Download PTU B.Tech 2021 Jan AE 5th Sem 78226 Numerical Methods Question Paper

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Total No. of Pages : 02
Total No. of Questions : 18

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B.Tech. (Automobile Engineering) (Sem. - 5)
NUMERICAL METHODS
Subject Code : BTAE-502-18
M.Code : 78226
Time : 3 Hrs. Max. Marks : 60

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INSTRUCTIONS TO CANDIDATES :

1. SECTION-A is COMPULSORY consisting of TEN questions carrying TWO marks each.
2. SECTION-B contains FIVE questions carrying FIVE marks each and students have to attempt any FOUR questions.
3. SECTION-C contains THREE questions carrying TEN marks each and students have to attempt any TWO questions.

SECTION-A

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Write briefly :

1. State Simpson’s three — Eighth rule
2. In four tosses of a coin, let x be the number of heads ~Calculate the expected value of x.
3. A sample of 20 items has a mean 42 units and S.D 5 units. Test the hypothesis that it is a random sample from a normal population with mean 45 units.
4. Find a real root of the equation x = ¢ using Newton Raphson method.

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5. Evaluate Δ tan^-1 x
6. Find positive real root of x^* — x =1 by bisection method, correct upto 2 decimal places between and 2.
7. State Merit’s of Lagrange’s formula
8. Define Spline function.
9. Define types of numerical instability.

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10. Prove that the absolute error in the common logarithm of a number is less than half the relative error of the given number.

SECTION-B

1. Solve the problem y” — x² + 3*= 0.y (0) = 1, y'(0) = 0 to evaluate y (0.1) using Taylor’s series methods.
2. Use Gauss elimination method to solve the following system of equations:
   2x+y+z=10, 3x+2y+3z=18, x+4y+9z=16

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3. Fit a poisson distribution to the following data and test the goodness of fit:
   

   X	f
   0	109
   1	65
   2	22
   3	3
   4	1

4. Use Adam’s Moulton-Bashforth method to find y (1.4) given dy/dx =x² (1+y), y (1) =1, y(1.1)= 1.233, y (1.2)= 1.548 and y (1.3) = 1.979.
5. a) Compute f'(3) from the following table:

   X	F(x)
   1	0
   2	1
   4	5
   8	21
   10	27

   
   b) Given the initial value problem : y’ = 1 + y², y(0) = 0, Find y (0.6) by Runge Kutta fourth order method taking h= 0.2

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SECTION-C

1. A river is 80m wide. The depth ‘y’ of the river at a distance ‘x” from one bank is given by following table:

   X	0	10	20	30	40	50	60	70	80
   y	0	4	7	9	12	15	14	8	3

   Find the approximate area of cross-section of the river using Simpson’s one — third rule.
2. a) A tank is discharging water through an orifice at a depth of x metre below the surface of the whose area is A m² Following are the values of x for the corresponding values of A.

   A	1.257	1.39	1.52	1.65	1.809	1.962	2.123	2.295	2.462	2.650
   X	1.5	1.65	1.8	1.95	2.1	2.25	2.4	2.55	2.7	2.85

   Using the formula (0.018) T = ∫_1.5^3.0 A/√x dx, calculate T, the time (in seconds) for the level of the water to drop from 3.0 m to 1.5 m above the orifice.
   b) Using Newton’s divided difference formula, calculate the value of f(6) from the following data;

   X	4	5	7	10	11	13
   F(x)	48	100	294	900	1210	2028

3. Find a positive value of (17)^1/3 correct to four decimal places by Newton’s Raphson’s method.

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NOTE : Disclosure of Identity by writing Mobile No. or Making of passing request on any page of Answer Sheet will lead to UMC against the Student.

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