MCQs for JSSC PGT Mathematics Exam

11

Mathematics • Coordinate Geometry

Find the rectangular coordinates of the point with cylindrical coordinates (2, 2pi/3, 1). Find the rectangular coordinates of the point with cylindrical coordinates (2, 2pi/3, 1).

A. (-1, sqrt(3), 1) (-1, sqrt(3), 1) B. (2, sqrt(3), -2) (2, sqrt(3), -2) C. (1, sqrt(3), 0) (1, sqrt(3), 0) D. (1, sqrt(3), -1) (1, sqrt(3), -1)

सही उत्तर: A Correct Answer: A

Using x = r cos(theta), y = r sin(theta), z = z, we get x = 2 cos(2pi/3) = -1, y = 2 sin(2pi/3) = sqrt(3), and z = 1.

12

Mathematics • Graph Theory

In a graph, the number of vertices with odd degrees is always: In a graph, the number of vertices with odd degrees is always:

A. Odd Odd B. Zero Zero C. Even Even D. Cannot be determined Cannot be determined

सही उत्तर: C Correct Answer: C

By the handshaking lemma, the sum of all vertex degrees is even. Therefore, vertices of odd degree must occur in an even number.

13

Mathematics • Graph Theory

If every point of G has even degree, then G is a: If every point of G has even degree, then G is a:

A. Planar Graph Planar Graph B. Tree Tree C. Eulerian Graph Eulerian Graph D. Pie Chart Pie Chart

सही उत्तर: C Correct Answer: C

A connected graph has an Euler circuit if and only if every vertex has even degree; hence it is Eulerian.

14

Mathematics • Dynamics

A solid homogeneous cone of height h and semi-vertical angle alpha oscillates about a diameter of its base. The length of the simple equivalent pendulum is: A solid homogeneous cone of height h and semi-vertical angle alpha oscillates about a diameter of its base. The length of the simple equivalent pendulum is:

A. (h/5)(2 + 5tan^2(alpha)) (h/5)(2 + 5tan^2(alpha)) B. (h/5)(2 + 3tan^2(alpha)) (h/5)(2 + 3tan^2(alpha)) C. (h/3)(2 + 3tan^2(alpha)) (h/3)(2 + 3tan^2(alpha)) D. (h/2)(2 + 5tan^2(alpha)) (h/2)(2 + 5tan^2(alpha))

सही उत्तर: B Correct Answer: B

For a physical pendulum, the equivalent length is I/(Md), where I is the moment of inertia about the axis and d is the distance of the centre of mass from it. For the stated cone this reduces to (h/5)(2 + 3tan^2(alpha)).

15

Mathematics • Complex Analysis

Which of the following statements is INCORRECT? Which of the following statements is INCORRECT?

A. The composite of two continuous functions is also continuous. The composite of two continuous functions is also continuous. B. Let f(x + iy) = u(x,y) + iv(x,y). If u and v both have continuous partial derivatives with respect to x and y at (x0,y0), then f is also differentiable at (x0,y0). Let f(x + iy) = u(x,y) + iv(x,y). If u and v both have continuous partial derivatives with respect to x and y at (x0,y0), then f is also differentiable at (x0,y0). C. If f is analytic at all points interior to and on a simple closed contour C, then the integral of f(z) dz over C is 0. If f is analytic at all points interior to and on a simple closed contour C, then the integral of f(z) dz over C is 0. D. If f is a continuous function defined on domain D, then the integral of f(z) dz from z1 to z2, where z1 and z2 lie in D, is the same along all contours connecting z1 and z2 and lying entirely in D. If f is a continuous function defined on domain D, then the integral of f(z) dz from z1 to z2, where z1 and z2 lie in D, is the same along all contours connecting z1 and z2 and lying entirely in D.

सही उत्तर: B Correct Answer: B

Continuity of the first partial derivatives of u and v alone does not ensure complex differentiability; the Cauchy-Riemann equations must also hold at the point. Option B is the answer marked in the source paper.

