Advanced Mathematics MCQs

21

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).

22

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.

23

Advanced Mathematics • Differential Equations

Consider a variable mu satisfying d^2(mu)/d(theta)^2 + mu = 2k cos(theta), with the conditions: (i) mu has the same value when theta = plus or minus pi/2; and (ii) integral from 0 to pi/2 of mu d(theta) = 0. The value of mu is (for arbitrary constant k): Consider a variable mu satisfying d^2(mu)/d(theta)^2 + mu = 2k cos(theta), with the conditions: (i) mu has the same value when theta = plus or minus pi/2; and (ii) integral from 0 to pi/2 of mu d(theta) = 0. The value of mu is (for arbitrary constant k):

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A. mu = k(theta sin(theta) + cos(theta)) mu = k(theta sin(theta) + cos(theta)) B. mu = k(sin(theta) + theta cos(theta)) mu = k(sin(theta) + theta cos(theta)) C. mu = k(theta sin(theta) - cos(theta)) mu = k(theta sin(theta) - cos(theta)) D. mu = k(sin(theta) - theta cos(theta)) mu = k(sin(theta) - theta cos(theta))

सही उत्तर: C Correct Answer: C

Substitution shows that k theta sin(theta) is a particular solution. Applying the two stated conditions to the complementary terms gives mu = k(theta sin(theta) - cos(theta)).

24

Advanced Mathematics • Vector Calculus

Find the length of the arc of the circular helix with vector equation r(t) = cos(t)i + sin(t)j + tk, from (1,0,0) to (1,0,2pi). Find the length of the arc of the circular helix with vector equation r(t) = cos(t)i + sin(t)j + tk, from (1,0,0) to (1,0,2pi).

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A. sqrt(2)pi sqrt(2)pi B. 2sqrt(2)pi 2sqrt(2)pi C. 1 1 D. pi pi

सही उत्तर: B Correct Answer: B

The magnitude of the first derivative of r(t) is sqrt(sin^2(t) + cos^2(t) + 1) = sqrt(2). Integrating the speed from 0 to 2pi gives 2sqrt(2)pi.

25

Advanced Mathematics • Graph Theory

What is the number of edges present in a complete graph with n vertices? What is the number of edges present in a complete graph with n vertices?

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A. n(n - 1) n(n - 1) B. n^2/2 n^2/2 C. n(n + 1)/2 n(n + 1)/2 D. n(n - 1)/2 n(n - 1)/2

सही उत्तर: D Correct Answer: D

Every unordered pair of distinct vertices forms exactly one edge, so the number of edges is C(n,2) = n(n - 1)/2.

26

Advanced Mathematics • Complex Analysis

Using the Residue Theorem, evaluate the integral of (5z - 2)/(z(z - 1)) dz over the contour C: |z| = 2. Using the Residue Theorem, evaluate the integral of (5z - 2)/(z(z - 1)) dz over the contour C: |z| = 2.

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A. 6pi i 6pi i B. 10pi i 10pi i C. 5pi i 5pi i D. 8pi i 8pi i

सही उत्तर: B Correct Answer: B

The contour contains the poles z = 0 and z = 1. Their residues are 2 and 3, whose sum is 5. Thus the integral is 2pi i x 5 = 10pi i.

27

Advanced Mathematics • Fluid Mechanics

For a viscous flow of a fluid through a pipe with circular cross-section given by r = a, the pressure gradient is constant, P = -partial(p)/partial(x). If u_x, u_y and u_z are the velocity components, respectively, then the velocity field is: For a viscous flow of a fluid through a pipe with circular cross-section given by r = a, the pressure gradient is constant, P = -partial(p)/partial(x). If u_x, u_y and u_z are the velocity components, respectively, then the velocity field is:

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A. (P/(4mu))(a^2 - r^2), 0, 0 (P/(4mu))(a^2 - r^2), 0, 0 B. (P/(2mu))(a^2 + r^2), 0, 0 (P/(2mu))(a^2 + r^2), 0, 0 C. (P/mu)(a^2 - r^2), 0, 0 (P/mu)(a^2 - r^2), 0, 0 D. (P/(3mu))(a^2 + r^2), 0, 0 (P/(3mu))(a^2 + r^2), 0, 0

सही उत्तर: A Correct Answer: A

For steady Hagen-Poiseuille flow in a circular pipe, the axial velocity profile is u_x = (P/(4mu))(a^2 - r^2), while the transverse velocity components are zero.

28

Advanced Mathematics • Mathematical Modelling

Which of the following is NOT a purpose of a model from political science? Which of the following is NOT a purpose of a model from political science?

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A. Providing insights into a general class of problems. Providing insights into a general class of problems. B. Produce obvious directions for further study. Produce obvious directions for further study. C. Forecast events or outcomes. Forecast events or outcomes. D. Organizing empirical generalizations or known facts. Organizing empirical generalizations or known facts.

सही उत्तर: B Correct Answer: B

Models are used to organize facts, provide insight and help forecast outcomes. Producing obvious directions for further study is marked as not being a purpose in the source paper.

29

Advanced Mathematics • Differential Equations

The order and degree of the differential equation x^2(dx)^2 + 2xy dxdy + y^2(dy)^2 - z^2(dz)^2 = 0 are respectively: The order and degree of the differential equation x^2(dx)^2 + 2xy dxdy + y^2(dy)^2 - z^2(dz)^2 = 0 are respectively:

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A. 1 and 1 1 and 1 B. 1 and 2 1 and 2 C. 2 and 2 2 and 2 D. 2 and 1 2 and 1

सही उत्तर: B Correct Answer: B

Only first-order differentials occur, while they appear polynomially to the second degree. Hence the order is 1 and the degree is 2.

30

Advanced Mathematics • Mathematical Modelling

Single-species population model is given by the equation (where the symbols have their usual meaning): Single-species population model is given by the equation (where the symbols have their usual meaning):

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A. dN/dt = B(t)D(t) + D(t)N(t) dN/dt = B(t)D(t) + D(t)N(t) B. dN/dt = B(t)N(t) + D(t)N(t) dN/dt = B(t)N(t) + D(t)N(t) C. dN/dt = B(t)/N(t) + N(t)/D(t) dN/dt = B(t)/N(t) + N(t)/D(t) D. dN/dt = B(t)N(t) - D(t)N(t) dN/dt = B(t)N(t) - D(t)N(t)

सही उत्तर: D Correct Answer: D

The rate of population change equals births minus deaths. With per-capita birth and death rates B(t) and D(t), dN/dt = B(t)N(t) - D(t)N(t).

Advanced Mathematics MCQs

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