B.Tech., IV Semester
Examination, November 2023
Grading System (GS)
Max Marks: 70 | Time: 3 Hours
Note:
i) Attempt any five questions.
ii) All questions carry equal marks.
a) Evaluate $\sqrt{12}$ to four decimal places by Newton Raphson Method. (Unit 1)
b) What is the rate of convergence of bisection method? (Unit 1)
a) Prove that
i) $\Delta = \frac{1}{2} \delta^2 + \delta \sqrt{1 + \left(\frac{\delta^2}{4}\right)}$
ii) $\Delta + \nabla = \frac{\Delta}{\nabla} - \frac{\nabla}{\Delta}$ (Unit 1)
b) Construct a backward difference table for $y = \log x$ given that
| $x$ | 10 | 20 | 30 | 40 | 50 |
|---|---|---|---|---|---|
| $y$ | 1 | 1.3010 | 1.4771 | 1.6021 | 1.6990 |
And find values of $\nabla^3 \log 40$ and $\nabla^4 \log 50$ (Unit 1)
a) Using following table, by Lagrange's Method find $f(x)$ as a polynomial in $x$:
| $x$ | -1 | 0 | 3 | 6 | 7 |
|---|---|---|---|---|---|
| $f(x)$ | 3 | -6 | 39 | 822 | 1611 |
(Unit 1)
b) Using Newton's divided difference formula, calculate the value of $f(6)$ from the following data:
| $x$ | 1 | 2 | 7 | 8 |
|---|---|---|---|---|
| $f(x)$ | 1 | 5 | 5 | 4 |
(Unit 1)
a) Find $f'(1.1)$ and $f''(1.1)$ from the following table:
| $x$ | 1.0 | 1.2 | 1.4 | 1.6 | 1.8 | 2.0 |
|---|---|---|---|---|---|---|
| $f(x)$ | 0.0 | 0.1280 | 0.5540 | 1.2960 | 2.4320 | 4.0 |
(Unit 2)
b) Find missing values in the following table:
| $x$ | 0 | 5 | 10 | 15 | 20 | 25 |
|---|---|---|---|---|---|---|
| $y$ | 6 | 10 | - | 17 | - | 31 |
(Unit 1)
a) Find $\int_0^6 \frac{e^x}{1+x} dx$ approximately using simpson's $\frac{3}{8}$th rule on integration. (Unit 2)
b) Solve the equation $\frac{dy}{dx} = 1 - y$ with initial condition $y(0) = 0$ using Euler's modified method and tabulate the solution at $x = 0.1, 0.2, 0.3$. (Unit 3)
a) Using Lagrange's interpolation formula to fit a polynomial to the data:
| $x$ | -1 | 0 | 2 | 3 |
|---|---|---|---|---|
| $f(x)$ | -8 | 3 | 1 | 12 |
(Unit 1)
b) Using Regula-Falsi method, find the real root of $x \log_{10} x = 1.2$, correct to four decimal places. (Unit 1)
a) Solve $\frac{dy}{dx} = x + y^2$ by using Runge-Kutta method of fourth order to find an approximate value of $y$ for $x = 0.2$, given that $y = 1$ when $x = 0$. (Take $h = 0.1$) (Unit 3)
b) Evaluate $L^{-1} \left[ \frac{s+7}{s^2 + 4s + 8} \right]$ (Unit 4)
a) Find the inverse Laplace transform of $\frac{s+4}{s(s-1)(s^2+4)}$. (Unit 4)
b) Prove that if $L\{f(t)\} = F(s)$ then $L \left\{ \frac{1}{t} f(t) \right\} = \int_s^\infty F(s) ds$ hence, evaluate $\frac{e^{at} - \cos bt}{t}$. (Unit 4)
