BT-301 Mathematics - III - Jun 2023 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
   a

   Find Solution using Newton's Divided Difference Interpolation formula.

   $x$	300	304	305	307
   $F(x)$	2.4771	2.4829	2.4843	2.4871

   How to answer

   • Construct divided difference table for given data points.
   • Write Newton's divided difference interpolation formula.
   • Substitute divided difference coefficients into formula.
   • Simplify the resulting polynomial.
2. 7 Marks
   bUsing Newton Raphson Method, find the real root of the equation $3x = \cos x + 1$.
   How to answer

   • Define $f(x) = 3x - \cos x - 1$ and calculate $f'(x) = 3 + \sin x$.
   • Find initial approximation $x_0$ where $f(x)$ changes sign.
   • Apply iteration formula $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$.
   • Iterate until root converges to desired accuracy (e.g. 3-4 decimal places).
3. 7 Marks
   a

   Find the first and second derivatives at $x = 1.1$ from the following table

   $x$	1	1.2	1.4	1.6	1.8	2.0
   $F(x)$	0	.1280	.5440	1.2960	2.4320	4.00

   How to answer

   • Set up finite difference forward table with $h = 0.2$.
   • Calculate $u = \frac{1.1 - 1}{0.2} = 0.5$.
   • Use derivative of Newton's forward interpolation formula for $f'(1.1)$.
   • Use second derivative formula for $f''(1.1)$.
   • Compute final numerical values.
4. 7 Marks
   b

   Find Solution using Simpson's 3/8 rule.

   $x$	0	0.1	0.2	0.3	0.4
   $y$	1	0.9975	0.99	0.9776	0.8604

   How to answer

   • Identify interval length $h = 0.1$.
   • State Simpson's 3/8 rule formula for 3 intervals (from $x=0$ to $0.3$) or apply combined rule if appropriate.
   • Substitute values of $y_0, y_1, y_2, y_3$ into $\frac{3h}{8}(y_0 + 3y_1 + 3y_2 + y_3)$.
   • Compute the numerical integration value.
5. 7 Marks
   aApply Runge-kutta method of fourth order to solve $$10 \frac{dy}{dx} = x^2 + y^2 ; y(0) = 1 \text{ for } x = 0.1$$
   How to answer

   • Rewrite as $\frac{dy}{dx} = f(x, y) = 0.1(x^2 + y^2)$ with $x_0 = 0, y_0 = 1, h = 0.1$.
   • Calculate $k_1 = h f(x_0, y_0)$.
   • Calculate $k_2 = h f(x_0 + h/2, y_0 + k_1/2)$ and $k_3 = h f(x_0 + h/2, y_0 + k_2/2)$.
   • Calculate $k_4 = h f(x_0 + h, y_0 + k_3)$.
   • Compute $y(0.1) = y_0 + \frac{1}{6}(k_1 + 2k_2 + 2k_3 + k_4)$.
6. 7 Marks
   bUse Milne's method to find the solution of differential equation $\frac{dy}{dx} = x - y^2$; given that $y = 0$ when $x = 0$ for $0 < x < 1$.
   How to answer

   • Generate starting values $y_1, y_2, y_3$ using Picard's or Taylor series or Runge-Kutta method.
   • Calculate derivative values $f_0, f_1, f_2, f_3$ where $f(x, y) = x - y^2$.
   • Apply Milne's predictor formula to compute predicted $y_4$.
   • Compute $f_4$ and apply Milne's corrector formula to obtain refined $y_4$.
7. 7 Marks
   aFind Laplace transform of the following data

   1. i)$t^2 \cos at$
   2. ii)$t^2 e^t \sin 4t$
   How to answer

   • For (i): Use $L\{\cos at\} = \frac{s}{s^2 + a^2}$ and apply $L\{t^2 f(t)\} = \frac{d^2}{ds^2}[F(s)]$.
   • Differentiate twice to obtain $\frac{2s(s^2 - 3a^2)}{(s^2 + a^2)^3}$.
   • For (ii): Find $L\{t^2 \sin 4t\}$ or use first shifting theorem on $L\{t^2 \sin 4t\}$ by substituting $s \to s-1$.
   • Simplify the resulting algebraic expression.
8. 7 Marks
   bSolve : $(D^2 + 1) y = t \cos 2t, y(0) = 0 = y'(0), t > 0$.
   How to answer

   • Take Laplace transform on both sides using initial conditions $y(0) = 0, y'(0) = 0$.
   • Compute $L\{t \cos 2t\} = \frac{s^2 - 4}{(s^2 + 4)^2}$.
   • Obtain $Y(s) = \frac{s^2 - 4}{(s^2 + 1)(s^2 + 4)^2}$.
   • Use partial fraction decomposition.
   • Take inverse Laplace transform to find $y(t)$.
9. 7 Marks
   aFind the Mean and Variance of the Poisson distribution.
   How to answer

