Mathematics MCQs

121

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.

122

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.

123

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.

124

Mathematics • Integral Calculus

The area under the curve given in polar coordinates by r=f(omega) is given by: The area under the curve given in polar coordinates by r=f(omega) is given by:

A. integral r^2 d(omega) integral r^2 d(omega) B. integral (r^3/2)dr integral (r^3/2)dr C. 2 integral r^2 d(omega) 2 integral r^2 d(omega) D. integral (r^2/2)d(omega) integral (r^2/2)d(omega)

सही उत्तर: D Correct Answer: D

The area element in polar coordinates is r dr d(omega). Integrating radially from 0 to r gives (1/2)r^2d(omega), hence area=(1/2)integral r^2d(omega).

125

Mathematics • Differential Equations

If the roots of the auxiliary equation of the differential equation d^4y/dx^4 - 5d^3y/dx^3 + 6d^2y/dx^2 + 4dy/dx - 8y=0 are 2,2,2 and -1, then the general solution is: If the roots of the auxiliary equation of the differential equation d^4y/dx^4 - 5d^3y/dx^3 + 6d^2y/dx^2 + 4dy/dx - 8y=0 are 2,2,2 and -1, then the general solution is:

A. y=c1e^(2x)+c2e^(-x) y=c1e^(2x)+c2e^(-x) B. y=c1e^(2x)+c2e^(-x)+c3e^x+c4e^(-2x) y=c1e^(2x)+c2e^(-x)+c3e^x+c4e^(-2x) C. y=(c1+c2x+c3x^2)e^(-x)+c4e^(2x) y=(c1+c2x+c3x^2)e^(-x)+c4e^(2x) D. y=(c1+c2x+c3x^2)e^(2x)+c4e^(-x) y=(c1+c2x+c3x^2)e^(2x)+c4e^(-x)

सही उत्तर: D Correct Answer: D

A root 2 of multiplicity three contributes (c1+c2x+c3x^2)e^(2x), and the simple root -1 contributes c4e^(-x).

126

Mathematics • Group Theory

What is the order of (2,6) in Z_6 x Z_12? What is the order of (2,6) in Z_6 x Z_12?

A. 1 1 B. 6 6 C. 10 10 D. 2 2

सही उत्तर: B Correct Answer: B

The order of 2 in Z_6 is 3 and the order of 6 in Z_12 is 2. The order of the pair is lcm(3,2)=6.

127

Mathematics • Conic Sections

Which conic does the equation R=5/(4-11cos(omega)) represent? Which conic does the equation R=5/(4-11cos(omega)) represent?

A. Hyperbola Hyperbola B. Parabola Parabola C. Ellipse Ellipse D. Circle Circle

सही उत्तर: A Correct Answer: A

Writing the denominator in standard polar-conic form shows eccentricity e=11/4. Since e>1, the conic is a hyperbola.

128

Mathematics • Linear Algebra

For any subset U of a vector space V, let U-perpendicular denote the set of vectors orthogonal to every vector in U. Find (U-perpendicular)-perpendicular, where U={(2,0,0),(1,0,3)}. For any subset U of a vector space V, let U-perpendicular denote the set of vectors orthogonal to every vector in U. Find (U-perpendicular)-perpendicular, where U={(2,0,0),(1,0,3)}.

A. R^3 R^3 B. {(2,0,0),(1,0,3)} {(2,0,0),(1,0,3)} C. {(x,y,z): y=0} {(x,y,z): y=0} D. {(0,0,0)} {(0,0,0)}

सही उत्तर: C Correct Answer: C

The two vectors are independent and span the xz-plane y=0. In finite-dimensional inner-product spaces, (U-perpendicular)-perpendicular equals the span of U.

129

Mathematics • Group Theory

GL_n(R) denotes the group of n x n non-singular matrices under matrix multiplication. Which one of the following gives an abelian subgroup? GL_n(R) denotes the group of n x n non-singular matrices under matrix multiplication. Which one of the following gives an abelian subgroup?

A. Set of all non-singular upper-triangular matrices. Set of all non-singular upper-triangular matrices. B. Set of all non-singular matrices with positive trace. Set of all non-singular matrices with positive trace. C. Set of all matrices with positive determinant. Set of all matrices with positive determinant. D. Set of all non-singular diagonal matrices. Set of all non-singular diagonal matrices.

सही उत्तर: D Correct Answer: D

Non-singular diagonal matrices are closed under multiplication and inversion, and diagonal matrices commute with one another. Hence they form an abelian subgroup.

130

Mathematics • Complex Analysis

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

A. tan(z) tan(z) B. sin(z) sin(z) C. cosh(z) cosh(z) D. cos(z) cos(z)

सही उत्तर: B Correct Answer: B

This is the standard Maclaurin expansion z-z^3/3!+z^5/5!-..., which equals sin(z).

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