BT-301 Mathematics - III - Nov 2022 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
   aFind a positive root of the equation by iteration method $3x = \cos x + 1$.
   How to answer

   • Rewrite the equation as $x = \phi(x) = \frac{1}{3}(\cos x + 1)$.
   • Verify convergence condition $|\phi'(x)| < 1$ near the root.
   • Choose an initial approximation $x_0$ (e.g., $x_0 = 0.5$ or $0.6$).
   • Perform successive iterations $x_{n+1} = \frac{1}{3}(\cos x_n + 1)$ in radians.
   • Stop when consecutive approximations agree to desired decimal places.
2. 7 Marks
   b

   Using Lagrange's interpolation formula, find $y(6)$ from the following data:

   X	3	5	7	9	11
   Y	6	24	58	108	74

   How to answer

   • State Lagrange's interpolation formula for 5 points ($x_0$ to $x_4$).
   • Substitute $x = 6$ and values $x_0=3, x_1=5, x_2=7, x_3=9, x_4=11$.
   • Calculate each Lagrange basis polynomial term multiplied by $y_i$.
   • Sum the evaluated terms to find $y(6)$.
3. 7 Marks
   aEvaluate $\int_{0}^{\pi} \sin x \, dx$ by dividing the range into 10 equal parts using: i) Trapezoidal rule ii) Simpson's 1/3 rule
   How to answer

   • Find step size $h = \frac{\pi - 0}{10} = \frac{\pi}{10}$.
   • Construct table of $x_i = i\pi/10$ and $y_i = \sin(x_i)$ for $i=0,1,\dots,10$.
   • Apply Trapezoidal formula and compute numerical value.
   • Apply Simpson's 1/3 formula and compute numerical value.
4. 7 Marks
   bUsing Gauss-Seidel iteration method to solve the system. $$10x + y + z = 12, \quad 2x + 10y + z = 13, \quad 2x + 2y + 10z = 14.$$
   How to answer

   • Verify diagonal dominance of the system matrix.
   • Express $x, y, z$ in iterative form using Gauss-Seidel schema.
   • Set initial approximations $x^{(0)}=0, y^{(0)}=0, z^{(0)}=0$.
   • Calculate subsequent iterations until values converge to desired accuracy (yielding $x=1, y=1, z=1$).
5. 7 Marks
   aGiven that $\sin 45 = 0.7071$, $\sin 50 = 0.7660$, $\sin 55 = 0.8192$, $\sin 60 = 0.8660$. Then find the $\sin 58$ by using Newton's backward interpolation formula.
   How to answer

   • Construct the backward difference table with values of $x$ and $y=\sin x$.
   • Identify $x_n = 60$, $h = 5$, and compute $u = \frac{58 - 60}{5} = -0.4$.
   • Write Newton's backward difference interpolation formula.
   • Substitute backward differences $\nabla y_n, \nabla^2 y_n, \nabla^3 y_n$ and $u$ to evaluate $\sin 58$.
6. 7 Marks
   bUsing Simpson's 3/8 rule, evaluate $\int_{0}^{6} \frac{dx}{1+x^2}$ by dividing the range into 6 equal parts.
   How to answer

   • Determine interval width $h = \frac{6-0}{6} = 1$.
   • Create a table of points $x = 0, 1, 2, 3, 4, 5, 6$ and calculate $y = \frac{1}{1+x^2}$.
   • State Simpson's 3/8 formula.
   • Substitute function values into the formula and compute the final numerical result.
7. 14 Marks
   Given $\frac{dy}{dx} = x + \sin y$ and $y(0) = 1$. Compute $y(0.2)$ and $y(0.4)$ with $h = 0.2$ using Euler's modified method.
   How to answer

   • State Modified Euler's (predictor-corrector) formulas.
   • Step 1: For $x_0 = 0, y_0 = 1$, compute initial predictor $y_1^{(0)} = y_0 + h f(x_0, y_0)$.
   • Iterate the corrector until $y(0.2)$ stabilizes.
   • Step 2: Use $x_1 = 0.2$ and computed $y_1$ to find predictor $y_2^{(0)}$.
   • Iterate corrector to determine $y(0.4)$.
8. 7 Marks
   aUsing Laplace Transform, Evaluate $\int_{0}^{\infty} \frac{\cos at - \cos bt}{t} \, dt$.
   How to answer

   • Find $\mathcal{L}\{\cos at - \cos bt\} = \frac{s}{s^2 + a^2} - \frac{s}{s^2 + b^2}$.
   • Apply division by $t$ rule: $\mathcal{L}\left\{\frac{\cos at - \cos bt}{t}\right\} = \int_s^{\infty} \left( \frac{u}{u^2 + a^2} - \frac{u}{u^2 + b^2} \right) du$.
   • Integrate to get $\frac{1}{2} \ln\left(\frac{s^2 + b^2}{s^2 + a^2}\right)$.
   • Take limit as $s \to 0$ to obtain the value $\ln(b/a)$.
9. 7 Marks
   bFind $$L^{-1} = \left\{ \frac{s^2}{(s^2 + a^2)^2} \right\}$$
   How to answer

