CSS 2025 Applied Mathematics Past Paper

Paper details

Examination

CSS

Year

2025

Subject

Applied Mathematics

Paper

Single Paper

Subject type

Optional

Mode

Subjective

Questions recorded from this paper

These 8 question wordings are transcribed from CSS Vista’s structured Applied Mathematics past-paper record for CSS 2025.

1. Q1 · Statics (10%)(a) (i) Prove that r n = nr n−2 r , where r = x i + yj + zk . (10) (ii) a  b  c = a  b  c , then prove that a and c are parallel. (b) Find the area of the region that is enclosed between the curves y = x 2 and y = x + 6 . (10)
2. Q4 · Statics (10%)(a) Forces 2 BC , CA, BA act along the sides of a triangle ABC . Show that their (10) resultant is 6DE . Where D bisects BC and E is a point on CE such that CE = CA . (b) Find the center of mass of the surface generated by the revolution of the arc of the (10) parabola, lying between the vertex and the latus rectum, about the x -axis.
3. Q2 · Tangential(a) Find the tangential and normal components of acceleration of a point describing (10) the ellipse x2 y 2 + = 1 with uniform speed V , when the particle is at ( 0, b ) . a 2 b2 (b) Find the solution of initial value problem by separation of variables (10) 1 − y 2 dx − 1 − x 2 dy = 0, y (0) =
4. Q3 · variation of parameters(a) Find the general solution of the given differential equation by variation of (10) parameters. 3 y − 6 y + 6 y = e x sec x (b) Find the power series solution of ( x 2 + 1) y + xy − y = 0 (10)
5. Q5 · boundary value problems(a) Obtain the Fourier series over the indicated interval for the given function. (10) f ( x) = 3 + 2 x, −   x  0, =  + 2 x, 0 x (b) Solve the boundary value problem (10) uxx + u yy = 0, 0  x  a, 0  y  b, u (0, y ) = 0, u (a, y ) = 0, 0  y  b u ( x, 0) = 0, u ( x, b) = f ( x), 0  x  a.
6. Q7 · boundary value problems(a) Use Euler’s method to approximate the solution of the initial value problem. (10) y = 1 + y / x, 1  x  2, y (1) = 2, with h = 0.25 (b) Using Green’s theorem, evaluate  F (r ). d r counter clock wise around the (10) 1 1 4  boundary curve C of the region R , where F =  xy 4 , x y , R the rectangle 2 2  with vertices ( 0, 0) , ( 3, 0) , ( 3, 2) , ( 0, 2) .
7. Q6 · Solution of a system of linear equations(a) Use Newton’s Raphson method to find the solution accurate to within 10-4 (10) (corrected upto four decimal places) for the given problem. x − cos x = 0, 0,  / 2 . (b) Solve the system of linear equations using Gauss Seidel method (with three digit (10) rounding arithmetic ) 3x1 + 4 x2 − x3 = 8 5 x1 + 3 x2 + 2 x3 = 17 − x1 + x2 − 3x3 = −8
8. Q8 · trapezoidal rule(a) Evaluate the Integral  dx , Using Trapezoidal Rule for five points (corrected (10) x2 upto two decimal places). (b) Find the D’Alembert solution of the wave equation u xx = 2 utt subject to the (10) c Cauchy Initial conditions u( x, 0) = f ( x), ut ( x, 0) = g ( x) .

Topic-wise Applied Mathematics past-paper analysis.