BT-401 Mathematics-III - Jun 2026 Question Paper

Under each question, How to answer lists the points an answer should cover, outlined by CampusPrep from the syllabus. Check the details against your textbook.

1. 7 Marks
   aProve that :

   1. i)$\frac{\Delta^2}{E}x^3 = 6xh^2$
   2. ii)$\mu^2 = 1 + \frac{\delta^2}{4}$
   How to answer

   • Define forward difference Delta, shift E with Ex = x+h
   • Expand Delta^2/E x^3 using E^-1 and binomial to get 6xh^2
   • Define averaging mu and central difference delta
   • Derive mu^2 = 1 + delta^2/4 from mu = (E^{1/2}+E^{-1/2})/2
2. 7 Marks
   bApply Newton-Raphson method to evaluate approximately $\sqrt{12}$.
   How to answer

   • Set f(x)=x^2-12=0 for sqrt(12)
   • Write Newton-Raphson iteration x_{n+1}=x_n-f/f'
   • Choose initial approximation near 3.5
   • Iterate to convergence and give approximate value
3. 7 Marks
   a

   Fit a cubic polynomial which takes the following value :

   x	0	1	2	3
   y	1	0	1	10

   Hence find $y(4)$.

   How to answer

   • Form Newton forward/backward or Lagrange cubic using given data
   • Determine coefficients of cubic polynomial
   • Verify polynomial at x=0,1,2,3
   • Evaluate at x=4 to get y(4)
4. 7 Marks
   bCalculate $\log_e 7$ by Simpson's 1/3 rule correct to four decimal places.
   How to answer

   • Write log_e 7 as definite integral of 1/x
   • Choose interval and even number of subintervals for Simpson 1/3
   • Apply Simpson 1/3 formula with ordinates
   • Compute to four decimal places
5. 7 Marks
   aSolve system of Linear equations by Jacobi's method : $$4x+y+z=7,\quad x+5y+z=-8,\quad x+y+6z=6$$
   How to answer

   • Check diagonal dominance and rewrite equations for iteration
   • Write Jacobi iteration formulas for x, y, z
   • Assume initial values and iterate
   • Present converged solution
6. 7 Marks
   bExpand $\frac{e^z}{z(z+1)}$ as Taylor's series about $z = 2$.
   How to answer

   • Write f(z)=e^z/[z(z+1)] analytic near z=2
   • Compute derivatives or use known expansions about z=2
   • Form Taylor series f(2)+f'(2)(z-2)+...
   • Give region of validity
7. 7 Marks
   aTabulated by Milne's method the numerical solution of $\frac{dy}{dx} = x + y$ with $x_0 = 0, y_0 = 1$ from $x = 0.2$ to $x = 0.3$.
   How to answer

   • Identify ODE dy/dx=x+y with x0=0, y0=1
   • Obtain starting values by Picard/Taylor/Runge-Kutta for Milne
   • Apply Milne predictor and corrector formulas
   • Tabulate values from x=0.2 to x=0.3
8. 7 Marks
   bSolve : $\nabla^2 u = -81xy, 0 < x, y < 1, h = \frac{1}{3}$. Given, $$u(0,y) = u(x,0) = 0, u(1,y) = u(x,1) = 100.$$
   How to answer

   • Discretize Poisson equation with h=1/3 giving 4 interior nodes
   • Use 5-point formula with source -81xy
   • Apply boundary values 0 on x=0,y=0 and 100 on x=1,y=1
   • Solve resulting linear system for interior u
9. 7 Marks
   aObtain y for x=0.1 from the differential equation $\frac{dy}{dx} = x^2 + y$, given $y = -1$ when $x = 0$ by using Runge-Kutta formula of fourth order.
   How to answer

   • Identify f(x,y)=x^2+y with x0=0,y0=-1,h=0.1
   • Compute k1,k2,k3,k4 of RK fourth order
   • Apply y1=y0+(k1+2k2+2k3+k4)/6
   • Obtain y at x=0.1
10. 7 Marks
    bSolve the equation $\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}$ in $0 < x < 5, t \ge 5$ by Crank-Nicholson method. Given that $u(x,0) = 20, u(0,t) = 0, u(5,t) = 100$ and $h = 1$.
    How to answer

    • Write heat equation ut=uxx with given initial/boundary data
    • Form Crank-Nicolson implicit scheme with h=1
    • Assemble tridiagonal system for next time level
    • Solve for u at interior points
11. 7 Marks
    aProve that : $\int_{t=0}^{\infty} \int_{u=0}^{t} e^{-t} \frac{\sin u}{u} du dt = \frac{\pi}{4}$ and also find the Laplace Transform of $\cos h$ $at$ $\sin bt$.
    How to answer

    • Use Laplace property L{sin u /u} or interchange of integration
    • Evaluate double integral to pi/4
    • Recall L{sin bt} and shifting for cosh(at)
    • Write L{cosh(at) sin(bt)} formula
12. 7 Marks
    bUsing Convolution theorem, prove that $$L^{-1}\left[\frac{1}{p^3(p^2+1)}\right] = \frac{t^2}{2} + \cos t -1$$
    How to answer

    • Write 1/[p^3(p^2+1)] as product F(p)G(p)
    • Find inverse of each: t^2/2 and sin t
    • Apply convolution integral from 0 to t
    • Simplify to t^2/2 + cos t -1
13. 7 Marks
    aSolve the differential equation by Laplace Transform : $$(D^2 - 2D + 1)y = e^t,\quad y(0) = 2, y'(0) = -1.$$
    How to answer

    • Take Laplace of both sides using y(0), y'(0)
    • Solve for Y(p) = (p+...)/(p-1)^3 form
    • Take inverse Laplace via partial fractions
    • Obtain y(t) solution
14. 7 Marks
    bFind the Fourier cosine transform of $e^{-x^2}$.
    How to answer

    • Write definition of Fourier cosine transform
    • Insert f(x)=e^{-x^2} from 0 to infinity
    • Evaluate integral using Gaussian integral
    • Express result in terms of s
15. 7 Marks
    aIf the probability of hitting a object is 10% and 10 shots are fixed independently. What is the probability that the object will be hit at least once?
    How to answer

    • Model as binomial with p=0.1, n=10
    • Probability at least once = 1 - P(none)
    • Compute (0.9)^10
    • Subtract from 1
16. 7 Marks
    bIn a distribution exactly normal, 7% of the items are under 35 and 89% are under 63. What are the mean and standard deviation of the distribution?
    How to answer

    • Assume normal with mean m and SD sigma
    • Convert 35 and 63 to standard normal z values for 7% and 89%
    • Form two equations (35-m)/sigma and (63-m)/sigma
    • Solve for mean and standard deviation