Advanced Mathematics MCQs

81

Advanced Mathematics • Partial Differential Equations

The integral curves of the equations dx/[x^2(y^2-z^2)] = dy/[y^2(z^2-x^2)] = dz/[z^2(x^2-y^2)] are given by, where c1 and c2 are arbitrary constants: The integral curves of the equations dx/[x^2(y^2-z^2)] = dy/[y^2(z^2-x^2)] = dz/[z^2(x^2-y^2)] are given by, where c1 and c2 are arbitrary constants:

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A. x^2 + y^2 - z^2 = c1 and x^-1 + y^-1 - z^-1 = c2 x^2 + y^2 - z^2 = c1 and x^-1 + y^-1 - z^-1 = c2 B. x^2 - y^2 + z^2 = c1 and x^-1 - y^-1 + z^-1 = c2 x^2 - y^2 + z^2 = c1 and x^-1 - y^-1 + z^-1 = c2 C. x^2 - y^2 - z^2 = c1 and x^-1 - y^-1 - z^-1 = c2 x^2 - y^2 - z^2 = c1 and x^-1 - y^-1 - z^-1 = c2 D. x^2 + y^2 + z^2 = c1 and x^-1 + y^-1 + z^-1 = c2 x^2 + y^2 + z^2 = c1 and x^-1 + y^-1 + z^-1 = c2

सही उत्तर: D Correct Answer: D

Using suitable multipliers in the Lagrange auxiliary system yields the two independent first integrals x^2+y^2+z^2=c1 and 1/x+1/y+1/z=c2.

82

Advanced Mathematics • Fluid Mechanics

In the one-dimensional equation of motion, Bernoulli's equation is given by, where v^2/2 is kinetic energy, p/rho is flow energy and gz is potential energy, all per unit mass: In the one-dimensional equation of motion, Bernoulli's equation is given by, where v^2/2 is kinetic energy, p/rho is flow energy and gz is potential energy, all per unit mass:

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A. v^2/2 - p/rho - gz = constant v^2/2 - p/rho - gz = constant B. v^2/2 + p/rho - gz = constant v^2/2 + p/rho - gz = constant C. v^2/2 - p/rho + gz = constant v^2/2 - p/rho + gz = constant D. v^2/2 + p/rho + gz = constant v^2/2 + p/rho + gz = constant

सही उत्तर: D Correct Answer: D

Bernoulli's equation states that kinetic energy, pressure-flow energy and gravitational potential energy per unit mass sum to a constant along a streamline.

83

Advanced Mathematics • Linear Algebra

What is the dimension of the vector space of all 2 x 2 symmetric real-valued matrices? What is the dimension of the vector space of all 2 x 2 symmetric real-valued matrices?

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

सही उत्तर: A Correct Answer: A

A symmetric 2 x 2 matrix has the form [[a,b],[b,c]], determined by three independent real parameters. Its dimension is 3.

84

Advanced Mathematics • Probability

Suppose (X,Y) is jointly distributed with density f(s,t) = 1/pi if s^2 + t^2 <= 1, and 0 otherwise. Find Cov(X,Y). Suppose (X,Y) is jointly distributed with density f(s,t) = 1/pi if s^2 + t^2 <= 1, and 0 otherwise. Find Cov(X,Y).

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A. -1 -1 B. 1 1 C. 0 0 D. pi pi

सही उत्तर: C Correct Answer: C

By symmetry of the uniform distribution on the unit disk, E(X)=E(Y)=0 and E(XY)=0. Hence Cov(X,Y)=0.

85

Advanced Mathematics • Graph Theory

Every n-connected graph with p vertices has at least: Every n-connected graph with p vertices has at least:

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A. pn/2 lines pn/2 lines B. n(p + 1)/2 lines n(p + 1)/2 lines C. pn lines pn lines D. 2pn lines 2pn lines

सही उत्तर: A Correct Answer: A

An n-connected graph has minimum degree at least n. By the handshaking lemma, twice the number of edges is at least pn, so it has at least pn/2 edges.

86

Advanced Mathematics • Probability

Which one of the following gives a pair of independent random variables? Which one of the following gives a pair of independent random variables?

