Mathematics MCQs

111

Mathematics • Group Theory

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

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.

112

Mathematics • Real Analysis

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

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.

113

Mathematics • Linear Programming

Select the correct dual of the LPP: Maximize z=5x1+9x2, subject to 2x1+5x2>=37, 6x1-7x2>=56, and x1,x2>=0. Select the correct dual of the LPP: Maximize z=5x1+9x2, subject to 2x1+5x2>=37, 6x1-7x2>=56, and x1,x2>=0.

A. Minimize L=-37y1-56y2; subject to -2y1-6y2>=5, -5y1+7y2>=9, y1,y2>=0 Minimize L=-37y1-56y2; subject to -2y1-6y2>=5, -5y1+7y2>=9, y1,y2>=0 B. Minimize L=-5y1-9y2; subject to 2y1+5y2>=37, 6y1-7y2>=56, y1,y2>=0 Minimize L=-5y1-9y2; subject to 2y1+5y2>=37, 6y1-7y2>=56, y1,y2>=0 C. Minimize L=5y1+9y2; subject to 2y1+5y2>=56, 6y1-7y2>=37, y1,y2>=0 Minimize L=5y1+9y2; subject to 2y1+5y2>=56, 6y1-7y2>=37, y1,y2>=0 D. Minimize L=37y1+56y2; subject to 2y1+6y2>=5, 5y1-7y2>=9, y1,y2>=0 Minimize L=37y1+56y2; subject to 2y1+6y2>=5, 5y1-7y2>=9, y1,y2>=0

सही उत्तर: A Correct Answer: A

After converting the primal constraints to the appropriate standard sign form, transposing the coefficient matrix gives the dual displayed in option A, which is also the answer marked in the source.

114

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.

115

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.

116

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)!.

117

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.

118

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).

119

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.

120

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.

Mathematics MCQs

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