MCQs for JSSC

2181

Advanced Mathematics • Complex Analysis

Compute the integral of cos(z/2) dz from 0 to pi + 2i. Compute the integral of cos(z/2) dz from 0 to pi + 2i.

A. e + 1/e e + 1/e B. sin(pi/4) sin(pi/4) C. 2pi i 2pi i D. 0 0

सही उत्तर: A Correct Answer: A

An antiderivative is 2sin(z/2). At z = pi + 2i this is 2sin(pi/2 + i) = 2cosh(1) = e + 1/e; the lower endpoint contributes zero.

2182

Geometry • Conic Sections

Find the focus of the conic x^2 + 19 - 8x - y = 0. Find the focus of the conic x^2 + 19 - 8x - y = 0.

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A. (4, 3.25) (4, 3.25) B. (2.75, 4) (2.75, 4) C. (3.25, 4) (3.25, 4) D. (4, 2.75) (4, 2.75)

सही उत्तर: A Correct Answer: A

Rewriting gives (x - 4)^2 = y - 3. Comparing with (x-h)^2 = 4a(y-k), we get a = 1/4, so the focus is (4, 3 + 1/4) = (4, 3.25).

2183

Advanced Mathematics • Dynamics

Which one of the following is a correct statement? Which one of the following is a correct statement?

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A. Holonomic systems are systems for which all constraints are integrable into positional constraints. Holonomic systems are systems for which all constraints are integrable into positional constraints. B. Non-holonomic systems are systems where velocities and other derivatives of position are not constraints. Non-holonomic systems are systems where velocities and other derivatives of position are not constraints. C. Holonomic systems are systems for which all constraints are non-integrable into positional constraints. Holonomic systems are systems for which all constraints are non-integrable into positional constraints. D. Non-holonomic systems are systems that have constraints that are integrable into positional constraints. Non-holonomic systems are systems that have constraints that are integrable into positional constraints.

सही उत्तर: A Correct Answer: A

Holonomic constraints can be expressed, or integrated, as relations involving generalized coordinates and time. Therefore option A is correct.

2184

Advanced Mathematics • Linear Programming

In the simplex method, which variables are introduced to convert constraint conditions into equations? In the simplex method, which variables are introduced to convert constraint conditions into equations?

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A. Feasible variables Feasible variables B. Basic variables Basic variables C. Degenerate variables Degenerate variables D. Slack variables Slack variables

सही उत्तर: D Correct Answer: D

Slack variables are added to less-than-or-equal-to constraints to convert the inequalities into equations.

2185

Advanced Mathematics • Partial Differential Equations

The integral curves of the equations dx/[x^2(y^2-z^2)] = dy/[y^2(z^2-x^2)] = dz/[z^2(x^2-y^2)] are given by, where c1 and c2 are arbitrary constants: The integral curves of the equations dx/[x^2(y^2-z^2)] = dy/[y^2(z^2-x^2)] = dz/[z^2(x^2-y^2)] are given by, where c1 and c2 are arbitrary constants:

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A. x^2 + y^2 - z^2 = c1 and x^-1 + y^-1 - z^-1 = c2 x^2 + y^2 - z^2 = c1 and x^-1 + y^-1 - z^-1 = c2 B. x^2 - y^2 + z^2 = c1 and x^-1 - y^-1 + z^-1 = c2 x^2 - y^2 + z^2 = c1 and x^-1 - y^-1 + z^-1 = c2 C. x^2 - y^2 - z^2 = c1 and x^-1 - y^-1 - z^-1 = c2 x^2 - y^2 - z^2 = c1 and x^-1 - y^-1 - z^-1 = c2 D. x^2 + y^2 + z^2 = c1 and x^-1 + y^-1 + z^-1 = c2 x^2 + y^2 + z^2 = c1 and x^-1 + y^-1 + z^-1 = c2

सही उत्तर: D Correct Answer: D

Using suitable multipliers in the Lagrange auxiliary system yields the two independent first integrals x^2+y^2+z^2=c1 and 1/x+1/y+1/z=c2.