   • Define Poisson distribution PMF: $P(X=r) = \frac{e^{-\lambda} \lambda^r}{r!}$.
   • Derive Mean: $E[X] = \sum_{r=0}^{\infty} r \frac{e^{-\lambda} \lambda^r}{r!} = \lambda$.
   • Derive $E[X^2] = E[X(X-1)] + E[X] = \lambda^2 + \lambda$.
   • Calculate Variance: $Var(X) = E[X^2] - (E[X])^2 = \lambda$.
10. 7 Marks
    bA fair coin is tossed 10 times, what are the probability of getting exactly 6 heads and at least six heads.
    How to answer

    • Identify binomial model with $n = 10, p = 0.5, q = 0.5$.
    • Compute $P(X = 6) = \binom{10}{6} (0.5)^6 (0.5)^4 = \frac{210}{1024} = \frac{105}{512}$.
    • Compute $P(X \ge 6) = P(X=6) + P(X=7) + P(X=8) + P(X=9) + P(X=10)$.
    • Evaluate sum: $\frac{210 + 120 + 45 + 10 + 1}{1024} = \frac{386}{1024} = \frac{193}{512}$.
11. 7 Marks
    aUse Taylor series method to obtain the value of $y$ at $x = 0.1$, For the differential equation $\frac{dy}{dx} = x + y; y(0) = 1$.
    How to answer

    • Write Taylor series expansion about $x_0 = 0$.
    • Find successive derivatives: $y' = x + y$, $y'' = 1 + y'$, $y''' = y''$, $y^{(4)} = y'''$.
    • Evaluate at $x_0 = 0, y_0 = 1$: $y'_0 = 1, y''_0 = 2, y'''_0 = 2, y^{(4)}_0 = 2$.
    • Substitute into Taylor series for $x = 0.1$ and calculate numerical value.
12. 7 Marks
    bSolve the following equation by Gauss-Seidel method $$\begin{aligned} 20x + y - 2z &= 17 \\ 3x + 20y - z &= -18 \\ 2x - 3y + 20z &= 25 \end{aligned}$$
    How to answer

    • Check diagonal dominance of the system.
    • Express $x, y, z$ in iterative form.
    • Start with initial guess $(x^{(0)}, y^{(0)}, z^{(0)}) = (0, 0, 0)$.
    • Perform iterations updating each variable immediately.
    • Iterate until values converge to stable solution (approx $x=1, y=-1, z=1$).
13. 7 Marks
    aSolve the following system of equation by Crout's method $$\begin{aligned} 2x + 3y - z &= 5 \\ 3x + 2y + z &= 10 \\ x - 5y + 3z &= 0 \end{aligned}$$
    How to answer

    • Write system in matrix form $AX = B$.
    • Decompose $A = LU$ where $L$ is lower triangular and $U$ is upper triangular with diagonal elements equal to 1.
    • Solve $LY = B$ using forward substitution for $Y$.
    • Solve $UX = Y$ using backward substitution for $X = [x, y, z]^T$.
14. 7 Marks
    bFind the Fourier Cosine Transform of $e^{-x}$
    How to answer

    • State definition of Fourier cosine transform: $F_c\{f(x)\} = \sqrt{\frac{2}{\pi}} \int_0^\infty f(x) \cos(sx) dx$ or $\int_0^\infty f(x) \cos(sx) dx$.
    • Substitute $f(x) = e^{-x}$.
    • Evaluate integral using standard formula $\int_0^\infty e^{-x} \cos(sx) dx = \frac{1}{1 + s^2}$.
    • Write final transform expression.
15. 7 Marks
    aFind $y(0.2)$ by Euler's modified method $$\frac{dy}{dx} = \log_{10} (x + y), y(0) = 1$$
    How to answer

    • Identify $f(x, y) = \log_{10}(x + y)$, $x_0 = 0, y_0 = 1$, choose step size $h = 0.2$ (or two steps of $0.1$).
    • Compute initial estimate using standard Euler's method $y_1^{(0)}$.
    • Apply modified Euler's formula iteratively until convergence.
    • State final value of $y(0.2)$.
16. 7 Marks
    bFind the root of the equation $x^3 - 9x + 1 = 0$ between $x = 2$ and $x = 4$ by the method of bisection.
    How to answer

    • Let $f(x) = x^3 - 9x + 1$; calculate $f(2) = 8 - 18 + 1 = -9 < 0$ and $f(4) = 64 - 36 + 1 = 29 > 0$.
    • Compute midpoint $x_1 = \frac{2 + 4}{2} = 3$ and evaluate $f(3) = 27 - 27 + 1 = 1 > 0$.
    • Update interval to $[2, 3]$ since $f(2) < 0$ and $f(3) > 0$.
    • Perform further bisection iterations until desired accuracy is achieved.