Important Instructions:

MCQs: 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  and  be roots of  and  and  be the roots of . If , , form a geometric progression. Then ratio  is

(A)

(B)

(C)

(D) .

 

Question 2:

Let  and . Then  is equal to

(A)

(B)

(C) 1

(D) 0.

 

Question 3:

Let  be a twice differentiable function on (1, 6). If  and  " , for all , then

(A)

(B)

(C)

(D) .

 

Question 4:

Let  be given ellipse, length of whose latus rectum is 10. If its eccentricity is the maximum value of the function, , then  is equal to

(A) 126

(B) 135

(C) 116

(D) 145.

 

Question 5:

The mean and variance of 8 observations are 10 and 13.5, respectively. If 6 of these observations are , then the absolute difference of the remaining two observations is

(A) 3

(B) 5

(C) 7

(D) 9.

 

Question 6:

Let  be the point of local maxima of  where  and . Then the value of  at  is

(A) -4

(B) -22

(C) -30

(D) 14.

 

Question 7:

If , where , then  at  is

(A)

(B)

(C)

(D)

 

Question 8:

Two vertical poles  and  are standing apart on a horizontal ground with points  and  on the ground. If  is the point of intersection of  and , then the height of  (in  ) above the line  is

(A)

(B) 6

(C) 5

(D) .

 

Question 9:

If , then an ordered pair (α,β)  is equal to

(A)

(B)

(C)

(D) .

 

Question 10:

Given the following two statements:

 is a tautology.

 is a fallacy. Then

(A) both  and  are correct

(B) only  is correct

(C)both  and  are incorrect

(D) only  is correct.

 

Question 11:

Let  and . If the curve represented by  intersects the -axis at the points  and  where , then the value of  is

(A) 2

(B)

(C)

(D) 4.

 

Question 12:

 triangle  lying in the first quadrant has two vertices as  and . If , and ar⁡(△ABC)=55  sq. units, then the abscissa of the vertex  is

(A)

(B)

(C)

(D) .

 

Question 13:

Let  be a point on the hyperbola, . If the normal to it at  intersects the -axis at  and  is its eccentricity, then the ordered pair  is equal to

(A)

(B)

(C)

(D) .

 

Question 14:

The integral  is equal to (where  is constant of integration)

(A)

(B)
(C)

(D) .

 

Question 15:

If  and , where , then, which one of the following is not true?

(A)

(B)

(C)

(D) .

 

Question 16:

The value of  is equal to

(A)

(B)

(C)

(D) .

 

Question 17:

Let  denote the greatest integer . Then the equation in  has

(A) infinitely many solutions

(B) exactly two solutions

(C) no integral solution

(D) exactly four integral solutions.

 

Question 18:

Let . Then  is equal to
(A)

(B)

(C)

(D) .

 

Question 19:

A survey shows that  of the people in a city read newspaper A whereas  read newspaper . if  of the people read both the newspapers, then a possible value of  can be

(A) 29

(B) 55

(C) 37

(D) 65.

 

Question 20:

Let  be the solution of the differential equation, . If , then  is equal to

(A)

(B)

(C)

(D) .

 

Numerical Value Based Questions: Each question will have 4 marks for correct response and there will not be any negative marking for the wrong answer in this section.

Question 21:

Let . Then  is equal to __________ .

 

Question 22:

If the equation of a plane , passing through the intersection of the planes,    and  is  for some , then the distance of the point  from the plane  is __________.

 

Question 23:

If the system of equations: x – 2y + 3z = 9, 2x + y + z = b & x – 7y + az = 24, has infinitely many solutions, then a – b is equal to __________.

 

Question 24:

Suppose a differentiable function f(x) satisfies the identity . If , then  is equal to __________.

 

Question 25:

The probability of a man hitting a target is 1/10. The least number of shots required, so that the probability of his hitting the target at least once is greater than 1/4, is __________.

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