Mathematics MCQs

251

Advanced Mathematics • Dynamics

A solid homogeneous cone of height h and semi-vertical angle alpha oscillates about a diameter of its base. The length of the simple equivalent pendulum is: A solid homogeneous cone of height h and semi-vertical angle alpha oscillates about a diameter of its base. The length of the simple equivalent pendulum is:

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A. (h/5)(2 + 5tan^2(alpha)) (h/5)(2 + 5tan^2(alpha)) B. (h/5)(2 + 3tan^2(alpha)) (h/5)(2 + 3tan^2(alpha)) C. (h/3)(2 + 3tan^2(alpha)) (h/3)(2 + 3tan^2(alpha)) D. (h/2)(2 + 5tan^2(alpha)) (h/2)(2 + 5tan^2(alpha))

सही उत्तर: B Correct Answer: B

For a physical pendulum, the equivalent length is I/(Md), where I is the moment of inertia about the axis and d is the distance of the centre of mass from it. For the stated cone this reduces to (h/5)(2 + 3tan^2(alpha)).

252

Advanced Mathematics • Complex Analysis

Which of the following statements is INCORRECT? Which of the following statements is INCORRECT?

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A. The composite of two continuous functions is also continuous. The composite of two continuous functions is also continuous. B. Let f(x + iy) = u(x,y) + iv(x,y). If u and v both have continuous partial derivatives with respect to x and y at (x0,y0), then f is also differentiable at (x0,y0). Let f(x + iy) = u(x,y) + iv(x,y). If u and v both have continuous partial derivatives with respect to x and y at (x0,y0), then f is also differentiable at (x0,y0). C. If f is analytic at all points interior to and on a simple closed contour C, then the integral of f(z) dz over C is 0. If f is analytic at all points interior to and on a simple closed contour C, then the integral of f(z) dz over C is 0. D. If f is a continuous function defined on domain D, then the integral of f(z) dz from z1 to z2, where z1 and z2 lie in D, is the same along all contours connecting z1 and z2 and lying entirely in D. If f is a continuous function defined on domain D, then the integral of f(z) dz from z1 to z2, where z1 and z2 lie in D, is the same along all contours connecting z1 and z2 and lying entirely in D.

सही उत्तर: B Correct Answer: B

Continuity of the first partial derivatives of u and v alone does not ensure complex differentiability; the Cauchy-Riemann equations must also hold at the point. Option B is the answer marked in the source paper.

253

Advanced Mathematics • Vector Calculus

Find the directional derivative D_u f(x,y) at (x,y) = (1,2), where f(x,y) = x^3 - 3xy + 4y^2 and u is the unit vector in the direction theta = pi/6. Find the directional derivative D_u f(x,y) at (x,y) = (1,2), where f(x,y) = x^3 - 3xy + 4y^2 and u is the unit vector in the direction theta = pi/6.

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A. (13 - 3sqrt(3))/2 (13 - 3sqrt(3))/2 B. 10 10 C. 3sqrt(3) 3sqrt(3) D. 0 0

सही उत्तर: A Correct Answer: A

At (1,2), grad f = (3x^2 - 3y, -3x + 8y) = (-3,13). Dotting it with (cos(pi/6), sin(pi/6)) gives (13 - 3sqrt(3))/2.

254

Advanced Mathematics • Differential Equations

The integrating factor for the differential equation y(xy + 2x^2y^2) dx + x(xy - x^2y^2) dy = 0 is: The integrating factor for the differential equation y(xy + 2x^2y^2) dx + x(xy - x^2y^2) dy = 0 is:

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A. 1/(xy) 1/(xy) B. 1/(x^2y^2) 1/(x^2y^2) C. 1/(3x^3y^3) 1/(3x^3y^3) D. 1/(xy^3) 1/(xy^3)

सही उत्तर: C Correct Answer: C

Multiplying by 1/(3x^3y^3) makes the differential form exact; this is the integrating factor marked in the source paper.

255

Advanced Mathematics • Graph Theory

Circumference of a cyclic graph is defined as: Circumference of a cyclic graph is defined as:

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A. Twice the sum of its total degree Twice the sum of its total degree B. Length of the longest cycle Length of the longest cycle C. Length of the shortest cycle Length of the shortest cycle D. Thrice the sum of its total degree Thrice the sum of its total degree

सही उत्तर: B Correct Answer: B

The circumference of a graph is the length, measured by number of edges, of its longest cycle.

