MCQs for JSSC PGT Mathematics Exam

41

Advanced Mathematics • Differential Equations

Which of the following can be reduced to a Clairaut equation? Which of the following can be reduced to a Clairaut equation?

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A. y' = e^x + e^y + y^2 y' = e^x + e^y + y^2 B. y' = sin(x) + sin(y) y' = sin(x) + sin(y) C. y = x^4p - p^2x, where p = dy/dx y = x^4p - p^2x, where p = dy/dx D. y = 2xp + y^2p^3, where p = dy/dx y = 2xp + y^2p^3, where p = dy/dx

सही उत्तर: D Correct Answer: D

Option D is the equation marked in the source paper as reducible, after suitable substitution, to Clairaut form.

42

Advanced Mathematics • Applied Mechanics

The equation for finding the centre of pressure of a rocket is (where (C_N alpha)_n denotes the normal-force coefficient derivative of the nose): The equation for finding the centre of pressure of a rocket is (where (C_N alpha)_n denotes the normal-force coefficient derivative of the nose):

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A. (C_N alpha)_n = 3 (C_N alpha)_n = 3 B. (C_N alpha)_n = 2 (C_N alpha)_n = 2 C. (C_N alpha)_n = 1 (C_N alpha)_n = 1 D. (C_N alpha)_n = 4 (C_N alpha)_n = 4

सही उत्तर: B Correct Answer: B

For the rocket nose contribution used in the centre-of-pressure relation, the normal-force coefficient derivative is 2.

43

Advanced Mathematics • Digital Logic

Which is called the universal gate? Which is called the universal gate?

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A. AND AND B. OR OR C. NAND NAND D. XNOR XNOR

सही उत्तर: C Correct Answer: C

NAND is a universal gate because every Boolean function can be implemented using NAND gates alone.

44

Advanced Mathematics • Vector Calculus

Evaluate the line integral over C of (x^4 dx + xy dy), where C is the triangular curve consisting of line segments from (0,0) to (1,0), from (1,0) to (1,1), and from (1,1) to (0,0). Evaluate the line integral over C of (x^4 dx + xy dy), where C is the triangular curve consisting of line segments from (0,0) to (1,0), from (1,0) to (1,1), and from (1,1) to (0,0).

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A. 0 0 B. 12 12 C. 1/6 1/6 D. 1 1

सही उत्तर: C Correct Answer: C

Evaluating the integral along the three directed line segments and adding the results gives 1/6.

45

Advanced Mathematics • Complex Analysis

If a function f(z), defined on some domain D of the complex plane, is differentiable on D, it is said to be: If a function f(z), defined on some domain D of the complex plane, is differentiable on D, it is said to be:

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A. Analytic function Analytic function B. Non-singular function Non-singular function C. Bounded function Bounded function D. Harmonic function Harmonic function

सही उत्तर: A Correct Answer: A

A complex function differentiable at every point of a domain is analytic, or holomorphic, on that domain.

46

Advanced Mathematics • Complex Analysis

Let f(r cos(theta) + i r sin(theta)) = u(r,theta) + iv(r,theta). For f to be differentiable at (r,theta), the Cauchy-Riemann equations state that: Let f(r cos(theta) + i r sin(theta)) = u(r,theta) + iv(r,theta). For f to be differentiable at (r,theta), the Cauchy-Riemann equations state that:

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A. u_r = -v_r/r, v_r = -u_theta/r u_r = -v_r/r, v_r = -u_theta/r B. u_r = v_theta/r, v_r = -u_theta/r u_r = v_theta/r, v_r = -u_theta/r C. u_r = -v_theta/r, v_theta = -u_theta/r u_r = -v_theta/r, v_theta = -u_theta/r D. u_r = -v_theta/r, v_r = -u_theta/r u_r = -v_theta/r, v_r = -u_theta/r

सही उत्तर: B Correct Answer: B

In polar coordinates the Cauchy-Riemann equations are u_r = v_theta/r and v_r = -u_theta/r.

47

Advanced Mathematics • Linear Programming

A solution to a linear programming problem that satisfies all the constraints except the non-negativity constraint is referred to as: A solution to a linear programming problem that satisfies all the constraints except the non-negativity constraint is referred to as:

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A. Feasible solution Feasible solution B. Basic solution Basic solution C. Optimal solution Optimal solution D. Basic feasible solution Basic feasible solution

सही उत्तर: B Correct Answer: B

A basic solution satisfies the system of equality constraints but need not satisfy non-negativity. If it also satisfies non-negativity, it becomes a basic feasible solution.

48

Advanced Mathematics • Partial Differential Equations

The equation of the characteristic curve of the one-parameter family (x - a)^2 + (y - a)^2 + z^2 = 1 is: The equation of the characteristic curve of the one-parameter family (x - a)^2 + (y - a)^2 + z^2 = 1 is:

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A. (x - y)^2 = 2(1 - z^2) (x - y)^2 = 2(1 - z^2) B. (x - y)^2 = 1 - z^2 (x - y)^2 = 1 - z^2 C. (x - z)^2 = 2(1 - y^2) (x - z)^2 = 2(1 - y^2) D. (x - z)^2 = 1 - y^2 (x - z)^2 = 1 - y^2

सही उत्तर: A Correct Answer: A

Differentiating with respect to the parameter gives a = (x + y)/2. Eliminating a from the family yields (x - y)^2 = 2(1 - z^2).

49

Advanced Mathematics • Complex Analysis

If C is a positively oriented simple closed contour within and on which f is analytic, except for a finite number of singular points z_k (k = 1,2,...,n) interior to C, then the Residue Theorem states that: If C is a positively oriented simple closed contour within and on which f is analytic, except for a finite number of singular points z_k (k = 1,2,...,n) interior to C, then the Residue Theorem states that:

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A. Integral_C f(z)dz = e pi i (sum_k Res f(z_k)/k!) Integral_C f(z)dz = e pi i (sum_k Res f(z_k)/k!) B. Integral_C f(z)dz = 2pi i (sum_k Res f(z_k)) Integral_C f(z)dz = 2pi i (sum_k Res f(z_k)) C. Integral_C f(z)dz = 2pi i (sum_k Res f(z_k)/k!) Integral_C f(z)dz = 2pi i (sum_k Res f(z_k)/k!) D. Integral_C f(z)dz = sum_k Res f(z_k) Integral_C f(z)dz = sum_k Res f(z_k)

सही उत्तर: B Correct Answer: B

By the Residue Theorem, the contour integral equals 2pi i times the sum of the residues at all singularities inside C.

50

Advanced Mathematics • Probability

Consider 10 tosses of a biased coin with probability of obtaining a head equal to 1/3. Let Y denote the total number of tails observed. Find E(5Y + 2). Consider 10 tosses of a biased coin with probability of obtaining a head equal to 1/3. Let Y denote the total number of tails observed. Find E(5Y + 2).

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A. 10 10 B. 56/3 56/3 C. 106/3 106/3 D. 10/3 10/3

सही उत्तर: C Correct Answer: C

The probability of a tail is 2/3, so E(Y) = 10 x 2/3 = 20/3. Thus E(5Y + 2) = 5E(Y) + 2 = 106/3.

MCQs for JSSC PGT Mathematics Exam

MCQs for JSSC PGT Mathematics Exam

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