CSS 2019 Applied Mathematics Past Paper

Paper details

Examination

CSS

Year

2019

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 2019.

1. Q1 · Vector Calculus (10%)the direction of the vector i + 2 j + 2k ? ∫ ( xy + y )dx + x dy where c is bounded by the line y = x and the (b) Evaluate (10) c curve y = x2
2. Q2 · equilibrium of forces in three dimensions(a) Find the constants a, b, and c so that (10) F=(x+2y+az) i + (bx – 3y – z ) j + (4x + cy + 2 z) k is irrotational and hence find the function ψ such that F = ∇ ψ (b) The forces F 1 ,F 2 ,F 3 ,F 4 ,F 5 and F 6 act along the sides of a regular hexagone taken (10) in order. Verify that all the forces will be in equilibrium if, ∑ F= 0, and F 1 – F 4 = F 3 –F 6 =F 5 –F 2 .
3. Q3 · simple harmonic motion(a) A system of forces acts on a plate in the form of an equilateral triangle of side 2a. (10) The moment of the forces about the three vertices are M 1 , M 2 and M 3 respectively. Find the magnitudes of the resultant. (b) If a particle P move with a velocity V given by V2 = n2 (ax2 + 2bx + c). Show that (10) P executes a simple harmonic motion. Find the centre, the amplitude and the time period of the motion?
4. Q7 · Fourier series and partial differential equations (20%)(a) Define even function and odd function with examples. Verify that the Fourier (10) f (x) =  0 When 0 < x < π Series for the function 1 When π < x < 2 π3 is f ( x) = − (sin x + sin 3 x + sin 5 x...............) 2 π 3 5 (b) Solve the following partial differential equation by using method of separable (10) variable. ∂u ∂u = 2 +u , given u ( x, o) = 6e − 3x ∂x ∂t
5. Q6 · Solution of a system of linear equations(a) Determine the second degree polynomials by using Newton’s method. Also (10) estimate the value of f (0.1) and f (0.5) for the data. x 0.0 0.2 0.4 0.6 f (x) 15.0 21.0 30.0 51.0 (b) Does the dominate diagonal is necessary for finding the numerical solution of (10) system of linear equations by using Gauss Jacobi’s and Gauss Seidal methods. Explain the reason. In what conditions a numerical method is used instead of analytical method? Find the solution of the following system by performing three itrations of Gauss Seidal method. 6x – 3y + z = 11 2x + y – 8z = 15 x – 7y + z = 10
6. Q8 · trapezoidal rule(a) The Trapezoidal rule applied to ∫ f ( x)dx gives the value 4, and the Simpson’s rule gives value 2, what is the value of f (1) ? (b) Find the first two derivatives at x=1.1 and x =1 from the following data table. (10) x 1 1.2 1.4 1.6 1.8 2.0 f (x) 0.000 0.1280 0.5440 1.2960 2.4320 4.000
7. Q5 · Gaussian integration formulas(a) Solve the equation (10) 1 1 2 1 π 0= + x − x sin x − cos 2 x with x0 = (b) Derive two point Gaussian integration formula for the following integral and use it (10) to solve the integral. 1.6 2x ∫ 1 x2 − 4 dx
8. Q4 · Numerical solution of an ordinary differential equation(a) What is the difference between linear differential equation and Bernoulli’s (10) equation? Also find the solution of the following differential equation.  dy  x + y = 1− y  dx  (b) Use the method of undetermined coefficient to solve the following differential (10) equation. y” – 3y’ + 2y = 2x3 – 9 x2 + 6 x

Topic-wise Applied Mathematics past-paper analysis.