Federal Public Service Commission (FPSC)
Competitive Examination for Recruitment to BPS-17 Posts under the Federal Government
Time Allowed: 3 Hours
APPLIED MATHEMATICS, PAPER-I
MAXIMUM MARKS: 100 Marks
Attempt FIVE questions in all by selecting at least TWO Questions From SECTION-A and THREE Questions from SECTION-B, All questions carry EQUAL marks.
(iii)Use of Scientific Calculator is Allowed.
SECTION-A
Q. 1. Solve the following differential equations:
(a) y”’ – 3y” + 2y’ = eˣ / (1 + e⁻ˣ) (10)
(b) y’ = (2xy e^{(x/y)²}) / (y² + y² e^{(x/y)³} + 2x² e^{(x/y)²}) (10)
Q. 2. (a) Find the series solution of the following differential equation:
y” – xy = 0 (10)
(b) Use the method of Fourier integrals to find the solution of initial value problem with the partial differential equation.
∂u/∂t = c² (∂²u/∂x²) ; (-∞ < x < ∞)
And with initial condition u(x,0) = f(x) (10)
Q. 3. (a) Solve x²y” – 3xy’ + 5y = x² sin(ln x) (10)
(b) Find the solution of wave equation
∂²u/∂t² = c² (∂²u/∂x²) with boundary and initial conditions
u(0,t) = u(l,t) = 0, u(x,0) = f(x), ∂u(x,t)/∂t = g(x) (10)
SECTION-B
Q. 4. Discuss the following terms: (5×4=20)
(i) Tensors
(ii) Kronecker delta
(iii) Contraction
(iv) Metric Tensor
(v) Contravariant tensor of order two
Q. 5.
(a) Prove that {i \ j j} = ∂/∂xⁱ (log √g) (10)
(b) Prove that Δ = | δ_m1 δ_m2 δ_m3 |
| δ_n1 δ_n2 δ_n3 | = ε_mnp and ε_ijk ε_mnp = | δ_im δ_in δ_ip |
| δ_p1 δ_p2 δ_p3 | | δ_jm δ_jn δ_jp |
| δ_km δ_kn δ_kp |
Hence prove that ε_ijk ε_mnp = δ_im δ_jn – δ_in δ_jm (10)
SECTION-C
Q. 6. (a) (i) What is the difference between secant and false position method? Show also graphically. (5+5=10)
(ii) Prove that x_{n+1} = x_n – f(x_n)/f'(x_n)
(b) Solve the following system by Jacobi method. (Up to four decimal places).
8x + y – z = 8
2x + y + 9z = 12
x – 8y + 12z = 35 (10)
Q. 7. (a) Evaluate by 3/8 Simpson’s rule
∫₀¹ x√(1 + x³) dx ; with n = 6
Also calculate the absolute error. (10)
(b) The amount A of a substance remaining in a reacting system after an interval of time t in a certain chemical experiment is given by following data:
A: 94.8 87.9 81.3 68.7
t: 2 5 8 14
Find t when A=80. (10)
Q. 8. (a) If f(x) = x³, show that f(a,b,c) = a + b + c (10)
(b) Solve by trapezoidal rule
∫₀^{2π} x sin x dx ; with n = 8 (10)
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