16

Mathematics • Vector Calculus

Find the directional derivative D_u f(x,y) at (x,y) = (1,2), where f(x,y) = x^3 - 3xy + 4y^2 and u is the unit vector in the direction theta = pi/6. Find the directional derivative D_u f(x,y) at (x,y) = (1,2), where f(x,y) = x^3 - 3xy + 4y^2 and u is the unit vector in the direction theta = pi/6.

A. (13 - 3sqrt(3))/2 (13 - 3sqrt(3))/2 B. 10 10 C. 3sqrt(3) 3sqrt(3) D. 0 0

सही उत्तर: A Correct Answer: A

At (1,2), grad f = (3x^2 - 3y, -3x + 8y) = (-3,13). Dotting it with (cos(pi/6), sin(pi/6)) gives (13 - 3sqrt(3))/2.

17

Mathematics • Differential Equations

The integrating factor for the differential equation y(xy + 2x^2y^2) dx + x(xy - x^2y^2) dy = 0 is: The integrating factor for the differential equation y(xy + 2x^2y^2) dx + x(xy - x^2y^2) dy = 0 is:

A. 1/(xy) 1/(xy) B. 1/(x^2y^2) 1/(x^2y^2) C. 1/(3x^3y^3) 1/(3x^3y^3) D. 1/(xy^3) 1/(xy^3)

सही उत्तर: C Correct Answer: C

Multiplying by 1/(3x^3y^3) makes the differential form exact; this is the integrating factor marked in the source paper.

18

Mathematics • Graph Theory

Circumference of a cyclic graph is defined as: Circumference of a cyclic graph is defined as:

A. Twice the sum of its total degree Twice the sum of its total degree B. Length of the longest cycle Length of the longest cycle C. Length of the shortest cycle Length of the shortest cycle D. Thrice the sum of its total degree Thrice the sum of its total degree

सही उत्तर: B Correct Answer: B

The circumference of a graph is the length, measured by number of edges, of its longest cycle.

19

Mathematics • Mathematical Modelling

If x(t) denotes the number of susceptibles and y(t) denotes the number of infectives at time t, in a homogeneous group of (n + 1) individuals, the simple epidemic model without removal is given by the equation (where beta > 0 is the contact rate): If x(t) denotes the number of susceptibles and y(t) denotes the number of infectives at time t, in a homogeneous group of (n + 1) individuals, the simple epidemic model without removal is given by the equation (where beta > 0 is the contact rate):

A. dx/dt = beta x(n - 1 + x) dx/dt = beta x(n - 1 + x) B. dx/dt = -beta x(n + 1 - x) dx/dt = -beta x(n + 1 - x) C. dx/dt = -beta x(n - 1 + x) dx/dt = -beta x(n - 1 + x) D. dx/dt = beta x(n + 1 - x) dx/dt = beta x(n + 1 - x)

सही उत्तर: B Correct Answer: B

With no removal, x + y = n + 1 and susceptibles decrease at rate dx/dt = -beta xy. Substituting y = n + 1 - x gives dx/dt = -beta x(n + 1 - x).

20

Mathematics • Partial Differential Equations

The steady-state solution mu(x,t) = mu_s(x) of the heat equation mu_t = a^2 mu_xx satisfying the conditions mu(0,t) = -1 and mu(2,t) = 1 is: The steady-state solution mu(x,t) = mu_s(x) of the heat equation mu_t = a^2 mu_xx satisfying the conditions mu(0,t) = -1 and mu(2,t) = 1 is:

A. mu_s(x) = 1 - x mu_s(x) = 1 - x B. mu_s(x) = -1 + x mu_s(x) = -1 + x C. mu_s(x) = -1 + 3x mu_s(x) = -1 + 3x D. mu_s(x) = 1 + 2x mu_s(x) = 1 + 2x

सही उत्तर: B Correct Answer: B

At steady state, mu_xx = 0, so mu_s(x) = Ax + B. The boundary values give B = -1 and A = 1; hence mu_s(x) = -1 + x.

MCQs for JSSC PGT Mathematics Exam

MCQs for JSSC PGT Mathematics Exam

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