Advanced Mathematics MCQs

61

Advanced 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.

62

Advanced 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?

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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.

63

Advanced 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:

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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.

64

Advanced 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:

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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.

65

Advanced 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:

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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.

66

Advanced Mathematics • Group Theory

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

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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.

67

Advanced 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:

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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.

68

Advanced Mathematics • Linear Algebra

Obtain the dimension of the dual space for {(x,y,z): x + 2y + 3z = 0}. Obtain the dimension of the dual space for {(x,y,z): x + 2y + 3z = 0}.

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A. 3 3 B. 2 2 C. 4 4 D. 1 1

सही उत्तर: B Correct Answer: B

The given set is the null space of one nonzero linear equation in R^3, so it has dimension 2. A finite-dimensional vector space and its dual have the same dimension.

69

Advanced Mathematics • Linear Algebra

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

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A. For any vector space, a basis set gives a minimal spanning set. For any vector space, a basis set gives a minimal spanning set. B. A basis set for a vector space is not unique. A basis set for a vector space is not unique. C. Any set of vectors containing the null vector is linearly dependent. Any set of vectors containing the null vector is linearly dependent. D. For any linear transformation T:V to V and any basis {v1,...,vk} of V, the set {T(vi): 1 <= i <= k} also gives a basis set. For any linear transformation T:V to V and any basis {v1,...,vk} of V, the set {T(vi): 1 <= i <= k} also gives a basis set.

सही उत्तर: D Correct Answer: D

The image of a basis under an arbitrary linear transformation need not be a basis; this is guaranteed only when the transformation is invertible.

70

Advanced Mathematics • Mathematical Modelling

A forest can support a herd of 500 deer whose population has a linear growth rate of 20% per year. The number of deer that can be killed by hunting each year to maintain the herd size at 300 is: A forest can support a herd of 500 deer whose population has a linear growth rate of 20% per year. The number of deer that can be killed by hunting each year to maintain the herd size at 300 is:

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A. 28 28 B. 32 32 C. 24 24 D. 20 20

सही उत्तर: C Correct Answer: C

Using logistic growth with carrying capacity 500, annual growth at population 300 is 0.2 x 300 x (1 - 300/500) = 24. Hunting 24 deer per year maintains the population at 300.

Advanced Mathematics MCQs

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