MCQs for JSSC PGT Mathematics Exam

81

Advanced 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

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.

83

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).

84

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.

85

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.

86

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).

87

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.

88

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.

89

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.

90

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.

MCQs for JSSC PGT Mathematics Exam

MCQs for JSSC PGT Mathematics Exam

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