Advanced Mathematics MCQs

31

Advanced Mathematics • Queueing Theory

In a take-away food joint with a single service counter, customers are served on a first-come-first-served basis. The arrival rate follows a Poisson distribution and service time follows an exponential distribution. If 2 customers arrive every 10 minutes and 18 customers are served per hour, what is the expected number of customers in the joint at any given time? In a take-away food joint with a single service counter, customers are served on a first-come-first-served basis. The arrival rate follows a Poisson distribution and service time follows an exponential distribution. If 2 customers arrive every 10 minutes and 18 customers are served per hour, what is the expected number of customers in the joint at any given time?

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A. 2 2 B. 5 5 C. 9 9 D. 1 1

सही उत्तर: A Correct Answer: A

For the M/M/1 queue, lambda = 12 per hour and mu = 18 per hour. The expected number in the system is lambda/(mu - lambda) = 12/6 = 2.

32

Advanced Mathematics • Abstract Algebra

The set of units in Z_14, under the usual operations of addition and multiplication modulo 14, is: The set of units in Z_14, under the usual operations of addition and multiplication modulo 14, is:

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A. Z_14 Z_14 B. {1} {1} C. {1, 13, 3, 5, 11, 9} {1, 13, 3, 5, 11, 9} D. {1, -1, 3, 5, 2, 7} {1, -1, 3, 5, 2, 7}

सही उत्तर: C Correct Answer: C

The units modulo 14 are exactly the residue classes relatively prime to 14: 1, 3, 5, 9, 11 and 13.

33

Advanced Mathematics • Limits

Calculate lim (x^2 - 4)/(x^2 + 5x - 14) as x approaches 2. Calculate lim (x^2 - 4)/(x^2 + 5x - 14) as x approaches 2.

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A. Undefined Undefined B. 2/7 2/7 C. 4/5 4/5 D. 4/9 4/9

सही उत्तर: D Correct Answer: D

Factor and cancel x - 2: (x^2 - 4)/(x^2 + 5x - 14) = (x + 2)/(x + 7). At x = 2 this equals 4/9.

34

Advanced Mathematics • Digital Logic

Which logic gate has the following truth table? Inputs (A,B) and output C: (0,0)->0, (0,1)->1, (1,0)->1, (1,1)->1. Which logic gate has the following truth table? Inputs (A,B) and output C: (0,0)->0, (0,1)->1, (1,0)->1, (1,1)->1.

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A. OR with 2 inputs OR with 2 inputs B. NAND NAND C. AND with 2 inputs AND with 2 inputs D. NOR NOR

सही उत्तर: A Correct Answer: A

The output is 1 whenever at least one input is 1, which is the truth table of a two-input OR gate.

35

Advanced Mathematics • Group Theory

Which one of the following is a group? Which one of the following is a group?

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A. The set of all 2 by 2 matrices under multiplication. The set of all 2 by 2 matrices under multiplication. B. (R+, x), where a x b = sqrt(ab) for all a, b in R+. (R+, x), where a x b = sqrt(ab) for all a, b in R+. C. (C[0,1], x), where (f x g)(x) = f(x)g(x). (C[0,1], x), where (f x g)(x) = f(x)g(x). D. The set of odd integers under addition. The set of odd integers under addition.

सही उत्तर: C Correct Answer: C

Option C is the answer marked in the source paper. It represents the stated collection of continuous functions under pointwise multiplication.

36

Advanced Mathematics • Dynamics

The moment of inertia of a uniform thin rod of mass M and length L about a perpendicular axis of rotation at its end is: The moment of inertia of a uniform thin rod of mass M and length L about a perpendicular axis of rotation at its end is:

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A. (1/8)ML^2 (1/8)ML^2 B. (1/2)ML^2 (1/2)ML^2 C. (1/3)ML^2 (1/3)ML^2 D. (1/4)ML^2 (1/4)ML^2

सही उत्तर: C Correct Answer: C

The moment of inertia of a uniform rod about its centre is ML^2/12. By the parallel-axis theorem, about an end it is ML^2/12 + M(L/2)^2 = ML^2/3.

37

Advanced Mathematics • Differential Equations

Which of the following can be reduced to a Clairaut equation? Which of the following can be reduced to a Clairaut equation?

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A. y' = e^x + e^y + y^2 y' = e^x + e^y + y^2 B. y' = sin(x) + sin(y) y' = sin(x) + sin(y) C. y = x^4p - p^2x, where p = dy/dx y = x^4p - p^2x, where p = dy/dx D. y = 2xp + y^2p^3, where p = dy/dx y = 2xp + y^2p^3, where p = dy/dx

सही उत्तर: D Correct Answer: D

Option D is the equation marked in the source paper as reducible, after suitable substitution, to Clairaut form.

38

Advanced Mathematics • Applied Mechanics

The equation for finding the centre of pressure of a rocket is (where (C_N alpha)_n denotes the normal-force coefficient derivative of the nose): The equation for finding the centre of pressure of a rocket is (where (C_N alpha)_n denotes the normal-force coefficient derivative of the nose):

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A. (C_N alpha)_n = 3 (C_N alpha)_n = 3 B. (C_N alpha)_n = 2 (C_N alpha)_n = 2 C. (C_N alpha)_n = 1 (C_N alpha)_n = 1 D. (C_N alpha)_n = 4 (C_N alpha)_n = 4

सही उत्तर: B Correct Answer: B

For the rocket nose contribution used in the centre-of-pressure relation, the normal-force coefficient derivative is 2.

39

Advanced Mathematics • Digital Logic

Which is called the universal gate? Which is called the universal gate?

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A. AND AND B. OR OR C. NAND NAND D. XNOR XNOR

सही उत्तर: C Correct Answer: C

NAND is a universal gate because every Boolean function can be implemented using NAND gates alone.

40

Advanced Mathematics • Vector Calculus

Evaluate the line integral over C of (x^4 dx + xy dy), where C is the triangular curve consisting of line segments from (0,0) to (1,0), from (1,0) to (1,1), and from (1,1) to (0,0). Evaluate the line integral over C of (x^4 dx + xy dy), where C is the triangular curve consisting of line segments from (0,0) to (1,0), from (1,0) to (1,1), and from (1,1) to (0,0).

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A. 0 0 B. 12 12 C. 1/6 1/6 D. 1 1

सही उत्तर: C Correct Answer: C

Evaluating the integral along the three directed line segments and adding the results gives 1/6.

Advanced Mathematics MCQs

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