Mathematics MCQs

291

Mathematics • Coordinate Geometry

The Cartesian equation of the surface x = mu cos(nu), y = mu sin(nu), z = mu cot(nu) is: The Cartesian equation of the surface x = mu cos(nu), y = mu sin(nu), z = mu cot(nu) is:

A. x^2 + z^2 = y^2 tan(nu) x^2 + z^2 = y^2 tan(nu) B. x^2 + y^2 = z^2 cot(nu) x^2 + y^2 = z^2 cot(nu) C. x^2 + y^2 = z^2 tan^2(nu) x^2 + y^2 = z^2 tan^2(nu) D. x^2 + z^2 = y^2 cot(nu) x^2 + z^2 = y^2 cot(nu)

सही उत्तर: C Correct Answer: C

From x^2 + y^2 = mu^2 and z = mu cot(nu), we have mu = z tan(nu). Therefore x^2 + y^2 = z^2 tan^2(nu).

292

Mathematics • Solid Geometry

Revolving a circular area of radius R through 360 degrees about the x-axis generates a complete torus. The distance between the centre of the circle and the x-axis is d. The surface area of the torus is: Revolving a circular area of radius R through 360 degrees about the x-axis generates a complete torus. The distance between the centre of the circle and the x-axis is d. The surface area of the torus is:

A. 3pi^2Rd 3pi^2Rd B. 4pi^2Rd 4pi^2Rd C. pi^2Rd pi^2Rd D. 2pi^2Rd 2pi^2Rd

सही उत्तर: B Correct Answer: B

By Pappus' centroid theorem, surface area = circumference of the generating circle x distance travelled by its centroid = 2pi R x 2pi d = 4pi^2Rd.

293

Mathematics • Probability

The density of a random variable X is f_X(x) = a + bx^2 for x in [0,1], and 0 otherwise. Find (a,b) if E(X) = 3/5. The density of a random variable X is f_X(x) = a + bx^2 for x in [0,1], and 0 otherwise. Find (a,b) if E(X) = 3/5.

A. (3/5, -3/5) (3/5, -3/5) B. (3, -3) (3, -3) C. (1, 2) (1, 2) D. (3/5, 6/5) (3/5, 6/5)

सही उत्तर: D Correct Answer: D

Normalization gives a + b/3 = 1, while E(X) = a/2 + b/4 = 3/5. Solving yields a = 3/5 and b = 6/5.

294

Mathematics • Queueing Theory

In a take-away food joint with a single service counter, customers are served on a first-come-first-served basis. Arrivals follow a Poisson distribution and service times follow an exponential distribution. If 2 customers arrive every 10 minutes and 18 customers are served per hour, what is the probability that an incoming customer waits for more than 30 minutes before being served? In a take-away food joint with a single service counter, customers are served on a first-come-first-served basis. Arrivals follow a Poisson distribution and service times follow an exponential distribution. If 2 customers arrive every 10 minutes and 18 customers are served per hour, what is the probability that an incoming customer waits for more than 30 minutes before being served?

A. (2/3)e^-4 (2/3)e^-4 B. (2/3)e^-2.5 (2/3)e^-2.5 C. (2/3)e^-3 (2/3)e^-3 D. (2/3)e^-2 (2/3)e^-2

सही उत्तर: C Correct Answer: C

Here lambda = 12 per hour, mu = 18 per hour and rho = 2/3. For an M/M/1 queue, P(Wq > t) = rho e^(-(mu-lambda)t). At t = 0.5 hour this is (2/3)e^-3.

295

Mathematics • Digital Logic

The number of Boolean functions that can be generated by n variables is: The number of Boolean functions that can be generated by n variables is:

A. 2^(n-1) 2^(n-1) B. 2^n 2^n C. 2^(2^n) 2^(2^n) D. 2^(2n) 2^(2n)

सही उत्तर: C Correct Answer: C

There are 2^n possible input combinations, and each can independently map to 0 or 1. Hence the number of Boolean functions is 2^(2^n).

296

Mathematics • Multiple Integrals

Evaluate the double integral over D of xy dA, where D is the region bounded by the line y = x - 1 and the curve y^2 = 2x + 6. Evaluate the double integral over D of xy dA, where D is the region bounded by the line y = x - 1 and the curve y^2 = 2x + 6.

A. 100 100 B. 36 36 C. 10 10 D. 6 6

सही उत्तर: B Correct Answer: B

The curves meet at y = -2 and y = 4. Writing x from (y^2 - 6)/2 to y + 1 and integrating xy first with respect to x and then y gives 36.

297

Mathematics • Partial Differential Equations

The partial differential equation obtained by eliminating the arbitrary function f from z = e^(mx)f(x + y) is: The partial differential equation obtained by eliminating the arbitrary function f from z = e^(mx)f(x + y) is:

A. p - q = mz p - q = mz B. q - r = z q - r = z C. p - t = z p - t = z D. p + q = mz p + q = mz

सही उत्तर: A Correct Answer: A

Differentiating gives p = mz + e^(mx)f'(x+y) and q = e^(mx)f'(x+y). Hence p - q = mz.

298

Mathematics • Topology

Let A = {A1, A2, A3, ...} be an infinite collection of closed sets. Which of the following is true? Let A = {A1, A2, A3, ...} be an infinite collection of closed sets. Which of the following is true?

A. Arbitrary intersection of sets from A is open Arbitrary intersection of sets from A is open B. Finite union of sets from A is closed Finite union of sets from A is closed C. None of the options are true None of the options are true D. Arbitrary union of sets from A is closed Arbitrary union of sets from A is closed

सही उत्तर: B Correct Answer: B

A finite union of closed sets is always closed. An arbitrary union of closed sets need not be closed, while an arbitrary intersection of closed sets is closed rather than necessarily open.

299

Mathematics • Linear Algebra

A is a square matrix with each row sum equal to 1. Which statement regarding the eigenvalues of A is always true? A is a square matrix with each row sum equal to 1. Which statement regarding the eigenvalues of A is always true?

A. 0 is an eigenvalue of A. 0 is an eigenvalue of A. B. 1 is an eigenvalue of A. 1 is an eigenvalue of A. C. Both 0 and 1 are eigenvalues of A. Both 0 and 1 are eigenvalues of A. D. No statement can be made about an eigenvalue of A from the given information. No statement can be made about an eigenvalue of A from the given information.

सही उत्तर: B Correct Answer: B

If e is the column vector whose entries are all 1, then Ae = e because every row sum is 1. Therefore 1 is an eigenvalue.

300

Mathematics • Mathematical Modelling

Let V represent the temperature in degrees Fahrenheit of an object in a room whose temperature is kept constant at 60 degrees Fahrenheit. If the object cools from 100 degrees to 90 degrees in 10 minutes, how much more time, in minutes, will it take for its temperature to decrease to 80 degrees? Let V represent the temperature in degrees Fahrenheit of an object in a room whose temperature is kept constant at 60 degrees Fahrenheit. If the object cools from 100 degrees to 90 degrees in 10 minutes, how much more time, in minutes, will it take for its temperature to decrease to 80 degrees?

A. 12.08 12.08 B. 15.09 15.09 C. 14.09 14.09 D. 13.09 13.09

सही उत्तर: C Correct Answer: C

By Newton's law of cooling, V - 60 = 40e^(-kt). The first 10 minutes give e^(-10k) = 3/4. Solving for the additional time needed to reach V = 80 gives approximately 14.09 minutes.

Mathematics MCQs

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