Mathematics MCQs

321

Mathematics • Linear Programming

If at least one of the basic variables is zero in a basic feasible solution of an LPP, the solution is said to be: If at least one of the basic variables is zero in a basic feasible solution of an LPP, the solution is said to be:

A. Optimal Optimal B. Singular Singular C. Unbounded Unbounded D. Degenerate Degenerate

सही उत्तर: D Correct Answer: D

A basic feasible solution is degenerate when one or more of its basic variables has value zero.

322

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.

323

Mathematics • 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.

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).

324

Mathematics • Dynamics

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

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.

325

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?

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.

326

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:

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.

327

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:

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.

328

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?

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.

329

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).

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.

330

Mathematics • Graph Theory

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

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.

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