3. Let N = {1, 2, 3, . . .} be the set of natural numbers. For each n ∈ N , deﬁne An = {(n + 1)k : k ∈ N }. Then A1 ∩ A2 equals (A) A3 , (B) A4 , (C) A5 , (D) A6 . 4. Let S = {a, b, c} be a set such that a, b and c are distinct real numbers. Then min{max{a, b}, max{b, c}, max{c, a}} is always (A) the highest number in S, (B) the second highest number in S, (C) the lowest number in S, (D) the arithmetic mean of the three numbers in S. 5. The sequence < −4−n >, n = 1, 2, · · · , is (A) Unbounded and monotone increasing, (B) Unbounded and monotone decreasing, (C) Bounded and convergent, (D) Bounded but not convergent. ∫ x 6. 7x2 +2 dx equals 1 (A) 14 ln(7x2 + 2)+ constant, (B) 7x2 + 2, (C) ln x+ constant, (D) None of the above.

7. The number of real roots of the equation 2(x − 1)2 = (x − 3)2 + (x + 1)2 − 8 is (A) Zero, (B) One, (C) Two, (D) None of the above. 2

8. The three vectors [0, 1], [1, 0] and [1000, 1000] are (A) Dependent, (B) Independent, (C) Pairwise orthogonal, (D) None of the above. 9. The function f (.) is increasing over [a, b]. Then [f (.)]n , where n is an odd integer greater than 1, is necessarily (A) Increasing over [a, b], (B) Decreasing over [a, b], (C) Increasing over [a, b] if and only if f (.) is positive over [a, b], (D) None of the above. 10. The determinant of the matrix (A) 21, (B) -16, (C) 0, (D) 14. 11. In what ratio should a given line be divided into two parts, so that the area of the rectangle formed by the two parts as the sides is the maximum possible? (A) 1 is to 1, (B) 1 is to 4, (C) 3 is to 2, (D) None of the above. 12. Suppose (x∗ , y ∗ ) solves: M inimize ax + by, subject to xα + y α = M, and x, y ≥ 0, where a > b > 0, M > 0 and α > 1. Then, the solution is, 3 1 2 3 4 5 6 7 8 9 is

(A)

(B) x = 0, y ∗ = M α , 1 (C) y ∗ = 0, x∗ = M α , (D) None of the above. 1

x∗α−1 y∗α−1 ∗

= a, b

13. Three boys and two girls are to be seated in a row for a photograph. It is desired that no two girls sit together. The number of ways in which they can be so arranged is (A) 4P2 × 3!, (B) 3P2 × 2! (C) 2! × 3! (D) None of the above. √ √ √ 14. The domain of x for which x + 3 − x + x2 − 4x is real is, (A) [0,3], (B) (0,3), (C) {0}, (D) None of the above. 15. P (x) is a quadratic polynomial such that P (1) = P (-1). Then (A) The two roots sum to zero, (B) The two roots sum to 1, (C) One root is twice the other, (D) None of the above. √ √ √ √ 16. The expression 11 + 6 2 + 11 − 6 2 is (A) Positive and an even integer, (B) Positive and an odd integer, (C) Positive and irrational, (D) None of the above. 17. What is the maximum value of a(1 − a)b(1 − b)c(1 − c), where a, b, c vary over all positive fractional values? A 1, B 1 , 8

4

C D

1 , 27...