CSS 2021 Applied Mathematics Past Paper

Paper details

Examination

CSS

Year

2021

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 2021.

1. Q1 · Vector Calculus (10%)⃑⃑dS where A⃑ = zi⃑ + xj⃑ − 3y 2 zk⃑⃑ and S is the (a) Evaluate the surface integral ∫∫ A⃑. n (10) portion of the cylinder x 2 + y 2 = 8 lying in the first octant between z = 0 and z = 4. (b) Prove that (10) f ′ ( r) ∇(f(r)) = r⃑, r where r⃑ = xi⃑ + yj⃑ + zk⃑⃑ and r = |r⃑|.
2. Q2 · Statics (10%)(a) The greatest resultant that two forces can have is of magnitude P and the least is (10) of magnitude Q.Show that, when they act at an angle α, their resultant is of α α
3. Q3 · simple harmonic motion(a) Show that the law of force towards the pole, of a particle describing the curve (10) rn = an cos nθ is given by (n + 1)h2 a2n f= . r2n+3 The maximum velocity that a particle executing simple harmonic motion of (10) (b) amplitude a attains, is v. If it is disturbed in such a way that its maximum velocity becomes nv. Find the change in the amplitude and the time-period of motion.
4. Q7 · separable equations, exact equations(a) Use Simpson’s 3/8 rule to estimate the integral (10) ∫(x 3 − 2x 2 + 7x − 5)dx . By comparing your answer with exact value, find the error. (b) Solve the system of equations by Jacobi iterative method. (10) 10x + 3y + z = 19, 3x + 10y + 2z = 29, x + 2y + 10z = 35
5. Q5 · Equations with constant coefficients(a) Solve the equation by using method of undetermined coefficients (10) y ′′ − y ′ + y = 2 cos 3x. (b) Use the method of Frobenius to find two linear independent series solutions in (10) powers of x of the DE. x 2 y ′′ − (x 2 + x )y ′ + y = 0.
6. Q4 · Ordinary and singular points of a differential equation(a) Define ordinary and singular points of the differential equation (10) a2 (x )y ′′ + a1 (x)y ′ + a0 (x)y = 0. When a singular point is said to be regular and irregular? Find regular and irregular singular points of the differential equation (x 2 − 4)2 y ′′ + (x − 2)y ′ + y = 0. (b) Show that (10) 2 sin x J3 ⁄2 = √ [ − cos x]. πx x
7. Q6 · solution by the method of separation of variables(a) Classify general second order partial differential equation (PDE) into elliptic, (10) parabolic and hyperbolic form. Discuss the nature of the PDE (1 − x 2 )uxx − 2xyu xy + (1 − y 2 )uyy = 0 at each (x, y) ∈ R 2 . (b) Use the method of separation of variables to find the solution u(x, t): [0, T] × (10) [0, L] → R to the initial/boundary value problem u t (x, t) = u xx (x, t) for 0 < t ≤ T and 0 ≤ x ≤L, u (x, 0) = f(x ), for 0 ≤ x ≤ L, u (0, t) = u (L, t) = 0, for 0 < t ≤ T, where f: [0, L] → R is a known function.
8. Q8 · Simpson’s rules(a) In the following table values of y = x + sin x 2 are tabulated (10) x 1.0 1.1 1.2 1.3 1.4 1.5 1.6 f (x) 1.84147 2.03562 2.19146 2.29290 2.32521 2.27807 2.14935 Construct a difference table and estimate f(1.04) and f(1.57). (b) π ⁄2 (10) Use trapezoidal and Simpson’s 1/3 rules to approximate ∫0 sin2 (x )dx. Find a maximum bound for the error in each case. Compare your approximations with the actual result.

Topic-wise Applied Mathematics past-paper analysis.