0

19) If the roots of the equation x2 - bx + c = 0 be two consecutive integers, then b2 - 4c equals

1 )   2

2 )   1

3 )   -2

4 )   3

20) The system of equations αx + y + z = α - 1 x + αy + z = α - 1 x + y + αz = α - 1 has no solution, if α is

1 )   not -2

2 )   1

3 )   -2

4 )   either -2 or 1

21) The value of is

1 )   56C3

2 )   56C4

3 )   55C4

4 )   55C3

22) If then which one of the following holds for all n >= 1, by the principle of mathematical induction.

1 )   An = nA + (n - 1)I

2 )   An = 2n-1A + (n - 1)I

3 )   An = nA - (n - 1)I

4 )   An = 2n-1A - (n - 1)I

23) If the letters of word SACHIN are arranged in all possible ways and these words are written out as in dictionary, then the word SACHIN appears at serial number

1 )   603

2 )   602

3 )   601

4 )   600

24) If where a, b, c are in A.P. and |a| < 1, |b| < 1, |c| < 1 then x, y, z are in

1 )   Arithmetic - Geometric Progression

2 )   HP

3 )   GP

4 )   AP

25) If x is so small that x3 and higher powers of x may be neglected, then may be approximated as

1 )   -(3/8)x2

2 )   x/2 - (3/8)x2

3 )   1 - (3/8)x2

4 )   3x + (3/8)x2

26) If cos-1x - cos-1(y/2) = α , then 4x2 - 4xy cosα + y2 is equal to

1 )   4sin2α

2 )   - 4sin2α

3 )   2sin 2α

4 )   4

27) If in a Δ ABC, the altitudes from the vertices A, B, C on opposite sides are in H.P., then sin A, sin B, sin C are in

1 )   Arithmetic - Geometric Progression

2 )   H.P.

3 )   G.P.

4 )   A.P.

28) In a triangle ABC, let ∠C = π/2. If r is the inradius and R is the circumradius of the triangle ABC, then 2(r + R) equals

1 )   a + b + c

2 )   c + a

3 )   b + c

4 )   a + b

29) A function is matched below against an interval where it is supposed to be increasing. Which of the following pairs is incorrectly matched?

1 )   ( -∞ , 1/3 ]

2 )   ( -∞ , -4 ]

3 )   ( -∞ , ∞ )

4 )   [ 2 , ∞ )

30) Let a and b be the distinct roots of ax2 + bx + c = 0, then is equal to

1 )   -(a2/2)(α - β)2

2 )   (1/2)(α - β)2

3 )   (a2/2)(α - β)2

4 )    0

31) The normal to the curve x = a (cosθ + θsinθ), y = a (sinθ + θcosθ) at any point 'θ' is such that

1 )   it passes through ( a π/2 , -a )

2 )   it is at a constant distance from the origin

3 )   it passes through the origin

4 )   it makes angle π/2 + θ with the x-axis

32) Let f be differentiable for all x. If f(1) = -2 and f'(x) ≥ 2 for x ∈ [1, 6], then

1 )   f(6) < 5

2 )   f(6) = 5

3 )   f(6) ≥ 8

4 )   f(6) < 8

33) If f is a real-valued differentiable function satisfying |f(x) - f(y)| ≤ (x - y)2, x, y ∈ R and f(0) = 0, then f(1) equals

1 )   2

2 )   1

3 )   -1

4 )     0

34) Suppose f(x) is differentiable at x = 1 and then f '(1) equals

1 )   5

2 )   6

3 )   3

4 )   4

35) is equal to

1 )

2 )

3 )

4 )

36) A spherical ball 10 cm in radius is coated with a layer of ice of uniform thickness that melts at a rate of 50 cm3/min. When the thickness of ice is 5 cm, then the rate at which the thickness of ice decreases, is

1 )   (1/54π)cm/min

2 )   (5/6π)cm/min

3 )   (5/36π)cm/min

4 )   (1/18π)cm/min

37) Let f(x) be a non-negative continuous function such that the area bounded by the curve y = f(x), x-axis and the ordinates x = π/4 and Then f(π/2) is

1 )   ( 1 - π/4 - )

2 )   ( 1 - π/4 + )

3 )   ( π/4 + - 1 )

4 )   ( π/4 - + 1 )

38) Let f : R → R be a differentiable function having f(2) = 6, f '(2) = ( 1/48 ) . Then equals

1 )   12

2 )   18

3 )   24

4 )   36

39) The area enclosed between the curve y = loge (x + e) and the coordinate axes is