2186

Advanced Mathematics • Fluid Mechanics

In the one-dimensional equation of motion, Bernoulli's equation is given by, where v^2/2 is kinetic energy, p/rho is flow energy and gz is potential energy, all per unit mass: In the one-dimensional equation of motion, Bernoulli's equation is given by, where v^2/2 is kinetic energy, p/rho is flow energy and gz is potential energy, all per unit mass:

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A. v^2/2 - p/rho - gz = constant v^2/2 - p/rho - gz = constant B. v^2/2 + p/rho - gz = constant v^2/2 + p/rho - gz = constant C. v^2/2 - p/rho + gz = constant v^2/2 - p/rho + gz = constant D. v^2/2 + p/rho + gz = constant v^2/2 + p/rho + gz = constant

सही उत्तर: D Correct Answer: D

Bernoulli's equation states that kinetic energy, pressure-flow energy and gravitational potential energy per unit mass sum to a constant along a streamline.

2187

Advanced Mathematics • Linear Algebra

What is the dimension of the vector space of all 2 x 2 symmetric real-valued matrices? What is the dimension of the vector space of all 2 x 2 symmetric real-valued matrices?

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A. 3 3 B. 1 1 C. 2 2 D. 4 4

सही उत्तर: A Correct Answer: A

A symmetric 2 x 2 matrix has the form [[a,b],[b,c]], determined by three independent real parameters. Its dimension is 3.

2188

Advanced Mathematics • Probability

Suppose (X,Y) is jointly distributed with density f(s,t) = 1/pi if s^2 + t^2 <= 1, and 0 otherwise. Find Cov(X,Y). Suppose (X,Y) is jointly distributed with density f(s,t) = 1/pi if s^2 + t^2 <= 1, and 0 otherwise. Find Cov(X,Y).

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A. -1 -1 B. 1 1 C. 0 0 D. pi pi

सही उत्तर: C Correct Answer: C

By symmetry of the uniform distribution on the unit disk, E(X)=E(Y)=0 and E(XY)=0. Hence Cov(X,Y)=0.

2189

Advanced Mathematics • Graph Theory

Every n-connected graph with p vertices has at least: Every n-connected graph with p vertices has at least:

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A. pn/2 lines pn/2 lines B. n(p + 1)/2 lines n(p + 1)/2 lines C. pn lines pn lines D. 2pn lines 2pn lines

सही उत्तर: A Correct Answer: A

An n-connected graph has minimum degree at least n. By the handshaking lemma, twice the number of edges is at least pn, so it has at least pn/2 edges.

2190

Advanced Mathematics • Probability

Which one of the following gives a pair of independent random variables? Which one of the following gives a pair of independent random variables?

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A. 10 coins are thrown at random. X denotes the total number of heads and Y denotes the total number of tails. 10 coins are thrown at random. X denotes the total number of heads and Y denotes the total number of tails. B. 10 coins are tossed at random. X denotes the total number of heads on even tosses (2nd, 4th, ..., 10th) and Y denotes the total number of tails on odd tosses (1st, 3rd, ..., 9th). 10 coins are tossed at random. X denotes the total number of heads on even tosses (2nd, 4th, ..., 10th) and Y denotes the total number of tails on odd tosses (1st, 3rd, ..., 9th). C. For 1 <= i <= 6, let X_i denote the number of times digit i appears in 10 tosses of a die. Consider (X_1,X_2). For 1 <= i <= 6, let X_i denote the number of times digit i appears in 10 tosses of a die. Consider (X_1,X_2). D. Two dice are thrown at random; X is the maximum and Y is the minimum of the digits appearing. Two dice are thrown at random; X is the maximum and Y is the minimum of the digits appearing.

सही उत्तर: B Correct Answer: B

X depends only on the even-numbered tosses and Y only on the disjoint odd-numbered tosses. Since all tosses are independent, X and Y are independent.

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