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A. 10 coins are thrown at random. X denotes the total number of heads and Y denotes the total number of tails. 10 coins are thrown at random. X denotes the total number of heads and Y denotes the total number of tails. B. 10 coins are tossed at random. X denotes the total number of heads on even tosses (2nd, 4th, ..., 10th) and Y denotes the total number of tails on odd tosses (1st, 3rd, ..., 9th). 10 coins are tossed at random. X denotes the total number of heads on even tosses (2nd, 4th, ..., 10th) and Y denotes the total number of tails on odd tosses (1st, 3rd, ..., 9th). C. For 1 <= i <= 6, let X_i denote the number of times digit i appears in 10 tosses of a die. Consider (X_1,X_2). For 1 <= i <= 6, let X_i denote the number of times digit i appears in 10 tosses of a die. Consider (X_1,X_2). D. Two dice are thrown at random; X is the maximum and Y is the minimum of the digits appearing. Two dice are thrown at random; X is the maximum and Y is the minimum of the digits appearing.

सही उत्तर: B Correct Answer: B

X depends only on the even-numbered tosses and Y only on the disjoint odd-numbered tosses. Since all tosses are independent, X and Y are independent.

87

Advanced Mathematics • Vector Calculus

For f(x,y) = x^2y - y^3, which one of the following gives the corresponding gradient vector field? For f(x,y) = x^2y - y^3, which one of the following gives the corresponding gradient vector field?

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A. x^2 i x^2 i B. 2xy i + (x^2 - 3y^2) j 2xy i + (x^2 - 3y^2) j C. x i + (x^2 - y^2) j x i + (x^2 - y^2) j D. (x^3 - y^2) j (x^3 - y^2) j

सही उत्तर: B Correct Answer: B

The partial derivatives are f_x = 2xy and f_y = x^2 - 3y^2. Hence grad f = 2xy i + (x^2 - 3y^2)j.

88

Advanced Mathematics • Vector Calculus

If f(x,y,z) = x sin(yz), find the gradient of f at the point (1,3,0). If f(x,y,z) = x sin(yz), find the gradient of f at the point (1,3,0).

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A. (1,0,3) (1,0,3) B. (1,-1,3) (1,-1,3) C. (0,0,3) (0,0,3) D. (0,0,0) (0,0,0)

सही उत्तर: C Correct Answer: C

grad f = (sin(yz), xz cos(yz), xy cos(yz)). At (1,3,0), this becomes (0,0,3).

89

Advanced Mathematics • Probability

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

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A. For two random variables X and Y, Cov(X,Y)=0 implies that X and Y are independent. For two random variables X and Y, Cov(X,Y)=0 implies that X and Y are independent. B. If V(X)=0, then P(X=0)=1. If V(X)=0, then P(X=0)=1. C. Let F be the distribution function of a random variable. Then lim as n approaches infinity of F(x + 1/n) always exists. Let F be the distribution function of a random variable. Then lim as n approaches infinity of F(x + 1/n) always exists. D. If P(X <= 100)=1, then E(X) always exists. If P(X <= 100)=1, then E(X) always exists.

सही उत्तर: C Correct Answer: C

A distribution function is right-continuous, so F(x+1/n) approaches F(x). The other statements are not true in general.

90

Advanced Mathematics • Limits

In which one of the following cases does the limit exist? In which one of the following cases does the limit exist?

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A. lim as (x,y,z)->(0,0,0) of (x^2 + 2y^2 + 3z^2)/(x^2 + y^2 + z^2) lim as (x,y,z)->(0,0,0) of (x^2 + 2y^2 + 3z^2)/(x^2 + y^2 + z^2) B. lim as (x,y)->(0,0) of (2x^2 + sin^2(y))/(x^2 + y^2) lim as (x,y)->(0,0) of (2x^2 + sin^2(y))/(x^2 + y^2) C. lim as (x,y)->(0,0) of 5x^2y/(x^2 + y^2) lim as (x,y)->(0,0) of 5x^2y/(x^2 + y^2) D. lim as (x,y,z)->(0,0,0) of (2x^2 + 2y^2 - z^2)/(x^2 + y^2 + z^2) lim as (x,y,z)->(0,0,0) of (2x^2 + 2y^2 - z^2)/(x^2 + y^2 + z^2)

सही उत्तर: C Correct Answer: C

For option C, |5x^2y/(x^2+y^2)| <= 5|y|, which tends to zero. The other expressions have different limits along different coordinate directions.

Advanced Mathematics MCQs

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