Mathematics MCQs

341

Mathematics • Linear Algebra

A is an n x n non-singular matrix. Which one of the following statements is true? A is an n x n non-singular matrix. Which one of the following statements is true?

A. The eigenvectors of A are orthogonal to each other. The eigenvectors of A are orthogonal to each other. B. All the eigenvalues are positive. All the eigenvalues are positive. C. 0 cannot be an eigenvalue of A. 0 cannot be an eigenvalue of A. D. tr(A) = sum from i=1 to n of a_ii is not equal to 0. tr(A) = sum from i=1 to n of a_ii is not equal to 0.

सही उत्तर: C Correct Answer: C

A matrix is non-singular exactly when its determinant is nonzero. Since the determinant is the product of its eigenvalues, zero cannot be an eigenvalue.

342

Mathematics • Fluid Mechanics

A heavy car lands at the bottom of a lake on its wheels. Its door is 1.2 m high and 1 m wide, and the top edge of the door is 8 m below the free surface of the water. The hydrostatic force on the door is: A heavy car lands at the bottom of a lake on its wheels. Its door is 1.2 m high and 1 m wide, and the top edge of the door is 8 m below the free surface of the water. The hydrostatic force on the door is:

A. 101.3 kN 101.3 kN B. 104.3 kN 104.3 kN C. 102.3 kN 102.3 kN D. 100.4 kN 100.4 kN

सही उत्तर: A Correct Answer: A

The centroid of the rectangular door is at depth 8.6 m and its area is 1.2 m^2. Thus F=rho g A h_c = 1000 x 9.81 x 1.2 x 8.6, approximately 101.3 kN.

343

Mathematics • Integral Calculus

The value of the integral of sin^4(x)cos^3(x) dx over [0, pi/2] is: The value of the integral of sin^4(x)cos^3(x) dx over [0, pi/2] is:

A. 2/35 2/35 B. pi/6 pi/6 C. 6/35 6/35 D. pi/25 pi/25

सही उत्तर: A Correct Answer: A

Write cos^3x=(1-sin^2x)cosx and substitute u=sinx. The integral becomes integral from 0 to 1 of (u^4-u^6)du = 1/5-1/7 = 2/35.

344

Mathematics • Sequences and Series

As n approaches infinity, the sequence s_n = (n^3 + 4)/(10 + 6n^3): As n approaches infinity, the sequence s_n = (n^3 + 4)/(10 + 6n^3):

A. Is undefined Is undefined B. Converges to 1/6 Converges to 1/6 C. Converges to 2/5 Converges to 2/5 D. Diverges Diverges

सही उत्तर: B Correct Answer: B

Dividing numerator and denominator by n^3 gives (1+4/n^3)/(10/n^3+6), whose limit is 1/6.

345

Mathematics • Linear Algebra

Consider the vector space of all polynomials with real coefficients and degree bounded by 2, and define the linear functional T as T(f)=f(0). Find the dimension of the null space of T. Consider the vector space of all polynomials with real coefficients and degree bounded by 2, and define the linear functional T as T(f)=f(0). Find the dimension of the null space of T.

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

सही उत्तर: A Correct Answer: A

The space has basis {1,x,x^2} and dimension 3. T has rank 1, so rank-nullity gives nullity 3-1=2.

346

Mathematics • Probability

What is the probability that, in a random arrangement of the English alphabet, the word MOTHER will appear? What is the probability that, in a random arrangement of the English alphabet, the word MOTHER will appear?

A. (1/6)^3 (1/6)^3 B. 20!/26! 20!/26! C. 21!/26! 21!/26! D. 22!/26! 22!/26!

सही उत्तर: C Correct Answer: C

Treat MOTHER as one fixed block. Together with the other 20 letters there are 21 objects, giving 21! favourable arrangements out of 26! total arrangements.

347

Mathematics • Dynamics

A rough uniform board of mass m and length 2a rests on a smooth horizontal plane. A man of mass M walks on it from one end to the other. The distance through which the board moves in this time is: A rough uniform board of mass m and length 2a rests on a smooth horizontal plane. A man of mass M walks on it from one end to the other. The distance through which the board moves in this time is:

A. 3Ma/(m+M) 3Ma/(m+M) B. 3Ma/(m+2M) 3Ma/(m+2M) C. 2Ma/(m+M) 2Ma/(m+M) D. Ma/(m+M) Ma/(m+M)

सही उत्तर: C Correct Answer: C

With no external horizontal force, the centre of mass remains fixed. If the man moves 2a relative to the board, the board displacement has magnitude 2Ma/(m+M).

348

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.

349

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.

350

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.

Mathematics MCQs

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