Mathematics MCQs

331

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?

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.

332

Mathematics • Solid Geometry

Find the volume of the solid bounded by the ellipsoid x^2/4 + y^2/9 + z^2/49 = 1. Find the volume of the solid bounded by the ellipsoid x^2/4 + y^2/9 + z^2/49 = 1.

A. 124 124 B. 176 176 C. 248 248 D. 312 312

सही उत्तर: B Correct Answer: B

The semiaxes are 2, 3 and 7. Thus volume = (4/3)pi(2)(3)(7) = 56pi, approximately 176.

333

Mathematics • Conic Sections

For the conic r = 1 + sin(omega), the slope of the tangent to the conic at omega = pi/3 is: For the conic r = 1 + sin(omega), the slope of the tangent to the conic at omega = pi/3 is:

A. -1/2 -1/2 B. 1/2 1/2 C. -1 -1 D. 1 1

सही उत्तर: C Correct Answer: C

For a polar curve, dy/dx = (r' sin(omega) + r cos(omega))/(r' cos(omega) - r sin(omega)), where r' here denotes dr/domega. Substituting r = 1 + sin(omega) at omega = pi/3 gives -1.

334

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?

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.

335

Mathematics • Coordinate Geometry

The equation x^2 - 6x + 6y + y^2 + 2 = 0 represents: The equation x^2 - 6x + 6y + y^2 + 2 = 0 represents:

A. An ellipse with latus rectum 4 and focus at (-3,3) An ellipse with latus rectum 4 and focus at (-3,3) B. A circle with radius 4 and centre at (-3,3) A circle with radius 4 and centre at (-3,3) C. A circle with radius 4 and centre at (3,-3) A circle with radius 4 and centre at (3,-3) D. An ellipse with latus rectum 4 and focus at (3,-3) An ellipse with latus rectum 4 and focus at (3,-3)

सही उत्तर: C Correct Answer: C

Completing squares gives (x-3)^2 + (y+3)^2 = 16. This is a circle with centre (3,-3) and radius 4.

336

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).

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).

337

Mathematics • Probability

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

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.

338

Mathematics • Limits

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

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.

339

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?

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).

340

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:

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.

Mathematics MCQs

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