Ms Quantitative Economics

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SYLLABUS AND SAMPLE QUESTIONS FOR MS(QE) 2012 Syllabus for ME I (Mathematics), 2012 Algebra: Binomial Theorem, AP, GP, HP, Exponential, Logarithmic Series, Sequence, Permutations and Combinations, Theory of Polynomial Equations (up to third degree). Matrix Algebra: Vectors and Matrices, Matrix Operations, Determinants. Calculus: Functions, Limits, Continuity, Differentiation of functions of one or more variables. Unconstrained Optimization, Definite and Indefinite Integrals: Integration by parts and integration by substitution, Constrained optimization of functions of not more than two variables. Elementary Statistics: Elementary probability theory, measures of central tendency; dispersion, correlation and regression, probability distributions, standard distributions–Binomial and Normal. Sample Questions for MEI (Mathematics), 2012 1. Kupamonduk, the frog, lives in a well 14 feet deep. One fine morning she has an urge to see the world, and starts to climb out of her well. Every day she climbs up by 5 feet when there is light, but slides back by 3 feet in the dark. How many days will she take to climb out of the well? (A) 3, (B) 8, (C) 6, (D) None of the above. 2. The derivative of f (x) = |x|2 at x = 0 is, (A) -1, (B) Non-existent, (C) 0, (D) 1/2.

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...
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