Applied Mathematics

FEDERAL PUBLIC SERVICE COMMISSION COMPETITIVE EXAMINATION FOR RECRUITMENT TO POSTS IN B.P.S.-17 UNDER THE FEDERAL GOVERNMENT, 2000. APPLIED MATHEMATICS. PAPER-II TIME ALLOWED: 3 HOURS MAXIMUM MARKS: 100 Note: Attempt FIVE questions in all, including QUESTION NUMBER 8 which is COMPULSORY, select TWO questions from each SECTION. All questions carry EQUAL marks. SECTION-I (a) Use the method of variation of parameters to find a general solution for the differential equation y? + y = tan2 x (b) Find the series solution of the differential equation y?+xy'+2y=0 with Centre of expansion at x0 =1. 2. (a) Find the general solution of y?' –2y? –y' + 2y = e3x (b) Define partial differential equation with examples. Solve 2p+3q=1. 3. (a) Use Monge’s method to solve q(1 +q)r–(l +2q)(l +p)s+(l +p)2 t=0. (b) Find the solution of the Laplace equation in three dimensions. SECTION-II 4. (a) Find the scalars that can be constructed from Cartesian tensors of rank land 2. (b) (i) If Ai li2…ir is a Cartesian tensor of rank r, then ? —Ai li2…ir is a tensor of rank r+ 1. ?x, (ii) Show that € ijk € ijk=6 5. (a) Show that Newton - Raphson is a quadratically convergent method. (b) Solve f(x) = e-x — Sin [nx] =0, using Regula Falsi Method. 2

6. (a) Solve the system of equations l0x1 + X2 + 2x3 = 44 x1 + 2x2 + 10x3 = 61 2x1 +l0x2 +X3 = 51 by Gauss- Seidel iterative method with an absolute convergence criterion of € = = 0.0001 (b) Use the Trapezoidal and simposon's rules to estimate the integral l² f(x)dx = l³ (x³ - 2x2 + 7x — 5)dx. Also find the extreme value of the error in each rule. 7. (a) Find the maximum value of Z = 7x1 +5x2 subject to the constraints X1 + 2x2 = 6, 4x1+3x2 =12, xI,x2 = 0 (b) Write notes on the followings: (i) Lagrange Interpolation (ii) Convergence of iterative method for solving non-linear equations. COMPULSORY QUESTION 8. Write only correct answers in the answer book. Do not reproduce the questions. (1) A homogeneous differential equation always: (a) Possesses non-trivial solution (b) Possesses trivial solution. (c) None of these. d2y (2) The differential equation —------ + Y2 = x3 dx2 (a) is non- – homogenous and Linear. (b) Homogeneous and non-linear. (c) Non — homogeneous and non-linear. (d) None of these. d2 y (3) The differential equation —------ = 0 has the primitive:

dx3 (a) y = kx³ + Bx² + C (b) y = kx7 + B (c) y = kx²+ BX + C (4) The differential equation x2 y? + xy’ + (x2 – k2)y =0 is: (a) The gauss equation. (b) The Legendre equation. (c) The Bessel equation. (d) None of these. ?2z ?2z ?2z (5) A partial differential equation A —— + B —— +C —— = x + 2y, A, B and ?x2 ?x?y ?y2 C being real constants, is: (a) Homogeneous (b) Non — homogeneous. (c) None of these. (6) Given that Aij is a symmetric tensor and Bij an anti —symmetric tensor, then (a) Aij Bij = 0 (b) Aij Bij = 0 (c) None of these. (7) If Ai1i2,…ir and Bi1i2,…ir are tensors of rank r, then Ai1i2,…ir ± Bi1i2,…ir are (a) Tensors of rank r + 1 (b) Tensors of rank r - 1 (c) Tensors of rank r (d) None of these.

(8) Relative error is equal to: (a) The approximate value + error error (b) ———— Truevalue (c) Truncation error. (d) None of these. (9) Gauss – Seidel method for solving the system of Linear equation is an (a) Iterative method. (b) Direct method. (c) None of these. 3 10. The error term – — h5 f(4) (?) is of the 80 (a) Trapezoidal Rule 1 (b) Simpson’s — Rule. 3 3 (c) Simpson’s —Rule. 8 (e) None of these d2u du 11. The equation [ ——]5 + [x6 ——]2 +x9u3 =f(x)is dx2 dx (a) an ordinary differential equation. (b) Is not an ordinary differential equation.

(c) None of these. (12) The equations y? + 4y’ + y = 0, y(0) = 1, y' (1) = 0 define (a) Initial value problem. (b) Boundary value problem. (c) None of these. dy x + 3y (13) The equation — = ——— is dx 3x+y (a) With homogeneous coefficients. (b) With Non-homogeneous coefficients. (c) None of these. (14) The method of undetermined coefficients is used for finding: (a) a general solution of non-homogeneous linear differential equation. (b) A particular solution of non-homogeneous linear differential equation. (c) None of these. (15) For the differential equation (x2 — 1) y? +2xy' + 6y = 0 (a) All the values of x are ordinary points. (b) All the values of x other than x = ± 1 are ordinary points. (c) None of these. (16) Let det (R) of the matrix R is equal to 1, then the transformation is called: (a) Proper transformation. (b) Improper transformation. (c) None of these. (17) Which one is correct: (a) djj = 3 (b) djj = 1 (c) djj = 0

(d) None of these. (18) Milne’s method is (a) a single step method. (b) A multi step method (c) None of these. (19) Lagrange interpolating polynomial is for (a) Equi-spaced data. (b) Unequally spaced data. (c) None of these. (20) Which method is called regula-falsi method: (a) Method of interpolation (b) Secant Method ‘ (c) Bisection Method. (d) Method of iteration. (e) None of these. ******************************