PART-A[SINGLE CORRECT CHOICE TYPE]

Q.1 to Q.10 has four choices (A), (B), (C), (D) out of which ONLY ONE is correct.

Q.1 Total number of ordered pairs (x, y) of real numbers satisfying

| tan y | + (sin x)2 = 0 and x2 + y2 2 is equal to(A) 3 (B) 6 (C*) 9 (D) 12

Q.2 If

0n

n2cos = 4, then the most general values of are given by

(A) n 3

(B) 2n

4

(C*) n

6

(D) 2n

6

(where n I).

Q.3 If tan2x = 335

13

, then the general values of x are

(A*) 12

n

(B) 8

n

(C) 12

5n

(D)

8

3n

where n I.

Q.4 The sum of all the solution(s) in x of the equation a2 2a + sec2 )xa( = 0 lying in [1, 100], is(A) 100 (B*) 5050 (C) 5049 (D) 1

Q.6 There are 7 persons which include a group of 3 friends F1, F

2, F

3. Number of ways they can be seated

on a round table if atleast two out of F1, F

2, F

3 are seated next to each other is

(A) 144 (B*) 576 (C) 360 (D) 432

Q.7 Total number of 5 digit numbers having all different digits and divisible by 4 that can be formed using the

digits {1, 3, 2, 6, 8, 9}, is equal to

(A*) 192 (B) 32 (C) 1152 (D) 384

Q.8 If the letters of the word LAUGH are arranged by a lexicographer then the number of words that appear

before the word LAUGH is

(A*) 76 (B) 77 (C) 78 (D) 79

Q.9 If the line y = 2x 4 does not intersect the parabola y = x2 + 2ax, then the true set of values of a, is

(A*) 1 < a < 3 (B) 2 < a < 4 (C) 3 < a < 3 (D) 3 < a < 5

Q.10 The number of integral values of 'm' for which atleast one solution of the inequality

x2 2mx + m2 4 0 satisfies the inequality x2 6x + 8 0, is(A) 4 (B) 5 (C) 6 (D*) 7

[MULTIPLE CORRECT CHOICE TYPE]Q.1 to6 has four choices (A), (B), (C), (D) out of which ONE OR MORE may be correct.

Q.1 Number of ways in which 10 distinct things can be distributed to 3 children if each child receiving none,

one or more number of things, is NOT equal to

(A*) the number of ways of all possible selections of one or more questions from 10 given questions,

each question having an alternative.

(B) the sum of all the coefficients in the expansion of the binomial (2p + q)10.

(C*) number of 10 digit numbers (containing at least one odd digit) that can be written, if each digit of the

number is selected from the set {1, 2, 3, 4, 5, 6}.

(D*) number of different signals that can be transmitted by making use of 3 different coloured flags

keeping one above the other, if 10 different flags are available.

Q.2 If m be the number of positive integral solutions of x + y + z + w = 10, then m is divisible by

(A*) 3 (B*) 4 (C) 5 (D*) 7

Q.3 An 'n' digit number is constructed using only the digit 1, 2, 3 or 4 such that it contain the digit 1 either an

even number of times or not at all. If the number of such 'n' digit numbers are 528 then the value of 'n' is

coprime with

(A) 5 (B*) 6 (C*) 7 (D*) 8

Q.4 If both roots of the quadratic equation f(x) = x2 px + q = 0 are confined in the interval (a, b)

where a < b, then which of the following condition(s) must hold good?

(A*) p2 4q (B*) f(a) > 0 (C*) f (b) > 0 (D*) a < 2

p < b

Q6

Q.5 For which of the following graphs of quadratic expression y = ax2 + bx + c, then product abc is

positive?

(A) (B*)

(C) (D*)

Q.6 If the real roots of the quadratic equation x2 ax + 1 = 0 differ by unity then [a] can be equal to

[Note: [a] denotes the greatest integer less than or equal to a .]

(A*) 2 (B) 3 (C*) 3 (D) 2

[INTEGER TYPE]

Q.1 to Q7 are "Integer Type" questions. (The answer to each of the questions are from 0-9 )

Q.1 Find the number of positive integral values of k for which kx2 + (k 3) x + 1 < 0 for atleast

one positive x.

Q.2 If both the roots of equation x2 2kx + (k2 + k 5) = 0 are smaller than 5, then find the largest integral

value of k.

Ans. 3

Q.4 Consider the quadratic equation P(x) = x2 mx + 4m + 20 = 0 where m is a real parameter.

Let A = {10, 9, 8,..............., 8, 9, 10} be the set of 21 integers.

If S is the set of all values of m for which P(x) = 0 has real and distinct roots then find the number of

elements in (A S).Ans. 0006

Q.3 Find the value of k R for which the quadratic equation x2 + (2k + 1)x + (k2 + 2) = 0 has two realroots such that one root is twice the other root.

Ans. 0004

Q.5 If the expression y = )ax(

)5x)(1x(

is capable of taking all real values for x R (the set of all real

numbers), then find the number of integral values of 'a'.

Ans. 0003

Q.7 If x2 11x + 30 and x2 9x + 18 are both distinct positive integers, then find the integral solution of the

equation 18x9x2 30x11x

= 30x11x2 18x9x

.

Ans. 7

Q.6 If , and are the roots of the equation 5x3 qx 1 = 0, (q R) then find the value of

333 222

.

Ans. 0003