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
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The probability of using 2 2 transports A, B, C, D by an individual is 1 , 9 , 4 , 9 respectively. The 9 9 probability that he arrives late at work if he uses transportation A, B, 4 C, D is 5 , 7 , 6 , and 6 respectively. What is the probability that he 7 7 7 used transport A if he reached oﬃce on time? A B C D 1 , 9 1 , 7 3 , 7 2 . 9 19. What is the least (strictly) positive value of the expression a3 +b3 +c3 − 3abc, where a, b, c vary over all strictly positive integers? (You may use ( ) the identity a3 +b3 +c3 −3abc = 1 (a+b+c) (a−b)2 +(b−c)2 +(c−a)2 .) 2 A 2, B 3, C 4, D 8. 20. If a2 + b2 + c2 = 1, then ab + bc + ca is, (A) −0.75, (B) Belongs to the interval [−1, −0.5], (C) Belongs to the interval [0.5, 1], (D) None of the above. 21. Consider the following linear programming problem: Maximize a + b subject to a + 2b ≤ 4, a + 6b ≤ 6, 5 a − 2b ≤ 2, a, b ≥ 0. An optimal solution is: (A) a=4, b=0, (B) a=0, b=1, (C) a=3,b=1/2, (D) None of the above. ∫ −1 1 22. The value of −4 x dx equals, (A) ln 4, (B) Undeﬁned, (C) ln(−4) − ln(−1), (D) None of the above. 23. Given x ≥ y ≥ z, and x + y + z = 9, the maximum value of x + 3y + 5z is (A) 27, (B) 42, (C) 21, (D) 18. 24. A car with six sparkplugs is known to have two malfunctioning ones. If two plugs are pulled out at random, what is the probability of getting at least one malfunctioning plug. (A) 1/15, (B) 7/15, (C) 8/15, (D) 9/15. 25. Suppose there is a multiple choice test which has 20 questions.

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