Table of Contents
KHYBER PAKHTUNKHWA, PUBLIC SERVICE COMMISSION
COMPETITIVE EXAMINATION FOR THE POSTS OF PROVINCIAL MANAGEMENT SERVICES (BPS-17) 2013
APPLIED MATHEMATICS, PAPER-II
TIME ALLOWED: 03 HOURS
MAX.MARKS: 100
Note:- Attempt any Five Questions, selecting two questions from Section-A, ONE question from Section-B and two questions from Section-C. All questions carry equal marks.
SECTION-A
| Q1. | (a) | Solve the differential equation \(\left(x^2y^3 – \frac{1}{1+9x^2}\right)dx + x^3y^2dy = 0\) |
| (b) | The slope of the tangent line to a curve at the point \((x,y)\) on the curve \(6x^3\sqrt{x^4+9}\). If the point \((2,250)\) lies on the curve, find an equation of the curve. |
| Q2. | (a) |
Use the method of variation of parameters to solve \[ \frac{d^2y}{dx^2} + y = \tan x \sec x \] |
| (b) | Solve \(\frac{d^2y}{dx^2} – 3\frac{dy}{dx} + 2y = e^{3x}\) |
| Q3. | (a) |
Find the solution of partial differential equation \[ y\frac{\partial u}{\partial x} – x\frac{\partial u}{\partial y} + yu = xy \] |
| (b) | Find the solution of the Cauchy Problem consisting of the partial differential equation \(\frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} = 0\) together with the initial condition \(u(0,y) = \sin y\). |
SECTION-B
| Q4. | (a) | Write Newton’s law in tensor form. |
| (b) |
Find \(g_{jk}\) and \(g^{jk}\) corresponding to \[ ds^2 = 3(dx^1)^2 + 2(dx^2)^2 + 4(dx^3)^2 – 6dx^1dx^3 \] |
| Q5. | (a) |
Prove that for an orthogonal coordinate system \[ g_{11} = \frac{1}{g^{11}} , \quad g_{22} = \frac{1}{g^{22}} , \quad g_{33} = \frac{1}{g^{33}} \] |
– 2 –
| (b) |
Prove that (i) \(\nabla \times (\nabla \times \vec{A}) = \nabla(\nabla \cdot \vec{A}) – \nabla^2\vec{A}\) (ii) \(\nabla^2 r^n = n(n+1)r^{n-2}\), where \(\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}\). *Note: The original text handwritten notation says “\(\nabla r^n = n r^{n-2} r\)” which mathematically corresponds to the Laplacian identity \(\nabla^2 r^n\). |
SECTION-C
| Q6. | (a) | Find a positive real root of \(e^x – 3x = 0\). |
| (b) | Find the root of \(x^3 – 3x – 3 = 0\) to four decimal places using Newton Raphson method, that lies near \(x = 2\). |
| Q7. | (a) | Using the method of Regula Falsi to find the positive real roots of \(x^3 + x^2 – 3x – 3 = 0\). |
| (b) | Evaluate \(\int_{0}^{\pi/2} \frac{\cos x}{1+x} dx\), using Simpson’s rule with four intervals, correct to 3 decimal places. |
| Q8. | (a) | Solve the following system of equations by using Gauss-Seidel method. |
|
\[ \begin{aligned} 4x_1 – 2x_2 + x_3 &= 12 \\ 2x_1 + 3x_2 – x_3 &= 7 \\ 2x_1 – 2x_2 + 2x_3 &= 8 \end{aligned} \] |
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| Q8. | (b) |
Solve the following system of equations by Jacobi’s method \[ \begin{aligned} x_1 + 6x_2 + 2x_3 &= 15 \\ x_1 + x_2 – 6x_3 &= -3 \\ 6x_1 + x_2 + x_3 &= 9 \end{aligned} \] |
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