1 )   3

2 )   4

3 )   1

4 )   2

40) The parabolas y2 = 4x and x2 = 4y divide the square region bounded by the lines x = 4, y = 4 and the coordinate axes. If S1, S2, S3 are respectively the areas of these parts numbered from top to bottom; then S1 : S2 : S3 is

1 )   3

2 )   4

3 )   1

4 )   2

41)

1 )   I3 = I4

2 )   I3 > I4

3 )   I2 > I1

4 )   I1 > I2

42) The line parallel to the x-axis and passing through the intersection of the lines ax + 2by + 3b = 0 and bx - 2ay - 3a = 0, where (a, b) ≠ (0, 0) is

1 )   above the x-axis at a distance of 3/2 from it

2 )   above the x-axis at a distance of 2/3 from it

3 )   below the x-axis at a distance of 3/2 from it

4 )   below the x-axis at a distance of 2/3 from it

43) If x (dy/dx) = y ( log y - log x + 1 ) , then the solution of the equation is

1 )   log ( y/x ) = cx

2 )   log ( x/y ) = cy

3 )   ylog ( x/y ) = cy

4 )   xlog ( y/x ) = cy

44) If a vertex of a triangle is (1, 1) and the mid points of two sides through the vertex are (-1, 2) and (3, 2), then the centriod of the triangle is

1 )   ( 1 , 7/3 )

2 )   ( 1/3 , 7/3 )

3 )   ( -1 , 7/3 )

4 )   ( -1/3 , 7/3 )

45) If the circles x2 + y2 + 2ax + cy + a = 0 and x2 + y2 - 3ax + dy - 1 = 0 intersect in two distinct points P and Q then the line 5x + by - a = 0 passes through P and Q for

1 )   infinitely many values of a

2 )   exactly two values of a

3 )   exactly one value of a

4 )   no value of a

46) If non-zero numbers a, b, c are in H.P., then the straight line x/a + y/b + 1/c = 0 always passes through a fixed point. That point is

1 )   ( 1, -2 )

2 )   ( 1, -1/2 )

3 )   ( -1, 2 )

4 )   ( -1, -2 )

47) If a circle passes through the point (a, b) and cuts the circle x2 + y2 = p2 orthogonally, then the equation of the locus of its centre is

1 )   x2 + y2 - 2ax - 3by + (a2 - b2 p2) = 0

2 )   2ax + 2by - (a2 + b2 + p2) = 0

3 )   x2 + y2 - 3ax - 4by + ( a2 + b2 - p2 ) = 0

4 )   2ax + 2by - ( a2 - b2 + p2 ) = 0

48) An ellipse has OB as semi minor axis, F and F' its focii and the angle FBF ' is a right angle. Then the eccentricity of the ellipse is

1 )   1/4

2 )   1/

3 )   1/

4 )   1/2

49) A circle touches the x-axis and also touches the circle with centre at (0, 3) and radius 2. The locus of the centre of the circle is

1 )   a hyperbola

2 )   a parabola

3 )   an ellipse

4 )   a circle

50) The angle between the lines 2x = 3y = -z and 6x = -y = - 4z is

1 )   45°

2 )   30°

3 )     0°

4 )   90°

51) If the angle q between the line ( x + 1 )/1 = ( y - 1 )/2 = ( z - 2 )/2 and the plane is such that sin θ = 1/3 the value of λ is

1 )   3/4

2 )   -4/3

3 )   5/3

4 )   -3/5

52) The locus of a point P(α , β) moving under the condition that the line y = αx + β is a tangent to the hyperbola x2/a2 - y2/b2 = 1 is

1 )   a parabola

2 )   a hyperbola

3 )   an ellipse

4 )   a circle

53) The distance between the line r = 2i - 2j + 3k + λ(i - j + 4k) and the plane r.(i + 5j + k) = 5 is

1 )   3/10

2 )   10/3

3 )   10/9

4 )   10/(10)

54) For any vector a, the value of (a - i )2 + (a - j)2 + (a - k)2 is equal to

1 )   2 a 2

2 )   4 a 2

3 )   3 a 2

4 )   a 2

55) If the plane 2ax - 3ay + 4az + 6 = 0 passes through the midpoint of the line joining the centres of the spheres x2 + y2 + z2 + 6x - 8y - 2z = 13 and x2 + y2 + z2 - 10x + 4y - 2z = 8 then a equals

1 )   -2

2 )   2

3 )   -1

4 )   1

56) Let a, b and c be distinct non-negative numbers. If the vectors ai + aj + ck, i + k and ci + cj + bk lie in a plane, then c is

1 )   equal to zero

2 )   the Harmonic Mean of a and b

3 )   the Geometric Mean a and b

4 )   the Arithmetic Mean of a and b

57) If a, b, c are non-coplanar vectors and λ is a real number then [ λ (a + b) λ2 b λ c ] = [ a b + c b ] for

