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Math Paper

Q.1:- In an ellipse, if the lines joining focus to the extremities of the minor axis from an

equilateral triangle with the minor axis, then the eccentricity of the ellipse is

  • (b)                            (c)                                (d)

Q.2:- The curve        has a vertical tangent at the point

(a)  (1, 1)                                              (b) (1, 0)

(c) (0, 1)                                                (d) none of these

Q.3:- If         = 4 , then the value of x2 + y2 + z2 + t2 is

(a)                                            (b) 1 2𝑥𝑦𝑧𝑡

(c) 4                                                                 (d) 6

Q.4:- Four dice are rolled. The number of possible outcomes in which at least one dice

shows 2 is

(a) 625                         (b) 671                           (c) 1023                         (d) 1296

Q.5:- If f( )  f(𝑥)  f(𝑦) for all 𝑥 and 𝑦 & if f(5)  2 and  f ‘(0)  3, then f ‘(5) is

(a)  0                       (b) 2                           (c) 5                             (d) 6

Q.6:- The equation of the curve satisfying the differential equation 𝑦2 (𝑥2  1) = 2𝑥𝑦1 passing

through the point (0, 1) and having slope of tangent at  = 0 as 3 is

(a) 𝑦 𝑥3  3𝑥  1                                      (b) 𝑦 𝑥3  3𝑥  1

(c) 𝑦 𝑥2  3𝑥  1                                       (d) 𝑦 𝑥2 3𝑥  1

Q.7:- For the function f(𝑥)  which of the following is true?

(a)  does not exist

(b)  does not exist

(c) Both limits exist and

(d) Both limits exist and

Q.8:- If a hyperbola passes through the foci of the ellipse   and its transverse

& conjugate axes coincide with the major and minor axes of the ellipse & product of

their eccentricities be 1, then the equation of hyperbola is

(a)                                                   (b)

(c)                                                    (d) none of these

Q.9:- If p, q, r are positive and are in A.P., then roots of the quadratic equation px2+qx+r=0

are complex for

(a)      4                                            (b)    4

(c) all p and r                                                (d) no p and r

Q.10:- For any two sets A and B if A 0 and  for some set X,

then

(a)  A                                               (b)

(c)  B                                                (d) none of these

Q.11:- If the coefficient of variation of a distribution is 45  and the mean is 12, then its

standard deviation is

(a) 5.2                                              (b) 5.3

(c) 5.4                                              (d) none of these

Q.12:- The largest term in the expansion of  where  is

(a)                                               (b)

(c)                                               (d) none of these

Q.13:- The curve described parametrically by  and  represents

(a) a pair of straight                               (b) an ellipse

(c) a parabola                                         (d) a hyperbola

Q.14:- Lets r be a relation from R (set of real numbers) to R  defined by

and  is an irrational number}. The relation r is

(a) an equivalence relation                                  (b) reflexive only

(c) symmetric only                                               (d) transitive only

Q.15:- The set  is

(a) a finite set                                                      (b) an infinite set

(c) an empty set                                                  (d) none of these

Q.16:- For  lies in the interval

(a)                                   (b)

(c)                                                        (d)

Q.17:- If  and if , then the value of  lies in the interval

(a)                                                        (b)

(c)                                                                (d)

Q.18:- If P is a point  on the line  such that P and the point  are on the opposite sides of the line  then

(a)                                                   (b)

(c)                                                    (d) none of these

Q.19:- In a sequence of 21 terms, the first 11 terms are in A.P. with common difference 2 & the last 11 terms are in G.P. with common ratio 2. If the middle term of A.P. be equal to the middle term of the G.P., then the middle term of the entire sequence is

(a)                         (b)                            (c)                              (d)

Q.20:- If n is an integer and if   , then n equals

(a) 1                                              (b) 1

(c) 2                                              (d) none of these

Q.21:- A person draws a card from a pack of playing cards, replaces it and shuffles the pack. He continues doing this until he draws a spade. The chance that he will fail the first two times is

(a)                         (b)                            (c)                              (d)

Q.22:- A particle is acted on by a force of 6 units in the direction  and is displaced from the point  to the point . The work done is

(a)  18                       (b) 15                           (c) 12                             (d) 9

Q.23:- A variable line through the point  cuts the coordinate axes in the points A and B. If the point P divides AB internally in the ratio 3 : 1, then the locus of P is

(a)                                               (b)

(c)                                                 (d)

Q.24:- The maximum value of  is

(a) 5                                              (b) 6

(c) 7                                              (d) none of these

Q.25:- The number of positive integers satisfying the inequality n+1Cn-2 n+1Cn-1  50 is

(a)  9                       (b) 8                           (c) 7                             (d) 6

Q.26:- The distance of the point on  which is nearest to the line  is

(a)                       (b)                            (c)                              (d)

Q.27:- Lets  be a differentiable function such that   . Then the value of  is

(a) 25                                              (b) 26

(c) 27                                              (d) none of these

Q.28:- If  be continuous function such that the area bounded by the curve  , the -axis and the lines  is  Value of  is

(a)                                   (b)                              (c)                               (d)

