Mathematics MCQs

361

Mathematics • Integral Calculus

The area under the curve given in polar coordinates by r=f(omega) is given by: The area under the curve given in polar coordinates by r=f(omega) is given by:

A. integral r^2 d(omega) integral r^2 d(omega) B. integral (r^3/2)dr integral (r^3/2)dr C. 2 integral r^2 d(omega) 2 integral r^2 d(omega) D. integral (r^2/2)d(omega) integral (r^2/2)d(omega)

सही उत्तर: D Correct Answer: D

The area element in polar coordinates is r dr d(omega). Integrating radially from 0 to r gives (1/2)r^2d(omega), hence area=(1/2)integral r^2d(omega).

362

Mathematics • Differential Equations

If the roots of the auxiliary equation of the differential equation d^4y/dx^4 - 5d^3y/dx^3 + 6d^2y/dx^2 + 4dy/dx - 8y=0 are 2,2,2 and -1, then the general solution is: If the roots of the auxiliary equation of the differential equation d^4y/dx^4 - 5d^3y/dx^3 + 6d^2y/dx^2 + 4dy/dx - 8y=0 are 2,2,2 and -1, then the general solution is:

A. y=c1e^(2x)+c2e^(-x) y=c1e^(2x)+c2e^(-x) B. y=c1e^(2x)+c2e^(-x)+c3e^x+c4e^(-2x) y=c1e^(2x)+c2e^(-x)+c3e^x+c4e^(-2x) C. y=(c1+c2x+c3x^2)e^(-x)+c4e^(2x) y=(c1+c2x+c3x^2)e^(-x)+c4e^(2x) D. y=(c1+c2x+c3x^2)e^(2x)+c4e^(-x) y=(c1+c2x+c3x^2)e^(2x)+c4e^(-x)

सही उत्तर: D Correct Answer: D

A root 2 of multiplicity three contributes (c1+c2x+c3x^2)e^(2x), and the simple root -1 contributes c4e^(-x).

363

Mathematics • Group Theory

What is the order of (2,6) in Z_6 x Z_12? What is the order of (2,6) in Z_6 x Z_12?

A. 1 1 B. 6 6 C. 10 10 D. 2 2

सही उत्तर: B Correct Answer: B

The order of 2 in Z_6 is 3 and the order of 6 in Z_12 is 2. The order of the pair is lcm(3,2)=6.

364

Mathematics • Conic Sections

Which conic does the equation R=5/(4-11cos(omega)) represent? Which conic does the equation R=5/(4-11cos(omega)) represent?

A. Hyperbola Hyperbola B. Parabola Parabola C. Ellipse Ellipse D. Circle Circle

सही उत्तर: A Correct Answer: A

Writing the denominator in standard polar-conic form shows eccentricity e=11/4. Since e>1, the conic is a hyperbola.

365

Mathematics • Linear Algebra

For any subset U of a vector space V, let U-perpendicular denote the set of vectors orthogonal to every vector in U. Find (U-perpendicular)-perpendicular, where U={(2,0,0),(1,0,3)}. For any subset U of a vector space V, let U-perpendicular denote the set of vectors orthogonal to every vector in U. Find (U-perpendicular)-perpendicular, where U={(2,0,0),(1,0,3)}.

A. R^3 R^3 B. {(2,0,0),(1,0,3)} {(2,0,0),(1,0,3)} C. {(x,y,z): y=0} {(x,y,z): y=0} D. {(0,0,0)} {(0,0,0)}

सही उत्तर: C Correct Answer: C

The two vectors are independent and span the xz-plane y=0. In finite-dimensional inner-product spaces, (U-perpendicular)-perpendicular equals the span of U.

366

Mathematics • Group Theory

GL_n(R) denotes the group of n x n non-singular matrices under matrix multiplication. Which one of the following gives an abelian subgroup? GL_n(R) denotes the group of n x n non-singular matrices under matrix multiplication. Which one of the following gives an abelian subgroup?

A. Set of all non-singular upper-triangular matrices. Set of all non-singular upper-triangular matrices. B. Set of all non-singular matrices with positive trace. Set of all non-singular matrices with positive trace. C. Set of all matrices with positive determinant. Set of all matrices with positive determinant. D. Set of all non-singular diagonal matrices. Set of all non-singular diagonal matrices.

सही उत्तर: D Correct Answer: D

Non-singular diagonal matrices are closed under multiplication and inversion, and diagonal matrices commute with one another. Hence they form an abelian subgroup.

367

Mathematics • Complex Analysis

The Taylor-series expansion sum of (-1)^n z^(2n+1)/(2n+1)! for n=0,1,2,... represents which analytic function? The Taylor-series expansion sum of (-1)^n z^(2n+1)/(2n+1)! for n=0,1,2,... represents which analytic function?

A. tan(z) tan(z) B. sin(z) sin(z) C. cosh(z) cosh(z) D. cos(z) cos(z)

सही उत्तर: B Correct Answer: B

This is the standard Maclaurin expansion z-z^3/3!+z^5/5!-..., which equals sin(z).

368

Mathematics • Real Analysis

Let f(x)=1/x^2-5 be defined on [1,4]. Which of the following is true? Let f(x)=1/x^2-5 be defined on [1,4]. Which of the following is true?

A. There exists a point x in [1,4] such that f is not differentiable at x. There exists a point x in [1,4] such that f is not differentiable at x. B. The function has a tangent line between 1 and 4 with slope -1/64. The function has a tangent line between 1 and 4 with slope -1/64. C. The function has a tangent line between 1 and 4 with slope 3/2. The function has a tangent line between 1 and 4 with slope 3/2. D. The function has a tangent line between 1 and 4 with slope -5/16. The function has a tangent line between 1 and 4 with slope -5/16.

सही उत्तर: D Correct Answer: D

The secant slope is [f(4)-f(1)]/(4-1)=[1/16-1]/3=-5/16. By the Mean Value Theorem, some tangent has this slope.

369

Mathematics • Dynamics

The moment of inertia of an ellipse of mass M and semiaxes a and b about a tangent is, where p is the perpendicular from the centre to the tangent: The moment of inertia of an ellipse of mass M and semiaxes a and b about a tangent is, where p is the perpendicular from the centre to the tangent:

A. (5M/4)p^2 (5M/4)p^2 B. (5M/6)p^2 (5M/6)p^2 C. (5M/2)p^2 (5M/2)p^2 D. (5M/3)p^2 (5M/3)p^2

सही उत्तर: A Correct Answer: A

Using the moment of inertia of the elliptic lamina about a parallel central axis together with the parallel-axis theorem gives (5M/4)p^2 for the stated tangent.

370

Mathematics • Statics

Two equal uniform rods AB and AC, each of length 2b, are freely joined at A and rest on a smooth vertical circle of radius a. If 2theta is the angle between them, then: Two equal uniform rods AB and AC, each of length 2b, are freely joined at A and rest on a smooth vertical circle of radius a. If 2theta is the angle between them, then:

A. a sin^3(theta)=b cos(theta) a sin^3(theta)=b cos(theta) B. b sin^2(theta)=a cos(theta) b sin^2(theta)=a cos(theta) C. a sin^2(theta)=b cos(theta) a sin^2(theta)=b cos(theta) D. b sin^3(theta)=a cos(theta) b sin^3(theta)=a cos(theta)

सही उत्तर: D Correct Answer: D

Applying the equilibrium condition to the symmetric rod configuration gives b sin^3(theta)=a cos(theta), as marked in the source paper.

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