CSS Applied Mathematics Past Paper 2025

Instructions specific to this paper

• First attempted Part-I (MCQS) on the separate OMR Answer Book which shall be taken back
• Overwriting/cutting of the options/answers will not be given credit.
• There is no negative marking. All MCQs must be attempted.
• = 3π + 2x, -π < x < 0; = 2x, 0 < x < π (10)
• , 0 ≤ x ≤ a (10)
• , uₜ(x, 0) = g(x).

Questions

1. Q. 210 marks

   (a) Find the tangential and normal components of acceleration of a point describing the ellipse x²/a² + y²/b² = 1 with uniform speed V⃗ , when the particle is at (0, b). (b) Find the solution of initial value problem by separation of variables: √(1−y²) dx − √(1−x²) dy = 0 (10)

2. Q. 310 marks

   (a) Find the general solution of the given differential equation by variation of parameters: 3y'' − 6y' + 6y = eˣ (10) (b) Find the power series solution of (x+1)y'' + xy' − y = 0. (10)

3. Q. 410 marks

   (a) Forces 2BC, CA, AB act along the sides of a triangle ABC. Show that their resultant is 0. Where D bisects BC and E is a point on CE such that CE = 1/3 CA. (b) Find the center of mass of the surface generated by the revolution of the arc of the parabola, lying between the vertex and the latus rectum, about the x-axis. (10)

4. Q. 510 marks

   (a) Obtain the Fourier series over the indicated interval for the given function: f(x) = 3π + 2x, -π < x < 0; = 2x, 0 < x < π (b) Solve the boundary value problem: uₓₓ + uᵧᵧ = 0, 0 < x < a, 0 < y < b u(x,0) = 0, u(a,y) = 0, 0 ≤ y ≤ b u(0,y) = 0, u(x,b) = f(x), 0 ≤ x ≤ a (10)

5. Q. 610 marks

   (a) Use Newton’s Raphson method to find the solution accurate to within 10⁻⁴ (corrected up to four decimal places) for the given problem: x − cos x = 0, [0, π/2] (b) Solve the system of linear equations using Gauss Seidel method (with three digit rounding arithmetic): 3x₁ + 4x₂ − x₃ = 8 5x₁ + 2x₂ + 2x₃ = 3 −x₁ + x₂ − 3x₃ = −8 (10)

6. Q. 710 marks

   (a) Use Euler’s method to approximate the solution of the initial value problem. y' = 1 + y / x, 1 ≤ x ≤ 2, y(1) = 2, with h = 0.25 (b) Using Green’s theorem, evaluate ∮ F(r) · dr counter clock wise around the (10) boundary curve C of the region R, where F = [1/2·xy⁴, 1/2·x²·y], the rectangle with vertices (0, 0), (3, 0), (3, 2), (0, 2).

7. Q. 810 marks

   (a) Evaluate the Integral ∫₁³ (1 / x²) dx. Using Trapezoidal Rule for five points (corrected upto two decimal places). (b) Find the D’Alembert solution of the wave equation uₓₓ = (1 / c²)·uₜₜ subject to the (10) Cauchy Initial conditions u(x, 0) = f(x), uₜ(x, 0) = g(x).

Standard FPSC instructions

Printed on this and every CSS paper.

• Part-II is to be attempted on the separate Answer Book.
• All the parts (if any) of each question must be attempted at one place instead of at different places.
• Write Q. No. in the Answer Book in accordance with Q. No. in the question paper.
• No page/space be left blank between the answers. All the blank pages of the Answer Book must be crossed.
• Extra attempt of any question or any part of the attempted question will not be considered.