Mathematics MCQs

81

Mathematics • Linear Algebra

Which one of the following is NOT a subspace? Which one of the following is NOT a subspace?

A. {(x,2x): x belongs to R} over R {(x,2x): x belongs to R} over R B. Q over Q Q over Q C. {(x,y): xy = 0} over R {(x,y): xy = 0} over R D. {(x,y): 2x + 3y = 0} over R {(x,y): 2x + 3y = 0} over R

सही उत्तर: C Correct Answer: C

The set xy = 0 is the union of the coordinate axes and is not closed under addition; for example, (1,0) and (0,1) belong to it but (1,1) does not.

82

Mathematics • Partial Differential Equations

The integral surface of yp + xq - z = 0 passing through the curve z = x^3, y = 0 is: The integral surface of yp + xq - z = 0 passing through the curve z = x^3, y = 0 is:

A. z(1 + x^2 - y^2) = (x^2 - y^2)(x + y + z) z(1 + x^2 - y^2) = (x^2 - y^2)(x + y + z) B. 1 + x^2 - y^2 = (x^2 - y^2)(x + y + z) 1 + x^2 - y^2 = (x^2 - y^2)(x + y + z) C. z(1 - x^2 - y^2) = (x^2 - y^2)(x + y + z) z(1 - x^2 - y^2) = (x^2 - y^2)(x + y + z) D. z(1 + x^2 - y^2) = (x^2 + y^2)(x + y + z) z(1 + x^2 - y^2) = (x^2 + y^2)(x + y + z)

सही उत्तर: A Correct Answer: A

Solving the Lagrange first-order PDE by its characteristic equations and imposing z = x^3 on y = 0 yields the surface in option A.

83

Mathematics • Partial Differential Equations

The particular integral of the equation (2D^2 - D')z = 10e^(x - 3y) is: The particular integral of the equation (2D^2 - D')z = 10e^(x - 3y) is:

A. 2e^(x - 3y) 2e^(x - 3y) B. e^(x - 3y) e^(x - 3y) C. 2e^(x + 3y) 2e^(x + 3y) D. e^(x + 3y) e^(x + 3y)

सही उत्तर: A Correct Answer: A

For e^(ax+by), replace D by a and D' by b. Here a = 1 and b = -3, so the operator value is 2(1)^2 - (-3) = 5. Thus the particular integral is (10/5)e^(x-3y) = 2e^(x-3y).

84

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.

85

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.

86

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).

87

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.

88

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.

89

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.

90

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.

Mathematics MCQs

श्रेणी अनुसार अभ्यास करें

विषय अनुसार अभ्यास करें

टॉपिक अनुसार अभ्यास करें

परीक्षा अनुसार अभ्यास करें