256

Advanced Mathematics • Mathematical Modelling

If x(t) denotes the number of susceptibles and y(t) denotes the number of infectives at time t, in a homogeneous group of (n + 1) individuals, the simple epidemic model without removal is given by the equation (where beta > 0 is the contact rate): If x(t) denotes the number of susceptibles and y(t) denotes the number of infectives at time t, in a homogeneous group of (n + 1) individuals, the simple epidemic model without removal is given by the equation (where beta > 0 is the contact rate):

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A. dx/dt = beta x(n - 1 + x) dx/dt = beta x(n - 1 + x) B. dx/dt = -beta x(n + 1 - x) dx/dt = -beta x(n + 1 - x) C. dx/dt = -beta x(n - 1 + x) dx/dt = -beta x(n - 1 + x) D. dx/dt = beta x(n + 1 - x) dx/dt = beta x(n + 1 - x)

सही उत्तर: B Correct Answer: B

With no removal, x + y = n + 1 and susceptibles decrease at rate dx/dt = -beta xy. Substituting y = n + 1 - x gives dx/dt = -beta x(n + 1 - x).

257

Advanced Mathematics • Partial Differential Equations

The steady-state solution mu(x,t) = mu_s(x) of the heat equation mu_t = a^2 mu_xx satisfying the conditions mu(0,t) = -1 and mu(2,t) = 1 is: The steady-state solution mu(x,t) = mu_s(x) of the heat equation mu_t = a^2 mu_xx satisfying the conditions mu(0,t) = -1 and mu(2,t) = 1 is:

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A. mu_s(x) = 1 - x mu_s(x) = 1 - x B. mu_s(x) = -1 + x mu_s(x) = -1 + x C. mu_s(x) = -1 + 3x mu_s(x) = -1 + 3x D. mu_s(x) = 1 + 2x mu_s(x) = 1 + 2x

सही उत्तर: B Correct Answer: B

At steady state, mu_xx = 0, so mu_s(x) = Ax + B. The boundary values give B = -1 and A = 1; hence mu_s(x) = -1 + x.

258

Advanced Mathematics • Abstract Algebra

Which one of the following statements is false? Which one of the following statements is false?

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A. A field (F, +, .) always has a multiplicative identity. A field (F, +, .) always has a multiplicative identity. B. M_nxn(R), the space of all n x n real-valued matrices under matrix addition and multiplication, forms an integral domain. M_nxn(R), the space of all n x n real-valued matrices under matrix addition and multiplication, forms an integral domain. C. The set of complex numbers forms an algebraically closed field. The set of complex numbers forms an algebraically closed field. D. For a group (G, .), the identity element is always unique. For a group (G, .), the identity element is always unique.

सही उत्तर: B Correct Answer: B

For n greater than 1, the ring of n x n real matrices has zero divisors and is noncommutative, so it is not an integral domain.

259

Advanced Mathematics • Linear Algebra

Consider the following real-valued maps defined over the vector space of 2 x 2 real-valued matrices. Which one of the following is not linear? Consider the following real-valued maps defined over the vector space of 2 x 2 real-valued matrices. Which one of the following is not linear?

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A. T(A) = tr(A) T(A) = tr(A) B. T(A) = a_11 T(A) = a_11 C. T(A) = A^t T(A) = A^t D. T(A) = det(A) T(A) = det(A)

सही उत्तर: D Correct Answer: D

The determinant is not a linear map because, in general, det(A + B) is not equal to det(A) + det(B), and det(cA) is not equal to c det(A).

260

Advanced Mathematics • Mathematical Modelling

The size of a population of small rodents was recorded as follows: Month: 0, 2, 6, 10; Population: 2, 5, 20, 109. The growth rate as a percentage per month and the population at the end of 12 months are respectively: The size of a population of small rodents was recorded as follows: Month: 0, 2, 6, 10; Population: 2, 5, 20, 109. The growth rate as a percentage per month and the population at the end of 12 months are respectively:

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A. 37.2428% and 234 37.2428% and 234 B. 38.2428% and 235 38.2428% and 235 C. 32.2428% and 131 32.2428% and 131 D. 39.2428% and 231 39.2428% and 231

सही उत्तर: D Correct Answer: D

Fitting the exponential population-growth model to the given observations gives the monthly growth rate and projected 12-month population shown in option D.

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