Download AKTU B-Tech 3rd Sem 2017-2018 RAS 301 Mathematics Iii Question Paper

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Printed pages: 02

Paper ID: 9019

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Roll No.

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Sub Code: RAS 301

B.Tech.

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(SEM III) THEORY EXAMINATION 2017-18

Time: 3 Hours

Total Marks: 70

Mathematics -III

Note: 1. Attempt all Sections. If require any missing data; then choose suitably.

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2. Any special paper specific instruction.

SECTION A

1. Attempt all questions in brief. 2 x 7 = 14

1. Define analytic function with an example.
2. Define the Binomial distribution with mean and variance.

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3. Write the normal equation for the curve ? = ? + ?
4. Give comparison between Regulafalsi method and Newton Raphson method
5. Write the relation between ?h divided difference and ?h forward difference.
6. What do you mean by initial value problem
7. Find ?-1{5?-15}

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SECTION B

2. Attempt any three of the following: 7 x 3 = 21

1. Give an example of a function in which Cauchy Riemann Equations are satisfied yet the function is not analytic at the origin. Justify your answer.
2. Find the measure of skewness and kurtosis based on moments for the following distribution and draw your conclusion.
   

   Marks	5-15	15-25	25-35	35-45	45-55
   No. of students	3	5	7	7	4

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3. Decompose ? = [1-217-2534] in the form LU, where L is lower triangular matrix and U is upper triangular matrix and hence solve the system of equations:
   5? - 2? + ? = 4
   7? + ? - 5? = 8
   3? + 7? + 4? = 10.
4. Express the function ? (?) = {1 ?h? |?|=10 ?h? |?|>1 as a Fourier Integral.
   

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   Hence evaluate ?08sin ? ? ?.
5. Given the initial value problem ? = ?3 - ?3, ?(0) = 1. Find the numerical solution of differential equation at x = 0.6 with h = 0.2 by using Runge-Kutta method of Fourth order.

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SECTION C

3. Attempt any one part of the following: 7x1=7

1. Evaluate the integration: ?0?4 ?.
2. State and prove the Cauchy Integral formula. Also evaluate ?(?2+4)2? = ?16 where C is the circle|z – i| = 2.

4. Attempt any one part of the following: 7x1=7

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1. Find Fourier cosine transform of 11+?2 and hence find Fourier sine transform of ?1+?2
2. Find the inverse Z-transform of F(z), where F(z) is given by
   (?) (?+2)(?+3)
   (?) (?-1)(?-2)(?+3) ?7?-11?2

5. Attempt any one part of the following: 7x1=7

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1. In a partially distributed laboratory record of an analysis of a correlation data, the following result are legible:
   Variance of ? = 9
   Regression equation: 8x – 10y = 66 = 0, 40x – 18y = 214.
   What were (a) the mean of x and y. (b) the standard deviation of y and the coefficient of x and y:
2. Find the mean and variance of normal distribution.

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6. Attempt any one part of the following: 7x1=7

1. Find the real root of the equation 2? - log10 ? + 5=0 by method of False position correct to three decimal places.
2. State and prove the Lagrange interpolation formula. Find the interpolating polynomial by By Lagrange interpolation formula for the given data

   X	5	6	9	11
   y	12	13	14	16

7. Attempt any one part of the following: 7x1=7

1. Apply Simpson's 3/8 th rule to obtain approximate value of (i) ?0?/2 ? ? ? (ii) ?00.3(2? - ?2)1/2 ? using Simpson's rule with 6 interval.

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2. Find x for which y is maximum and find the max value of y

   X	1.2	1.3	1.4	1.5	1.6
   y	0.9320	0.9636	0.9855	0.9975	0.9996