Mathematics MCQs

51

Mathematics • Graph Theory

A graph G with p vertices (p >= 3) is Hamiltonian if: A graph G with p vertices (p >= 3) is Hamiltonian if:

A. For all vertices v in G, degree(v) < p/2 For all vertices v in G, degree(v) < p/2 B. For all vertices v in G, degree(v) >= p/2 For all vertices v in G, degree(v) >= p/2 C. All vertices v in G have an even degree All vertices v in G have an even degree D. All vertices in G have an odd degree All vertices in G have an odd degree

सही उत्तर: B Correct Answer: B

By Dirac's theorem, a simple graph with p >= 3 vertices is Hamiltonian if every vertex has degree at least p/2.

52

Mathematics • Statics

If P and Q are two non-intersecting forces whose directions are perpendicular, then the ratio of the distances of the central axis from their lines of action is represented as: If P and Q are two non-intersecting forces whose directions are perpendicular, then the ratio of the distances of the central axis from their lines of action is represented as:

A. Q^2 : 2P^2 Q^2 : 2P^2 B. 2P^2 : Q^2 2P^2 : Q^2 C. Q^2 : P^2 Q^2 : P^2 D. P^2 : Q^2 P^2 : Q^2

सही उत्तर: C Correct Answer: C

For two perpendicular non-intersecting forces, the distances of the central axis from their respective lines of action are in the inverse-square ratio of the forces, giving Q^2 : P^2 in the stated order.

53

Mathematics • Complex Analysis

Let C denote the boundary of the circle with radius 3 around the centre z_1 = 2 + i. Calculate the integral I = integral_C e^(az)/(z - 2 - i) dz. Let C denote the boundary of the circle with radius 3 around the centre z_1 = 2 + i. Calculate the integral I = integral_C e^(az)/(z - 2 - i) dz.

A. 0 0 B. 2pi i e^(a(2 - i)) 2pi i e^(a(2 - i)) C. 2pi i e^(a(2 + i)) 2pi i e^(a(2 + i)) D. 2pi i e^(a(2 + 2i)) 2pi i e^(a(2 + 2i))

सही उत्तर: C Correct Answer: C

By Cauchy's integral formula, the integral equals 2pi i times e^(az) evaluated at z = 2 + i, giving 2pi i e^(a(2 + i)).

54

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).

55

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.

56

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.

57

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.

58

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).

59

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.

60

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.

Mathematics MCQs

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