MCQs for JSSC PGT Mathematics Exam

51

Advanced 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

Advanced 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:

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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

Advanced 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.

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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

Geometry • 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:

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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

Geometry • 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:

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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

Advanced 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.

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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

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. 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?

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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

Advanced 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:

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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

Advanced 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.

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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

Advanced 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:

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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.

MCQs for JSSC PGT Mathematics Exam

MCQs for JSSC PGT Mathematics Exam

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