­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  be a continuous function such that . If , then  is equal to :
(A) 4
(B) 10
(C) 11
(D) 16

 

Question 2:

Let  be the origin and  be the point . If  is the point , such that  is a right angled isosceles triangle with  as hypotenuse, then which of the following is NOT true?
(A)
(B)
(C)
(D)

 

Question 3:

If the system of linear equations.

has infinitely many solutions, then the distance of the point  from the plane  is :
(A)
(B) 4
(C)
(D)

 

Question 4:

Let  be a  matrix with  and . Then the sum of the diagonal elements of  can be :
(A) -1
(B) 2
(C) 1
(D)

 

Question 5:

The odd natural number a, such that the area of the region bounded by  is , is equal to :
(A) 3
(B) 5
(C) 7
(D) 9

 

Question 6:

Consider two G.P.'s.  and  of 60 and  terms respectively. If the geometric mean of all the  terms is , then  is equal to :
(A) 560
(B) 1540
(C) 1330
(D) 2600

 

Question 7:

If the function  is continuous at , then  is equal to :
(A) 1
(B) -1
(C) e
(D) 0

 

Question 8:

If  and  are continuous on , then  is equal to :
(A) -10
(B) 10
(C) 8
(D) -8

 

Question 9:

Let . Then the set of all values of , for which  has maximum value at , is :
(A)
(B)
(C)
(D)

 

Question 10:

If  and , then :
(A)
(B)
(C)
(D)

 

Question 11:

If  and , then the maximum value of  is :
(A)
(B)
(C)
(D)

 

Question 12:

A point  moves so that the sum of squares of its distances from the points  and  is 14. Let  be the locus of , which intersects the -axis at the points  and the -axis at the points . Then the area of the quadrilateral  is equal to :
(A)
(B)
(C)
(D) 9

 

Question 13:

Let the tangent drawn to the parabola  at the point  is perpendicular to the line . Then the normal to the hyperbola  at the point  does NOT pass through the point :
(A)
(B)
(C)
(D)

 

Question 14:

The length of the perpendicular from the point  on the line passing through  and parallel to the line  is :
(A)
(B)
(C)
(D) 1

 

Question 15:

Let  and . If the projection of  on the vector  is 30 , then  is equal to :
(A)
(B) 8
(C)
(D) 7

 

Question 16:

The mean and variance of a binomial distribution are  and  respectively. If , then  or 5  is equal to :
(A)
(B)
(C)
(D)

 

Question 17:

Let  be three mutually exclusive events such that  and . If the maximum and minimum values of  are  and , then  is equal to :
(A)
(B)
(C)
(D) 1

 

Question 18:

Let . Then  is equal to :
(A) 0
(B) -2
(C) -4
(D) 12

 

Question 19:  

 is equal to

(A) 1
(B) 2
(C)
(D)

 

Question 20:  

The statement  is 

(A) a tautology
(B) a contradiction
(C) equivalent
(D) equivalent

 

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:

 If for some , not all have same sign, one of the roots of the equation  is also a root of the equation , then  is equal to

 

Question 2:

The number of 5-digit natural numbers, such that the product of their digits is 36 , is.......

 

Question 3:

The series of positive multiples of 3 is divided into sets : . Then the sum of the elements in the  set is equal to

 

Question 4:

The number of distinct real roots of the equation  is

 

Question 5:

If the coefficients of  and  in the expansion of , are -3 and -5 respectively, then the coefficient of  is equal to

 

Question 6:

If , then  is equal to

 

Question 7:

Let a curve  pass through the point  and the area of the region under this curve, above the -axis and between the abscissae 3 and  be . If this curve also passes through the point  in the first quadrant, then  is equal to

 

Question 8:

The equations of the sides  and  of a triangle  are  and  respectively. If its orthocenter is , then  is equal to

 

Question 9:

Let the function , be decreasing in ( ) and increasing in (a, 4). A tangent to the parabola  at a point  on it passes through the point  but does not pass through the point . If the equation of the normal at  is , then  is equal to

 

Question 10:

Let  and  be two points on the line  at a distance  from the point , . Then the square of the area of the triangle  is                                                          

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