SPPU Numerical Methods and Optimization - May 2017 Exam Question Paper

SPPU Mechanical Engineering (Semester 6)
Numerical Methods and Optimization
May 2017

Total marks: --
Total time: --

INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary

Solve any one question from Q.1 & Q.2

1 Determine the maximum relative error for the function

$F=3x^{2}y^{2}+5y^{2}z^{2}-7x^{2}z^{2}+38\ \ \text{For}\ x=y=z=1 \ \ \text{and}\ \ \Delta x=-0.05, \Delta y=0.001 \ \ \text{and} \ \ \Delta z=0.02.$ /

6 M

2 Find real root of cos(x) -3(x)+5 =0. Correct to four decimal places using the False Position method.

6 M

Solve any one question from Q.3 & Q.4

3 Solve the following equations by Thomas Algortihm.

$\begin{align*}& 3x_{1}-x_{2}=5\\ &2x_{1}-3x_{2}+2x_{3}=5\\ &x_{2}+2x_{3}+5x_{4}=10\\ &x_{3}x_{4}=1\end{align*}$

6 M

4 Draw a flow chart for Gauss-Seidal Method.

6 M

Solve any one question from Q.5 & Q.6(a,b)

5 Two nproducts A and B are to be manufactured by a firm. Each of these product has to be processed on two machines M1 and M2. Product A requires 4 hours on machine M1 and 5 hours on machine M2. Product B requires 5 hours on machine M1 and 2 hours on machine M2. The available capacity per month is 100 hours and 80hour for machine M1 and M2 respectively. The profit per unit is Rs.10 and Rs.5 on product A and B resespectively. Estimate the number of units of each type to be produced per month for maximum profit by simplex Method.

8 M

6(a)

$\begin{align*} &\text{Minimize}Z = 80x_{1}+120x_{2}\\ &\text{Subject to}\ x_{1}+x_{2}\leq 9\\ &x_{1}\geq 2\\ &x_{2}\geq 3\\ &20x_{1}+50x_{2}\leq 300\\ &x_{1},x_{2}\geq 0\end{align*}$ /
(Use graphical method)

6 M

6(b) Explain following terms used in graphical method of optimization.
i) Constraints ii) Optimal solution

2 M

Solve any one question from Q.7(a,b) & Q.8(a,b)

7(a) From the given table find the value of x for y (x) = 0.390

x	20	25	30	35
f(x)	0.342	0.423	0.5	0.65

8 M

7(b) Fit the curve pv^y=k to the following data:

p(kg/cm²)	0.5	1	1.5	2	2.5	3
v(litres)	1620	1000	750	620	520	460

5 M

8(a) For the following data claculate difference and obtain forward difference polynamials. Interpolate at x = 0.25 and x = 0.35

x	0.1	0.2	0.3	0.4	0.5
y=f(x)	1.4	1.56	1.76	2.0	2.28

5 M

8(b) Fit a parabola

$y=ax^{2}+bx+c$ / in least square to the data.

x	10	12	15	23	20
y	14	17	23	25	21

5 M

Solve any one question from Q.9(a,b) & Q.10(a,b)

9(a) Evaluate

$\int_{2}^{6}\log _{10}xdx$ / by using trapezoidal rule, taking n = 8, correct to five decimal places.

8 M

9(b) Explain Simpon's 1/3 rule graphically and drive formula for integration of a function.

8 M

10(a) Explain what is meant by Simpson's strip for 1/3rd and 3/8th rule. Explain why Simpson's 3/8th rule give more accuracy compared to Trapezoidal and Simpson's 1/3rd rule with same number of strips.

8 M

10(b) Solve the Trapezoidal rule

$\int_{0}^{1}\int_{0}^{1}x^{2}y^{2}.dx.dy$ / Taking step length in x and y as 0.25.

8 M

Solve any one question from Q.11(a,b) & Q.12(a,b)

11(a) Draw flowchart for modified Euler's method.

8 M

11(b) Solve the Laplace equation

$\frac{\partial ^{2}T}{\partial x^{2}}+\frac{\partial ^{2}T}{\partial y^{2}}=0$ / for the square mesh as shown in diagram below.
!mage

10 M

12(a) Draw the flowchart for solving the Laplace equation.

8 M

12(b) Use Runge-Kutta method of fourth order to obtain the numercial solution of

$\frac{dy}{dx}=\sqrt{\left ( x^{2}+y \right )},$ / Find y at x = 0.4 given y(0) = 1, take h = 0.2.

10 M

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