   • Identify $F(s) = \frac{s}{s^2 + a^2}$ and $G(s) = \frac{s}{s^2 + a^2}$.
   • Apply Convolution Theorem: $\mathcal{L}^{-1}\{F(s)G(s)\} = \int_0^t \cos(au) \cos(a(t-u)) \, du$ (or use differentiation/partial fractions).
   • Use trigonometric identity $2 \cos A \cos B = \cos(A+B) + \cos(A-B)$ inside the integral.
   • Evaluate integral to obtain $\frac{1}{2a} (\sin at + at \cos at)$.
10. 14 Marks
    Using Laplace Transform, Solve $(D^2 + 2D + 5)y = e^{-t} \sin t$, given that $y(0) = 0, y'(0) = 1$.
    How to answer

    • Take the Laplace transform of both sides of the differential equation.
    • Substitute initial conditions $y(0) = 0$ and $y'(0) = 1$ to form an equation in $Y(s)$.
    • Solve algebraically for $Y(s) = \frac{1}{(s+1)^2 + 4} + \frac{1}{[((s+1)^2+1)((s+1)^2+4)]}$.
    • Use partial fractions or shifting theorem on $Y(s)$.
    • Take the inverse Laplace transform to obtain the final time-domain solution $y(t)$.
11. 7 Marks
    a20% of items produced from a factory are defective. Find the probability that in a sample of 5 chosen at random. i) None is Defective ii) One is Defective iii) $P(1 < x < 4)$.
    How to answer

    • Identify parameters: $n = 5$, $p = 0.2$, $q = 0.8$.
    • Compute $P(X = 0) = \binom{5}{0} (0.2)^0 (0.8)^5 = (0.8)^5 = 0.32768$.
    • Compute $P(X = 1) = \binom{5}{1} (0.2)^1 (0.8)^4 = 5(0.2)(0.4096) = 0.4096$.
    • Compute $P(1 < X < 4) = P(X = 2) + P(X = 3)$ and evaluate numerically.
12. 7 Marks
    bIf a random variable has a Poisson distribution such that $P(1) = P(2)$. Find out i) Mean of the distribution ii) $P(x \geq 1)$ iii) $P(1 < x < 4)$
    How to answer

    • Set $P(1) = P(2) \implies \frac{e^{-\lambda} \lambda}{1!} = \frac{e^{-\lambda} \lambda^2}{2!}$ and solve for $\lambda = 2$.
    • Mean of the distribution is $\lambda = 2$.
    • Calculate $P(x \geq 1) = 1 - P(x=0) = 1 - e^{-2}$.
    • Calculate $P(1 < x < 4) = P(x=2) + P(x=3) = \frac{e^{-2} 2^2}{2!} + \frac{e^{-2} 2^3}{3!} = e^{-2}\left(2 + \frac{4}{3}\right) = \frac{10}{3} e^{-2}$.
13. 7 Marks
    aDerive Mean of the Normal Distribution.
    How to answer

    • State PDF of normal distribution $f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$.
    • Write definition of mean $E[X] = \int_{-\infty}^{\infty} x f(x) \, dx$.
    • Substitute standard normal variable $z = \frac{x-\mu}{\sigma}$, so $x = \mu + \sigma z$ and $dx = \sigma dz$.
    • Split into two integrals: $\mu \int_{-\infty}^\infty \frac{1}{\sqrt{2\pi}} e^{-z^2/2} dz + \frac{\sigma}{\sqrt{2\pi}} \int_{-\infty}^\infty z e^{-z^2/2} dz$.
    • Evaluate the odd integral to 0 and total area to 1, showing $E[X] = \mu$.
14. 7 Marks
    bUsing Runge-Kutta method of fourth order solve $y' = xy$, $y(1) = 2$ at $x = 1.2$ with $h = 0.2$.
    How to answer

    • Define $f(x, y) = xy$, $x_0 = 1, y_0 = 2, h = 0.2$.
    • Calculate $k_1 = h f(x_0, y_0) = 0.2 (1)(2) = 0.4$.
    • Calculate $k_2 = 0.2 f(1.1, 2.2) = 0.484$.
    • Calculate $k_3 = 0.2 f(1.1, 2 + 0.242) = 0.2 (1.1)(2.242) = 0.49324$.
    • Calculate $k_4 = 0.2 f(1.2, 2 + 0.49324) = 0.2 (1.2)(2.49324) = 0.59838$.
    • Compute $y(1.2) = y_0 + \frac{1}{6}(k_1 + 2k_2 + 2k_3 + k_4)$.