SECTION - A

(One Options Correct Type)

This section contains 20 multiple choice questions. Each question has four choices (A), (B), (C) and (D), out of which ONLY ONE option is correct.

 

Question 1:

Let the position vectors of the points A, B, C and D be  ,    and . Let the set S ={   l Î R:  the points A, B, C and D are coplanar}. Then is equal to

(A)41

(B)25

(C)

(D)13

 

Question 2:

If , then  at (2, 2) is equal to:

(A)  

(B)

(C)

(D)

 

Question 3:

One vertex of a rectangular parallelepiped is at the origin O and the lengths of its edges along x, y and z axes are 3, 4 and 5 units respectively. Let P be the vertex (3, 4, 5) . Then the shortest distance between the diagonal OP and an edge parallel to z axis, not passing through O or P is :

(A)

(B)

(C)

(D)

 

Question 4:

The straight lines  and  pass through the origin and trisect the line segment of the line L :  between the axes. If m1 and m2 are the slopes of the lines  and , then the point of intersection of the line  with L lies on

(A)

(B)

(C)

(D)

 

Question 5:

If the equation of the plane passing through the line of intersection of the planes  ,  and parallel to the line  is , then  is equal to

 (A)

(B) 15

(C)

(D)

 

Question 6:

Let .  and  . If  is a vector perpendicular to both , and , then  is equal to

 (A)

(B)760

(C)

(D)

 

Question 7:

A pair of dice is thrown 5 times. For each throw, a total of 5 is considered a success. If the probability of at least 4 successes is  , then k is equal to

 (A)

(B)75

(C)

(D)

 

Question 8:

Let    , where [t] denotes greatest integer function. Then,

 (A)

(B)

(C)

(D)

 

Question 9:

Let . If  , then  is equal to 

(A)

(B)

(C)

(D)

 

Question 10:

The sum of all the roots of the equation  is  

(A)

(B)

(C)

(D)

 

Question 11:

From the top A of a vertical wall AB of height 30 m, the angles of depression of the top P and bottom of Q of a vertical tower PQ are 15 and 60 respectively, B and Q are on the same horizontal level. If C is a point on AB such that CB = PQ, then the area (in  ) of the quadrilateral BCPQ is equal to  

(A)

(B)

(C)

(D)

 

Question 12:

If the system of equations

 

 

 

has infinitely many solutions, then 2a+ 3b  is equal to

(A)

(B)

(C)

(D)

 

Question 13:

Statement  is logically equivalents to

(A)

(B)

(C)

(D) 

 

Question 14:

Let  be n positive consecutive terms of an arithmetic progression. If d > 0 is its common difference, then

 is

(A)

(B)

(C)

(D)

 

Question 15:

The mean and variance of a set of 15 numbers are 12 and 14 respectively. The mean and variance of another set of 15 numbers are 14 and 2 s respectively. If the variance of all the 30 numbers in the two sets is 13, then σ2  is equal to

(A) 9

(B)

(C)

(D)

 

Question 16:

The sum of the first 20 terms of the series 5+11+19+29+41+… is

(A) 3450

(B)

(C)

(D)

 

Question 17:

Let  , where all i, j and . Let a be the sum of all diagonal elements of A and  . Then  is equal to

(A) 7

(B)

(C)

(D)

 

Question 18:

Let .Then  is equal to :

(A)

(B)

(C) 

(D) 

 

Question 19:

If the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of is , then the third term from the beginning is :

(A) 

(B)

(C)

(D)

 

Question 20:

If  , the ratio  is  

(A) 

(B)

(C)

(D)

 

SECTION - B

(Numerical Answer Type)

This section contains 10 Numerical based questions. The answer to each question is rounded off to the nearest integer value.

Question 1:

Let  be a solution of the differential equation , 0<x<π2 . If y  ,then '  is equal to

 

Question 2:

Let A= .And B={0,1,2,3,4}. The number of elements The number of elements in the relation is

 

Question 3:

A circle passing through the point P , (a b) in the first quadrant touches the two coordinate axes at the points A and B. The point P is above the line AB. The point Q on the line segment AB is the foot of perpendicular from P on AB. If PQ is equal to 11 units, then the value of ab is………..

 

Question 4:

Let the point (p,p+1 ) lie inside the region

 If the set of all values of p is the interval (a, b), then  is equal to……….

 

Question 5:

Let the image of the point P (1,23) in the plane 2x -y+z=9 be Q. If the coordinates of the point R are (6,10,7) , then the square of the area of the triangle PQR is………..

 

Question 6:

Let the tangent to the curve  at the point P(1,3) on it meet the y-axis at A. Let the line passing through P and parallel to the line x-3y=6 meet the parabola  at B. If B lies on the line  then  is equal to…………

 

Question 7:

If the area of the region  is equal to , then the natural number n is equal to……….

 

Question 8:

Let  and [t] be the greatest integer £ t . Then the number of points, where the function    is not differentiable, is……….

 

Question 9:

The number of ways of giving 20 distinct oranges to 3 children such that each child gets at least one orange is………..

 

Question 10:

The coefficient of  in the expansion of …………..

**********