Advanced Mathematics MCQs

71

Advanced Mathematics • Differential Equations

If 1 and e^(-x) are two linearly independent solutions of the equation (1 + x)y'' + xy' - y = (1 + x)^2, then by the method of variation of parameters, the particular integral is: If 1 and e^(-x) are two linearly independent solutions of the equation (1 + x)y'' + xy' - y = (1 + x)^2, then by the method of variation of parameters, the particular integral is:

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A. x + 1 - x^2 x + 1 - x^2 B. x^2 - 1 - e^x x^2 - 1 - e^x C. x^2 + 1 - x x^2 + 1 - x D. x^2 + 1 - e^x x^2 + 1 - e^x

सही उत्तर: C Correct Answer: C

Applying variation of parameters with the two stated complementary solutions gives the particular integral x^2 + 1 - x, as marked in the source paper.

72

Advanced Mathematics • Real Analysis

The function sin(e^x), where x > 0: The function sin(e^x), where x > 0:

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A. Is bounded but does not converge Is bounded but does not converge B. Diverges to infinity as x approaches infinity Diverges to infinity as x approaches infinity C. Is strictly increasing but bounded Is strictly increasing but bounded D. Is bounded and converges to a real number as x approaches infinity Is bounded and converges to a real number as x approaches infinity

सही उत्तर: A Correct Answer: A

The sine function always lies between -1 and 1, but e^x grows without bound and its sine continues to oscillate. Thus the function is bounded but has no limit.

73

Advanced Mathematics • Real Analysis

Let f(x) be a differentiable function on R such that df/dx > 0 for all x and lim f(x) = 2 as x approaches minus infinity. Which of the following is true? Let f(x) be a differentiable function on R such that df/dx > 0 for all x and lim f(x) = 2 as x approaches minus infinity. Which of the following is true?

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A. f(3) > 2 f(3) > 2 B. f(3) >= 2 f(3) >= 2 C. f(3) <= 2 f(3) <= 2 D. f(3) = 2 f(3) = 2

सही उत्तर: A Correct Answer: A

Since f is strictly increasing and approaches 2 at minus infinity, every finite value of f must be strictly greater than 2. Therefore f(3) > 2.

74

Advanced Mathematics • Linear Algebra

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

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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.

75

Advanced 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:

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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.

76

Advanced 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:

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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).

77

Advanced 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:

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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.

78

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.

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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.

79

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.

80

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.

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