CSS past paper

CSS Applied Mathematics Past Paper 2020

CSS Applied Mathematics 2020 carries 100 marks over 3 hours. It has 8 questions.

Marks: 100
Time: 3 hours
Questions: 8

Years

How to attempt: Attempt ONLY FIVE questions. ALL questions carry EQUAL marks

Questions

Q. 1
- (a)
  Prove that $\nabla^{2}r^{n} = n(n + 1)\,r^{n-2}$
  (10)
- (b)
  Evaluate $\iint_{s} \underline{A}.\bar{n}\ ds$ where $\overline{A} = 18z\underline{i} - 12\underline{j} + 3y\underline{k}$ and S is that part of the plane $2x + 3y + 6z = 12$ which is located in the 1st octant.
  (10)
Q. 2
A particle P of mass m slides down a frictionless inclined plane AB of an angle $\alpha$ with the horizontal. If it starts from rest at the top A, find (a) the acceleration (b) the velocity and (c) the distance travelled after time t.
(20)
Q. 3
- (a)
  Discuss the motion of a particle moving in a straight line if it starts from rest at a distance ‘a’ from a point O and moves with an acceleration equal to $k$ times its distance from O.
  (10)
- (b)
  Find radial and transversal components of velocity and acceleration.
  (10)
Q. 4
- (a)
  Solve $\frac{d^{2}y}{dx^{2}} + y = Co\sec x$
  (10)
- (b)
  Solve $dy + \frac{y - Sinx}{x}dx = 0$
  (10)
Q. 5
- (a)
  Solve the initial value problem
  (10)
  $x(2 + x)\frac{dy}{dx} + 2(1 + x)y = 1 + 3x^{2}, \qquad y(-1) = 1$
- (b)
  Find the general solution of the equation
  (10)
  $(D^{3} - 2D + 1)y = 2x^{3} - 3x^{2} + 4x + 5$
Q. 6
- (a)
  Find the Fourier series of f:
  (10)
  $f(x) = \begin{cases} x, 0 < x < 1 \\ 0, 1 < x < 2 \end{cases}$
- (b)
  Solve the boundary value problem $\frac{\partial^{2}u}{\partial x^{2}} = \frac{1}{k}\frac{\partial u}{\partial t}$
  (10)
  Satisfying u(o,t)=u(l,t)=0 and $u(x,o) = lx - x^{2}$
Q. 7
- (a)
  By using regular Falsi method, solve
  (10)
  $Logx - Cosx = 0$
- (b)
  Find the value of $f(7.5)$ by using Newton Gregory Backward Difference Interpolation formula.
  (10)
  X: 5, 6.1, 6.9, 8, 8.6
  $f(x): 3.49, 4.82, 5.96, 7.5, 8.2$
Q. 8
- (a)
  Applying the Taylor series method, compute
  (10)
  $\int_{o}^{x} \frac{Sin\,t}{t}dt \ \ for\ \ x = 0\ (0.1)1$
- (b)
  Use fourth order RK method to solve
  (10)
  $\frac{dy}{dx} = t + y \ ;\ \ y(0) = 1 \ \ from\ t = 0 \ \ to\ t = 0.4\ taking\ \ h = 0.4$

Attempt ONLY FIVE questions. ALL questions carry EQUAL marks
All the parts (if any) of each Question must be attempted at one place instead of at different places.
Candidate must write Q. No. in the Answer Book in accordance with Q. No. in the Q.Paper.
No Page/Space be left blank between the answers. All the blank pages of Answer Book must be crossed.
Extra attempt of any question or any part of the attempted question will not be considered.
Use of Calculator is allowed.