CSS past paper

CSS Applied Mathematics Past Paper 2026

CSS Applied Mathematics 2026 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 in all. ALL questions carry EQUAL marks.

Questions

Q. 1
- (a)
  Prove $\nabla f(r) = \frac{f'(r)}{r}\vec{r}$ , and evaluate $U$ for $\nabla U = 2r^{4}\vec{r}$ .
  (8)
- (b)
  If $\vec{A}$ and $\vec{B}$ are differentiable vector function of position $(x, y, z)$ , find $\nabla . (\vec{A} \times \vec{B})$ .
  (7)
- (c)
  If $\vec{A} = x^{2}yzi - 2xz^{3}j + xz^{2}k$ and $\vec{B} = 2zi + yj - x^{2}k$ , find $\frac{\partial^{2}}{\partial x\,\partial y}(\vec{A} \times \vec{B})$ .
  (5)
Q. 2
- (a)
  What is the maximum range possible for a projectile fired from a cannon having muzzle velocity 1 mile/sec? What is the height reached in this case?
  (10)
- (b)
  Find the radial and transverse components of the velocity of the particle moving along the curve $ax^{2} + by^{2} = 1$ at any time $t$ , if the polar angle $\theta = ct^{2}$ .
  (10)
Q. 3
- (a)
  Verify that the linear differential equation
  (5)
  $(1 + x^{2})\frac{dy}{dx} + 2xy = e^{x^{2}}$
  can be solved by setting $y = ve^{x^{2}}$ . Find the equation satisfied by $v$ .
- (b)
  Explain how a first-order linear differential equation can be used to model pollution accumulation in a lake with constant inflow and outflow rates.
  (5)
- (c)
  Find the solution of the following differential equation
  (10)
  $y''' + 8y = 2x - 5 + 8e^{-2x};\ \ \text{with the conditions}$
  $y(0) = -5, \qquad y'(0) = 3, \qquad y''(0) = -4$
Q. 4
- (a)
  Find the general solution of the following differential equation on interval $(0, \infty)$
  (10)
  $16x^{2}y'' + 16xy' + (16x^{2} - 1)y = 0$
- (b)
  Solve the following PDEs using transformation
  (10)
  $\begin{array}{c} u_{t} + u_{x} + u = 0;\ \ for\ x > 0, t > 0, \\ u(0, t) = sin(t); u(x, 0) = 0. \end{array}$
Q. 5
- (a)
  Model the one-dimensional wave equation.
  (8)
- (b)
  A stretched string of length $L$ is lying along $x$ -axis and is fixed at both ends $x = 0$ and $x = L$ . Find the deflection $u(x, t)$ of the string at any time t, if initial displacement $u(x, 0) = f(x) = Lx - x^{2}$ and the initial velocity is
  (12)
  $u_{t}(x, 0) = g(x) = 4.$
Q. 6
- (a)
  Find the Fourier series of the function
  (10)
  $f(x) = \left(1 - \frac{|x|}{a}\right)H\left(1 - \frac{|x|}{a}\right),$
  Where $H(x)$ is the Heaviside unit step function defined by
  $H(x) = \left\{\begin{matrix} 1, & for\ x > 0 \\ 0, & for\ x < 0 \end{matrix}\right\}$
- (b)
  Evaluate the integral $I = \int_{0}^{3} \frac{1}{1+x^{4}}dx$ by using
  (10)
  - (i)
    Trapezoidal rule
  - (ii)
    Simpson’s 1/3 rule by taking h = ¼.
Q. 7
- (a)
  Use the Modified Euler method to solve the IVP:
  (10)
  $y' = y - x^{2} + 1; \quad y(0) = 0.5$
  for $0 \leq x \leq 2$ with step size $h = 0.5$ .
  Compare with the exact solution.
- (b)
  Implement appropriate technique to solve the following linear system.
  The first equation is printed '+ x − 3'; x₃ is presumably meant.
  (10)
  $\begin{array}{c} 3x_{1} - x_{2} + x - 3 = -1 \\ -x_{1} + 3x_{2} - x_{3} = 7 \\ x_{1} + x_{2} - 3x_{3} = -7 \end{array}$
Q. 8
- (a)
  Set up Newton’s scheme of iteration for finding the square root of a positive number N and evaluate $\sqrt{14}$ .
  (10)
- (b)
  Find the first four iterations of the equation $f(x) = x - 0.8 - 0.2sinx$ in the interval $\left[0, \frac{\pi}{2}\right]$ using Newton-Raphson Method.
  (10)

Attempt only FIVE questions in all. ALL questions carry EQUAL marks.
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 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.