Q.29:- A curve through and satisfying the differential equation

is

(a) a circle                                              (b) a parabola

(c) an ellipse                                           (d) a hyperbola

Q.30:-If  for all  and  is

(a) 8                       (b) 16                           (c) 32                             (d) 64

Q.31:- Equation  can have a real solution for

(a) all values of a                                          (b)

(c)                                            (d)

Q.32:- Let  The number of one to one function from A to B is

(a) 24                       (b) 60                           (c) 120                             (d) 360

Q.33:- The sum of the series  is

(a)                                            (b)

(c)                                            (d)

Q.34:- If  occurs in , then  must be of the form

(a) 3k                        (b) 3k                           (c) 3k                              (d) 3k

Q.35:- The equation of the circle which cuts orthogonally the circle , passes through (3, 0) and touches the axis of y is

(a)

(b)

(c)

(d) none of these

Q.36:- Let a relation R in the set N of natural number be defined as . The relation R is

(a) reflexive                                             (b) symmetric

(c) transitive                                            (d) an equivalence relation

is

(a)                                            (b)

(c)                                            (d) none of these

Q.38:- Solution set of inequality   is

(a)                                           (b)

(c)                                       (d)

Q.39:- Let z and w be two complex numbers such that  and  Then z equals

(a) 1 or                        (b)                            (c)                           (d)

Q.40:- If the system of equations

 

has a non-trivial solution, then the value of   is

(a) 0                       (b) 1                           (c) 2                             (d) 3

Q.41:- A determinant of second order is made with the elements 0, 1. What is the probability that the determinant is non-negative?

(a)                       (b)                            (c)                              (d)

Q.42:- If  and  then angle between  and  is

(a)                       (b)                             (c)                              (d)

Q.43:- A non-zero vector  is parallel to the line of intersection of the plane determined

by the vectors  and the plane determined by the vectors . The angle between  and  is

(a)                       (b)                                 (c)                              (d) none of these

Q.44:- Number of solutions of  is

(a) 2                       (b) 3                           (c) 4                             (d) none of these

Q.45:- If the slope of the one of the lines represents by  be the square of the other, then  is

(a) 3                      (b) 4                           (c) 5                             (d) 6

Q.46:- The value of  is

(a) 1                       (b)                            (c) 0                             (d) none of these

Q.47:- If Pm stands for mPm then  is equal to

(a)                                                  (b)

(c)                                       (d)

Q.48:- Let a and b be two integers such that .

The integer n such that  is

(a) 15                       (b) 65                           (c) 115                             (d) 165

Q.49:- The mean age of a combined group of men and women is 25 years. If the mean age of the group of men is 26 and that of the group of women is 21, then the percentage of men and women in the group is

(a) 46, 60                       (b) 80, 20                           (c) 20, 80                           (d) 60, 40

Q.50:- A variable chord is drawn through the origin to the circle . The locus of centre of circle drawn on this chord as diameter is

(a)                                                 (b)

(c)                                                  (d)

Q.51:- Let R and C denote the set of real numbers and complex numbers respectively. The function  defined by  is

(a) one to one                                           (b) onto

(c) injective                                               (d) neither one to one nor onto

Q.52:- If x is complex, the expression  takes all which lie in the interval (a, b) where

(a)                                            (b)

(c)                                               (d)

Q.53:- If  be an A.P. of non-zero terms, then   is equal to

(a)                                                           (b)

(c)                                                           (d) none of these

Q.54:- A and B are two events such that  and . If A & B be independent events then P(B) is

(a)                                                (b)

(c)                                                 (d) none of these

Q.55:- If , then for all

(a)                                            (b)

(c)                                               (d)

Q.56:- The straight lines L1, L2, L3, are parallel and lie in the same plane. A total of m points are takes on L1, n points on L2, k points on L3. The maximum number of triangles formed with vertices at these points is

(a) m+n+kC3                                            (b) m+n+kC3  mC3  nC3

(c) m+n+kC3  mC3  nC3                        (d) none of these

Q.57:- If  -1( ), then  is

(a) 1 in the whole plane                                    (b)  in the whole plane

(c) 1 in the 2nd and 3rd quadrants of the plane

(d) 1 in the 3rd and 4th quadrants of the plane

Q.58:- The range of function  7-xPx-3 is

(a) {1, 2, 3, 4}                                       (b) {3, 4, 5, 6}

(c) {0, 1, 2, 3, 4, 5}                               (d) {1, 2, 3}

Q.59:- If  and  have a common root where a, b, c  N (set of natural numbers), the least value of  is

(a) 13                                 (b) 11                               (c) 7                         (d) 9

Q.60:- The eccentricity of the ellipse  is

(a)                                                        (b)

(c)                                                          (d)

Q.61:- The sum of the series  is

(a)                                                             (b)

(c)                                              (d)

Q.62:- Range of the function  is

(a)                                            (b)

(c)                                               (d)

Q.63:-   equals

(a)                                                    (b)

(c)                                                       (d)

Q.64:-If  then the value of  is

(a)                                                      (b)

(c)                                                   (d)

Q.65:-

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