Download AKTU B-Tech 3rd Sem 2015-2016 NAS 301 Engineering Mathematics Iii Question Paper

NAS-301

(Following Paper ID and Roll No. to be filled in your Answer Book)

Paper ID: 199301

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Roll No.

B.Tech. (SEM. III) THEORY EXAMINATION, 2015-16

ENGINEERING MATHEMATICS-III

[Time: 3 hours]

[Maximum Marks: 100]

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Section-A

Q.1 Attempt all parts. All parts carry equal marks. Write answer of each part in shorts. (10×2=20)

1. Find inverse Z-transformation of ^8z-z³/_(4-z)²
2. If u(x, y) = x²-y², prove that the u satisfies Laplace equations.
3. Evaluate ?_c (z² +1)/_{z² +1} dz where C is circle |z| = 3/2.

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4. Expand ¹/_(z+1)(z+3) in the regions |z|<1
5. State Cauchy's integral theorem.
6. Prove that: ?log f(x) = log [1+^?f(x)/_f(x)]
7. Define regression lines.

Section-B

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Note: Attempt any five Questions from this section: (10×5=50)

Q.2 Find the Fourier transform of F(x) =

Q.3 Examine the nature of the function

f(z) = ^x²y/_(x+y)(x+y)

f(0)=0

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z?0

In the region including the origin.

Q.6 A survey of 240 families with 4 children shows the following distribution :

No. of boys 4 3 2 1 0
No. of families: 10 55 105 58 12

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Test the hypothesis that male and female births are equally probable.

(Given ?²_0.05 = 9.49 and 11.1 for 4 d.f. and 5 d.f. respectively)

Q.7 Solve the following differential equation using Runge-Kutta method:

Given that ^dy/_dx = ¹/_x+y with y(0) = 1, find y(2).

Q.9 Fit a straight line to the following data:

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t: 1.0 1.1 1.2 1.3 1.4
v: 43.1 47.7 52.1 56.4 60.8

Section-C

Attempt any two questions from this section: (15×2=30)

Q.10 a) Using Lagrange's interpolation formula, find y(10) from the following table.

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x: 5 6 9 11 12 13 14
y: 10 11 14 16

b) The first four moments about the working mean 28.5 of a distribution are 0.294, 7.144, 42.409 and 454.98. Calculate the moments about the mean. Also evaluate ß₁ and ß₂ and comment upon the skewness and kurtosis of the distribution.

Q.11 a) Solve x²-5x+3=0 by using Regula - Falsi method.

b) Using the Z-transform solve the following difference equations:

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Y_k+2 + 4Y_k+1+3y_k = 3^k given y(0) = 0, y(1) =1

Q.12 a) From the data given below, find the number of items n:

r = 0.5 SXY = 120, SX² = 90, s_y = 8 where x and y are deviations from the arithmetic mean.

b) If f(z)=u+iv is analytic function and u- v = e^x (cos y - sin y), find f(z) in terms of z.

c) Find ?₀^{1 dx}/_1+x² by using Simpson's ³/₈ rule on

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