Mathematics MCQs

21

Mathematics • Abstract Algebra

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

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.

22

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?

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).

23

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:

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.

24

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):

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)).

25

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).

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.

26

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?

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.

27

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.

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.

28

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:

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.

29

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?

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.

30

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:

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.

Mathematics MCQs

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