Numerical Analysis - BU 2022 Question Paper

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EW-224

B. Tech. IV^th Semester (New Scheme)

Examination, 2022

Inform. Tech.

Paper - IT-401

Numerical Analysis

Time : 3 Hours

[Maximum Marks : 60

Note :- Attempt any two parts each questions. All questions carry equal marks.

(a) Find the real root of the equation $xe^x - 3 = 0$ by Regula falsi method, correct to four decimal places.

(b) Apply Newton's forward interpolation formula to obtain $f(1)$, given that:

$$\sum^{10} f(x) = 500426, \sum^{10} f(x)^2 = 329240,$$ $$\sum^{7} f(x) = 175212 \text{ and } f(10)=40365.$$

$$x: -1 \quad 0 \quad 2 \quad 3 \quad 7 \quad 10$$ $$y: -11 \quad 1 \quad 1 \quad 1 \quad 141 \quad 561$$

hence find $f(8)$.

(a) Find the first and second derivative of $f(x)$ at $x = 1.5$ :

$$x: 1.5 \quad 2.0 \quad 2.5 \quad 3.0 \quad 3.5 \quad 4.0$$ $$y: 3.375 \quad 7.000 \quad 13.675 \quad 24.000 \quad 38.875 \quad 59.000$$

(b) A curve is drawn to pass through the following points.

$$x: 1 \quad 1.5 \quad 2.0 \quad 2.5 \quad 3.0 \quad 3.5 \quad 4.0$$ $$y: 2 \quad 2.4 \quad 2.7 \quad 2.8 \quad 3.0 \quad 2.6 \quad 2.1$$

(c) Estimate the area bounded by the curve, x-axis and the lines $x = 1, x = 4$. Also find the volume of solid generated by revolving this area using Weddle's rule.

(a) Solve by Gauss seidal method the equation (by taking initial approximation $x_0, y_0, z_0$ as zero):

$$28x + 4y - z = 32$$ $$x + 3y + 10z = 24$$ $$2x + 17y + 4z = 35$$

(b) Using Taylor's series find $y(2.1)$ correct to 5 decimal places. Given:

$$\frac{dy}{dx} = x - y, y(2) = 2.$$

$$\frac{dy}{dx} = x^2 + y^2, y(0) = 1.$$

(d) Given that $\frac{dy}{dx} = x - y$ and $y(0) = 0, y(0.2) = 0.0200, y(0.4) = 0.0795, y(0.6) = 0.1762$. Evaluate $y(0.8)$ by Adams-Bashforth method.

(a)

(i) Find the Laplace Transforms of $\int_0^t e^{-t} \cos t dt$.

(ii) Find the inverse Laplace Transforms of

$$\frac{(s^2+6)}{((s^2+1)(s^2+4))}.$$

(b) Solve the boundary value problem, by using Laplace transforms:

$$\frac{d^2y}{dt^2} - 3 \frac{dy}{dt} + 2y = 4t + e^{3t}$$ $$y(0) = 1, y'(0) = -1.$$

$$f(x) = \begin{cases} 1, & |x| \le 1 \\ 0, & |x| > 1 \end{cases}$$

Hence evaluate $\int_0^\infty \frac{\sin \lambda}{\lambda} d\lambda$.

(a) The chances that an aeroplane fails to return is 5%. Find the probability that.

(i) one plane does not return

(ii) at the most 5 planes do not return

(iii) what is the most probable number of returns?

(b) In a certain factory turning out razor blades, there is a small chance of $.002$ for any blade to be defective. The blades are supplied in packets of $10$. Use Poisson distribution to calculate the approximate number of packets containing no defective, one defective, two defective blades in a consignment of $10,000$ packets.

(c) When the mean of marks was $50\%$ and standard deviation $5\%$ then $60\%$ of the students failed in an examination. Determine the 'grace' marks to be awarded in order to show that $70\%$ of the students passed. Assume that the marks are normally distributed. Given that if

$$f(t) = \frac{1}{\sqrt{2\pi}} \int_0^t e^{-t^2/2} dt$$

then $f(0.25) = 0.1$, $f(0.52) = 0.2$, $f(0.1) = 0.039$ and $f(0.2) = 0.079$.