Important Instructions:

Each question is allotted 4 (four) marks for correct response, only one option is correct and for each incorrect response 1 mark i.e. ¼ (one-fourth) marks will be deducted.

Question 1:

Let  be the point of intersection of the common tangents to the parabola  and the hyperbola . If  and  denote the foci of the hyperbola where  lies on the positive -axis then  divides  in a ratio:

(A)

(B)

(C)

(D)

 

Question 2:

If three of the six vertices of a regular hexagon are chosen at random, then the probability that the triangle formed with these chosen vertices is equilateral is

(A)

(B)

(C)

(D)

 

Question 3:

Let  be a continuously differentiable function such that  and . If , then  is equal to

(A) 24

(B) 12

(C) 18

(D) 36.

 

Question 4:

The integral  is equal to : (Here  is a constant of integration)

(A)

(B)

(C)

(D)

 

Question 5:

The coefficient of  in the product  is

(A) 84

(B) 126

(C) -126

(D) -84

 

Question 6:

The number of ways of choosing 10 objects out of 31 objects of which 10 are identical and the remaining 21 are distinct, is

(A)

(B)

(C)

(D)

 

Question 7:

Let a random variable  have a binomial distribution with mean 8 and variance 4. If , then  is equal to

(A) 17

(B) 137

(C) 1

(D) 121

 

Question 8:

The equation , represents

(A) a circle of radius

(B) the line through the origin with slope 1

(C) a circle of radius 1

(D) the line through the origin with slope -1

 

Question 9:

If  is the minimum value of  for which the function  is increasing in the interval  and  is the maximum value of  in  when , then the ordered pair  is equal to

(A)

(B)

(C)

(D)

 

Question 10:

If the data  is such that the mean of first four of these is 11, the mean of the remaining six is 16 and the sum of squares of all of these is 2,000; then the standard deviation of this data is

(A)

(B) 2

(C) 4

(D)

 

Question 11:

If the area (in sq. units) of the region  is , then  is equal to

(A)

(B) 6

(C)

(D)

 

Question 12:

Consider the differential equation, . If value of y  is 1 when , then the value of  for which , is

(A)

(B)

(C)

(D)

 

Question 13:

If  and  are the roots of the equation , then  is equal to

(A)

(B)

(C)

(D)

 

Question 14:

A  ladder leans against a vertical wall. If the top of the ladder begins to slide down the wall at the rate ., then the rate (in .) at which the bottom of the ladder slides away from the wall on the horizontal ground when the top of the ladder is 1  above the ground is

(A) 25

(B)

(C)

(D)

 

Question 15:

The number of solutions of the equation  is

(A) 3

(B) 4

(C) 5

(D) 7

 

Question 16:

If  , the ordered pair  at  is equal to
(A)

(B)

(C)

(D)

 

Question 17:

The value of  is equal to

(A)

(B)

(C)

(D)

 

Question 18:

If the normal to the ellipse  at a point  on it is parallel to the line,  4 and the tangent to the ellipse at  passes through  then  is equal to

(A)

(B)

(C)

(D)

 

Question 19:

Let  and  be two vectors. If a vector perpendicular to both the vectors  and  has the magnitude 12 then one such vector is

(A)

(B)

(C)

(D)

 

Question 20:

The equation  represents a straight line lying in

(A) first, third and fourth quadrants

(B) first, second and fourth quadrants

(C) third and fourth quadrants only

(D) second and third quadrants only

 

Question 21:

For  let . If , then  is equal to

(A)

(B)

(C)

(D)

 

Question 22:

If the angle of intersection at a point where the two circles with radii 5 cm and 12 cm intersect is 90°, then the length (in cm) of their common chord is

(A)

(B)

(C)

(D)

 

Question 23:

If , then  is equal to

(A) 1

(B)

(C)

(D)

 

Question 24:

Let S_n denote the sum of the first n terms of an A.P. If S₄ = 16 and S₆ = –48, then S₁₀ is equal to

(A)

(B)

(C)

(D)

 

Question 25:

If  is the inverse of a 3 ´ 3 matrix A, then the sum of all values of a for which , is

(A) 0

(B)

(C) 1

(D) 2

 

Question 26:

If the line  intersects the plane  at a point P and the plane  at a point Q, then PQ is equal to

(A)

(B) 14

(C)

(D)

 

Question 27:

If the volume of parallelepiped formed by the vectors and  is minimum, then  is equal to

(A)

(B)

(C)

(D)  

 

Question 28:

If A is a symmetric matrix and B is a skew-symmetric matrix such that

, then  is equal to

(A)

(B)

(C)

(D)

 

Question 29:

If the truth value of the statement p→(~ q∨r) is false (F), then the truth values of the statement p, q, r, are respectively

(A)

(B)

(C)

(D)

 

Question 30:

For let  denotes the greatest integer function , then the sum of the series  is

(A)

(B)

(C)

(D)

**********