1 )   exactly three values of λ

2 )   exactly two values of λ

3 )   exactly one value of λ

4 )   no value of λ

58) Let a = i - k, b = xi + j + (1 - x)k and c = yi + xj + (1 + x - y)k. Then [ a, b, c ] depends on

1 )   both x and y

2 )   neither x nor y

3 )   only y

4 )   only x

59) Three houses are available in a locality. Three persons apply for the houses. Each applies for one house without consulting others. The probability that all the three apply for the same house is

1 )   8/9

2 )   7/9

3 )   2/9

4 )   1/9

60) A random variable X has Poisson distribution with mean 2. Then P( X > 1.5 ) equals

1 )   1 - 3/e2

2 )   3/e2

3 )   2/e2

4 )     0

61) Let A and B be two events such that where A stands for component of event A. Then events A and B are

1 )   independent but not equally likely

2 )   mutually exclusive and independent

3 )   equally likely and mutually exclusive

4 )   equally likely but not independent

62) A lizard, at initial distance of 21 cm behind an insect, moves from rest with an acceleration of 2 cm/s2 and pursues the insect which is crawling uniformly along a straight line at a speed of 20 cm/s. Then the lizard catch the insect after

1 )   21 s

2 )   24 s

3 )   20 s

4 )   1 s

63) The resultant R of two forces acting on a particle is at right angles to one of them and its magnitude is one third of the other force. The ratio of larger force to smaller one is

1 )   3 : 2

2 )   3 : 2

3 )   2 : 1

4 )   3 :

64) Two points A and B move from rest along a straight line with constant acceleration f and f ' respectively. If A takes m sec. more than B and describes 'n' units more than B in acquiring the same speed then

1 )   ( 1/2 ) ( f + f ' ) m = f f ' n2

2 )   ( f ' - f ) n = ( 1/2 ) f f ' m2

3 )   ( f - f ' ) m2 = f f ' n

4 )   ( f + f ' ) m2 = f f ' n

65) A and B are two like parallel forces. A couple of moment H lies in the plane of A and B and is contained with the. The resultant of A and B after combining is displaced through a distance

1 )   ( H )/( 2 ( A + B ) )

2 )   ( H )/( A - B )

3 )   ( 2H )/( A - B )

4 )   ( H )/( A + B )

66) The sum of the series : is

1 )

2 )

3 )

4 )

67) If a1, a2, a3, ...., an are in G.P., then the determinant is equal to

1 )   4

2 )   2

3 )   1

4 )     0

68) If both the roots of the quadratic equation x2 - 2kx + k2 + k - 5 = 0 are less than 5, then k lies in the interval

1 )   (- ∞ , 4)

2 )   [4, 5]

3 )   (5, 6]

4 )   (6, ∞)

69) If the equation anxn + an-1xn-1 + ... + a1x = 0, a1 ≠ 0, n >= 2, has a positive root x = α , then the equation nanxn-1 + (n - 1)an-1xn-2 + ... + a1 = 0 has a positive root, which is

1 )   greater than or equal to α

2 )   equal to α

3 )   greater than α

4 )   smaller than α

70) A real valued function f(x) satisfies the functional equation f(x - y) = f(x)f(y) - f(a - x)f(a + y), where a is a given constant and f(0) = 1, f(2a - x) is equal to

1 )   f(a) + f(a - x)

2 )   f(-x)

3 )   -f(x)

4 )   f(x)

71) The plane x + 2y - z = 4 cuts the sphere x2 + y2 + z2 - x + z - 2 = 0 in a circle of radius

1 )   2

2 )

3 )   3

4 )   1

72) If the pair of lines ax2 + 2(a + b)xy + by2 = 0 lie along diameters of a circle and divide the circle into four sectors such that the area of one of the sectors is thrice the area of another sector then

1 )   3a2 + 10ab + 3b2 = 0

2 )   3a2 + 2ab + 3b2 = 0

3 )   3a2 - 10ab + 3b2 = 0

4 )   3a2 - 2ab + 3b2 = 0

73) The value of a > 0 , is

1 )   π/3

2 )   2 π

3 )   a π

4 )   π / 2

74) A particle is projected from a point O with velocity u at an angle of 60° with the horizontal. When it is moving in a direction at right angles to its direction at O, its velocity then is given by

1 )   2 u / 3

2 )   u /

3 )   u / 3

4 )   u / 2

75) Let x1, x2, ..., xn be n observations such that and . Then a possible value of n among the following is

1 )   9

2 )   12

3 )   15

4 )   18

ANSWERS