Advanced Mathematics MCQs

91

Advanced Mathematics • Vector Calculus

Consider f(x,y) = xe^y. At the point (2,0), what is the maximum rate of change? Consider f(x,y) = xe^y. At the point (2,0), what is the maximum rate of change?

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A. sqrt(5) sqrt(5) B. 4 4 C. 1 1 D. 0 0

सही उत्तर: A Correct Answer: A

The maximum directional derivative is |grad f|. Here grad f=(e^y,xe^y), which at (2,0) is (1,2), of magnitude sqrt(5).

92

Advanced Mathematics • Mathematical Modelling

In a simple epidemic model dS/dt = -beta SI and dI/dt = beta SI, where S(t) and I(t) denote the numbers of susceptible and infected persons respectively, S(t) is given by: In a simple epidemic model dS/dt = -beta SI and dI/dt = beta SI, where S(t) and I(t) denote the numbers of susceptible and infected persons respectively, S(t) is given by:

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A. S(t) = n(n+1)/(n + e^((n+1)t)) S(t) = n(n+1)/(n + e^((n+1)t)) B. S(t) = n(n+1)/(n + e^((n+1)beta t)) S(t) = n(n+1)/(n + e^((n+1)beta t)) C. S(t) = (n+1)/(n e^(n beta t)) S(t) = (n+1)/(n e^(n beta t)) D. S(t) = n/(n + e^(n beta t)) S(t) = n/(n + e^(n beta t))

सही उत्तर: B Correct Answer: B

Using S+I=n+1 and separating dS/dt=-beta S(n+1-S), the solution with the stated standard initial conditions is the expression in option B.

93

Advanced Mathematics • Linear Algebra

A is an n x n non-singular matrix. Which one of the following statements is true? A is an n x n non-singular matrix. Which one of the following statements is true?

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A. The eigenvectors of A are orthogonal to each other. The eigenvectors of A are orthogonal to each other. B. All the eigenvalues are positive. All the eigenvalues are positive. C. 0 cannot be an eigenvalue of A. 0 cannot be an eigenvalue of A. D. tr(A) = sum from i=1 to n of a_ii is not equal to 0. tr(A) = sum from i=1 to n of a_ii is not equal to 0.

सही उत्तर: C Correct Answer: C

A matrix is non-singular exactly when its determinant is nonzero. Since the determinant is the product of its eigenvalues, zero cannot be an eigenvalue.

94

Advanced Mathematics • Fluid Mechanics

A heavy car lands at the bottom of a lake on its wheels. Its door is 1.2 m high and 1 m wide, and the top edge of the door is 8 m below the free surface of the water. The hydrostatic force on the door is: A heavy car lands at the bottom of a lake on its wheels. Its door is 1.2 m high and 1 m wide, and the top edge of the door is 8 m below the free surface of the water. The hydrostatic force on the door is:

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A. 101.3 kN 101.3 kN B. 104.3 kN 104.3 kN C. 102.3 kN 102.3 kN D. 100.4 kN 100.4 kN

सही उत्तर: A Correct Answer: A

The centroid of the rectangular door is at depth 8.6 m and its area is 1.2 m^2. Thus F=rho g A h_c = 1000 x 9.81 x 1.2 x 8.6, approximately 101.3 kN.

95

Advanced Mathematics • Integral Calculus

The value of the integral of sin^4(x)cos^3(x) dx over [0, pi/2] is: The value of the integral of sin^4(x)cos^3(x) dx over [0, pi/2] is:

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A. 2/35 2/35 B. pi/6 pi/6 C. 6/35 6/35 D. pi/25 pi/25

सही उत्तर: A Correct Answer: A

Write cos^3x=(1-sin^2x)cosx and substitute u=sinx. The integral becomes integral from 0 to 1 of (u^4-u^6)du = 1/5-1/7 = 2/35.

96

Advanced Mathematics • Linear Algebra

Consider the vector space of all polynomials with real coefficients and degree bounded by 2, and define the linear functional T as T(f)=f(0). Find the dimension of the null space of T. Consider the vector space of all polynomials with real coefficients and degree bounded by 2, and define the linear functional T as T(f)=f(0). Find the dimension of the null space of T.

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

सही उत्तर: A Correct Answer: A

The space has basis {1,x,x^2} and dimension 3. T has rank 1, so rank-nullity gives nullity 3-1=2.

97

Advanced Mathematics • Probability

What is the probability that, in a random arrangement of the English alphabet, the word MOTHER will appear? What is the probability that, in a random arrangement of the English alphabet, the word MOTHER will appear?

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A. (1/6)^3 (1/6)^3 B. 20!/26! 20!/26! C. 21!/26! 21!/26! D. 22!/26! 22!/26!

सही उत्तर: C Correct Answer: C

Treat MOTHER as one fixed block. Together with the other 20 letters there are 21 objects, giving 21! favourable arrangements out of 26! total arrangements.

98

Advanced Mathematics • Dynamics

A rough uniform board of mass m and length 2a rests on a smooth horizontal plane. A man of mass M walks on it from one end to the other. The distance through which the board moves in this time is: A rough uniform board of mass m and length 2a rests on a smooth horizontal plane. A man of mass M walks on it from one end to the other. The distance through which the board moves in this time is:

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A. 3Ma/(m+M) 3Ma/(m+M) B. 3Ma/(m+2M) 3Ma/(m+2M) C. 2Ma/(m+M) 2Ma/(m+M) D. Ma/(m+M) Ma/(m+M)

सही उत्तर: C Correct Answer: C

With no external horizontal force, the centre of mass remains fixed. If the man moves 2a relative to the board, the board displacement has magnitude 2Ma/(m+M).

99

Advanced Mathematics • Group Theory

Which of the following is a non-abelian group? Which of the following is a non-abelian group?

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A. The set of non-zero real numbers under multiplication. The set of non-zero real numbers under multiplication. B. The real numbers under addition. The real numbers under addition. C. The set of 2 by 2 matrices under addition. The set of 2 by 2 matrices under addition. D. The set of 2 by 2 matrices with determinant 1 under matrix multiplication. The set of 2 by 2 matrices with determinant 1 under matrix multiplication.

सही उत्तर: D Correct Answer: D

The determinant-one matrices form SL(2), and matrix multiplication in this group is generally noncommutative.

100

Advanced Mathematics • Real Analysis

Function f(x)=|x|+sin(x) is: Function f(x)=|x|+sin(x) is:

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A. Convex Convex B. Continuous and differentiable at x=0 Continuous and differentiable at x=0 C. Continuous but not differentiable at x=0 Continuous but not differentiable at x=0 D. Concave Concave

सही उत्तर: C Correct Answer: C

Both terms are continuous. At zero, the one-sided derivatives are -1+1=0 and 1+1=2, so the derivative does not exist.

Advanced Mathematics MCQs

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