Mathematics MCQs

351

Mathematics • Differential Equations

The differential equation for the family of surfaces x^3z + x^2y = c, where c is a parameter, is: The differential equation for the family of surfaces x^3z + x^2y = c, where c is a parameter, is:

A. (3xz+2y)dx - xdy - x^2dz = 0 (3xz+2y)dx - xdy - x^2dz = 0 B. (2xz+y)dx + xdy + x^2dz = 0 (2xz+y)dx + xdy + x^2dz = 0 C. (3xz+2y)dx + xdy + x^2dz = 0 (3xz+2y)dx + xdy + x^2dz = 0 D. (xz+2y)dx + xdy + xdz = 0 (xz+2y)dx + xdy + xdz = 0

सही उत्तर: C Correct Answer: C

Differentiating x^3z+x^2y=c gives (3x^2z+2xy)dx+x^2dy+x^3dz=0. Dividing by x gives option C.

352

Mathematics • Complex Analysis

Select the function f(z) from below that is analytic on the unit disk |z|<1. Select the function f(z) from below that is analytic on the unit disk |z|<1.

A. f(z)=e^x e^(-iy) f(z)=e^x e^(-iy) B. f(z)=cos(x)cosh(y)-i sin(x)sinh(y) f(z)=cos(x)cosh(y)-i sin(x)sinh(y) C. f(z)=3x+i3xy f(z)=3x+i3xy D. f(z)=2x+ixy^2 f(z)=2x+ixy^2

सही उत्तर: B Correct Answer: B

Option B is cos(x+iy)=cos(z), which is entire and therefore analytic on the unit disk.

353

Mathematics • Complex Analysis

The power-series expansion sum of z^(4n+1)/(2n)! for n=0,1,2,... represents which analytic function? The power-series expansion sum of z^(4n+1)/(2n)! for n=0,1,2,... represents which analytic function?

A. z sinh(z) z sinh(z) B. z sinh(z^2) z sinh(z^2) C. 1/z^2 1/z^2 D. z cosh(z^2) z cosh(z^2)

सही उत्तर: D Correct Answer: D

Since cosh(z^2)=sum z^(4n)/(2n)!, multiplying by z gives sum z^(4n+1)/(2n)!.

354

Mathematics • Mathematical Modelling

A population of shrimp has unlimited resources and its population size grows at a rate of 10% per month. The initial population size is 1 million. The number of shrimp that may be caught each month without ultimately exhausting their population is: A population of shrimp has unlimited resources and its population size grows at a rate of 10% per month. The initial population size is 1 million. The number of shrimp that may be caught each month without ultimately exhausting their population is:

A. 10,000 10,000 B. 1,01,000 1,01,000 C. 1,00,000 1,00,000 D. 10,00,000 10,00,000

सही उत्तर: C Correct Answer: C

Ten percent of the initial population of 1,000,000 is 100,000. A monthly harvest equal to this growth can be sustained under the stated unlimited-resource model.

355

Mathematics • Differential Equations

The particular integral of the differential equation d^2y/dx^2 + y = cos(x) is: The particular integral of the differential equation d^2y/dx^2 + y = cos(x) is:

A. (1/2)x cos(x) (1/2)x cos(x) B. (1/2)x sin(x) (1/2)x sin(x) C. x^2 cos(x) x^2 cos(x) D. x^2 sin(x) x^2 sin(x)

सही उत्तर: B Correct Answer: B

Because cos(x) is a complementary solution, resonance occurs. The standard particular integral is (x/2)sin(x).

356

Mathematics • Partial Differential Equations

Two ends A and B of a rod of length 20 cm have temperatures 40 degrees C and 90 degrees C until steady state. After steady state, the temperatures at A and B are changed to 45 degrees C and 95 degrees C respectively. In the stated temperature-distribution formula mu(x,t)=Ax+B+series terms, the values of A and B are: Two ends A and B of a rod of length 20 cm have temperatures 40 degrees C and 90 degrees C until steady state. After steady state, the temperatures at A and B are changed to 45 degrees C and 95 degrees C respectively. In the stated temperature-distribution formula mu(x,t)=Ax+B+series terms, the values of A and B are:

A. A=3/2, B=42 A=3/2, B=42 B. A=45, B=5/2 A=45, B=5/2 C. A=5/2, B=3/2 A=5/2, B=3/2 D. A=5/2, B=45 A=5/2, B=45

सही उत्तर: D Correct Answer: D

The new steady component is linear with mu(0)=45 and mu(20)=95. Hence B=45 and A=(95-45)/20=5/2.

357

Mathematics • Differential Equations

The general solution of the differential equation y'' - [x/(x-1)]y' + [1/(x-1)]y = 0 is: The general solution of the differential equation y'' - [x/(x-1)]y' + [1/(x-1)]y = 0 is:

A. y=c2x-c1 cos(x) y=c2x-c1 cos(x) B. y=c1(x^2-1)+c2x y=c1(x^2-1)+c2x C. y=c1e^x-c2x^2e^x y=c1e^x-c2x^2e^x D. y=c1x-c2e^x y=c1x-c2e^x

सही उत्तर: D Correct Answer: D

Direct substitution shows that x and e^x are linearly independent solutions, so the general solution is a linear combination of them, equivalent to option D.

358

Mathematics • Queueing Theory

In a car wash with one cleaning machine, if cars arrive at a rate of a1 per minute and the expected service time for each car is b1 minutes, the utilization factor is: In a car wash with one cleaning machine, if cars arrive at a rate of a1 per minute and the expected service time for each car is b1 minutes, the utilization factor is:

A. 1/(a1-b1) 1/(a1-b1) B. a1*b1 a1*b1 C. a1/b1 a1/b1 D. a1/(a1-b1) a1/(a1-b1)

सही उत्तर: B Correct Answer: B

Utilization is arrival rate divided by service rate. Since mean service time is b1, the service rate is 1/b1, so utilization is a1b1.

359

Mathematics • Abstract Algebra

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

A. Intersection of any two subgroups is a subgroup. Intersection of any two subgroups is a subgroup. B. For any ring (R,+,.), the relation a(b-c)=0 implies b=c. For any ring (R,+,.), the relation a(b-c)=0 implies b=c. C. If (G,.) is abelian and H,K are subgroups of G, then {hk: h in H, k in K} is a subgroup of G. If (G,.) is abelian and H,K are subgroups of G, then {hk: h in H, k in K} is a subgroup of G. D. Z4 under addition modulo 4 is cyclic. Z4 under addition modulo 4 is cyclic.

सही उत्तर: B Correct Answer: B

Cancellation need not hold in an arbitrary ring because rings can have zero divisors. Thus a(b-c)=0 does not always imply b=c.

360

Mathematics • Conic Sections

For what value of c is the line y=0.5x+c tangent to the parabola y^2=4ax? For what value of c is the line y=0.5x+c tangent to the parabola y^2=4ax?

A. a/2 a/2 B. a/4 a/4 C. 2a 2a D. a a

सही उत्तर: C Correct Answer: C

A tangent to y^2=4ax with slope m has equation y=mx+a/m. With m=0.5, the intercept is c=a/0.5=2a.

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