Mathematics MCQs

301

Mathematics • Mathematical Modelling

Which of the following statements is NOT correct? Which of the following statements is NOT correct?

A. Urban water uses involve transportation. Urban water uses involve transportation. B. Urban water uses involve water supply. Urban water uses involve water supply. C. Urban water uses involve power generation. Urban water uses involve power generation. D. Urban water uses involve snow-removal strategies. Urban water uses involve snow-removal strategies.

सही उत्तर: D Correct Answer: D

Snow-removal strategies are not classified as an urban water use in the context of the question; option D is marked as the incorrect statement.

302

Mathematics • Digital Logic

The logical expression y = A + A^cB is equivalent to: The logical expression y = A + A^cB is equivalent to:

A. y = A^c + B y = A^c + B B. y = A + B y = A + B C. y = AB y = AB D. y = A^cB y = A^cB

सही उत्तर: B Correct Answer: B

Using the absorption identity, A + A^cB = (A + A^c)(A + B) = 1(A + B) = A + B.

303

Mathematics • Linear Algebra

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

A. C[0,1] denotes the space of all continuous functions. T:C[0,1] to R defined as T(f) = 2f(1) + integral from 0 to 1 of f(s)ds is a linear functional. C[0,1] denotes the space of all continuous functions. T:C[0,1] to R defined as T(f) = 2f(1) + integral from 0 to 1 of f(s)ds is a linear functional. B. M_2x3(R), the space of all real-valued 2 x 3 matrices, is isomorphic to R^5. M_2x3(R), the space of all real-valued 2 x 3 matrices, is isomorphic to R^5. C. Let A,B:V to V be linear and I be the identity map. Then AB = I implies that A and B are invertible. Let A,B:V to V be linear and I be the identity map. Then AB = I implies that A and B are invertible. D. A linear transformation T is invertible if and only if T is one-one and onto. A linear transformation T is invertible if and only if T is one-one and onto.

सही उत्तर: B Correct Answer: B

The vector space of 2 x 3 real matrices has dimension 6, so it is isomorphic to R^6, not R^5.

304

Mathematics • Group Theory

Which of the following is true? Which of the following is true?

A. Every permutation can be written as a product of intersecting cycles. Every permutation can be written as a product of intersecting cycles. B. Every permutation can be written as a product of cycles of length 1. Every permutation can be written as a product of cycles of length 1. C. Every permutation can be written as a product of disjoint cycles. Every permutation can be written as a product of disjoint cycles. D. Every permutation can be written as a product of cycles that do not commute. Every permutation can be written as a product of cycles that do not commute.

सही उत्तर: C Correct Answer: C

Every permutation has a decomposition into disjoint cycles, unique up to the order in which those disjoint cycles are written.

305

Mathematics • Queueing Theory

In a newly opened bank, customers arrive at the rate of 2 customers per hour and are served by a single teller counter in about 20 minutes. The bank has a newly implemented policy that there should be no more than a 5% chance that a customer has to wait for more than 30 minutes in the queue before being served. How many additional teller counters should be opened to meet the policy requirements? In a newly opened bank, customers arrive at the rate of 2 customers per hour and are served by a single teller counter in about 20 minutes. The bank has a newly implemented policy that there should be no more than a 5% chance that a customer has to wait for more than 30 minutes in the queue before being served. How many additional teller counters should be opened to meet the policy requirements?

A. 0 (the existing counter is sufficient) 0 (the existing counter is sufficient) B. 3 3 C. 1 1 D. 2 2

सही उत्तर: C Correct Answer: C

Applying the multi-server queue waiting-time criterion to the stated arrival rate, service rate and 30-minute threshold shows that two counters in total are required. Therefore one additional counter is needed.

306

Mathematics • Fluid Mechanics

The metacentric height of a rectangular boat 6 x 6 x 12, a quarter portion of which is submerged in water, is: The metacentric height of a rectangular boat 6 x 6 x 12, a quarter portion of which is submerged in water, is:

A. 1.5 1.5 B. 3.5 3.5 C. 4.5 4.5 D. 2.5 2.5

सही उत्तर: B Correct Answer: B

Using the displaced volume, centre of buoyancy and water-plane moment of inertia for the rectangular boat gives a metacentric height of 3.5 units.

307

Mathematics • Graph Theory

For a tree with q lines, the number of vertices p equals: For a tree with q lines, the number of vertices p equals:

A. 2q 2q B. q q C. q + 2 q + 2 D. q + 1 q + 1

सही उत्तर: D Correct Answer: D

A tree with p vertices has exactly p - 1 edges. Therefore, if the number of lines or edges is q, then p = q + 1.

308

Mathematics • Real Analysis

Let f:[0,infinity) be such that df/dx is monotonically increasing and f(0) = 0. If g(x) = f(x)/x, then g is: Let f:[0,infinity) be such that df/dx is monotonically increasing and f(0) = 0. If g(x) = f(x)/x, then g is:

A. Concave function Concave function B. Convex function Convex function C. Monotonically decreasing Monotonically decreasing D. Monotonically increasing Monotonically increasing

सही उत्तर: D Correct Answer: D

An increasing derivative makes f convex. For a convex function with f(0)=0, the secant slope f(x)/x from the origin is monotonically increasing.

309

Mathematics • Group Theory

Which one of the following groups is cyclic? Which one of the following groups is cyclic?

A. (Q, +) (Q, +) B. (Q+, multiplication) (Q+, multiplication) C. GL_n(R) under matrix multiplication GL_n(R) under matrix multiplication D. {[[1,n],[0,1]] : n belongs to Z} under matrix multiplication {[[1,n],[0,1]] : n belongs to Z} under matrix multiplication

सही उत्तर: D Correct Answer: D

The matrix [[1,n],[0,1]] is the nth power of [[1,1],[0,1]]. Thus the group is generated by one element and is cyclic.

310

Mathematics • Fluid Mechanics

For a blood vessel of constant radius R, length L and driving force P = P1 - P2, where P1 and P2 denote the pressures at either end of length L, the average flow is equal to half the maximum velocity and the resistance (P2 - P1)/v is proportional to: For a blood vessel of constant radius R, length L and driving force P = P1 - P2, where P1 and P2 denote the pressures at either end of length L, the average flow is equal to half the maximum velocity and the resistance (P2 - P1)/v is proportional to:

A. L/R^3 L/R^3 B. L/R^4 L/R^4 C. L/R^2 L/R^2 D. L/R L/R

सही उत्तर: B Correct Answer: B

Poiseuille's law gives volumetric flow proportional to the pressure difference times R^4/L. Therefore hydraulic resistance is proportional to L/R^4.

Mathematics MCQs

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