Download JNTUA B.Tech 1-1 R13 2014 June Regular 13A54102 Mathematics II Question Paper

R13

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B.Tech I Year (R13) Regular Examinations June/July 2014

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MATHEMATICS - II

(Common to EEE, ECE, EIE, CSE & IT)

Part - A

(Compulsory Question)

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Answer the following: (10 X 02 = 20 Marks)

1. (a) Find the Fourier constant b_n for x sin x in [-p,p], when expressed as a Fourier series.
2. (b) Find the Fourier series f (x) = e^x defined in [-p,p].
3. (c) Find a₀, f(x) = |sin x|, [-p, p].
4. (d) Write the complete solution of z = px + qy + pq.

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5. (e) Eliminate the arbitrary constants a and b from z = ax + by + a² + b².
6. (f) Find the rank of 4 -2 2 5 3 2 2 4 1
7. (g) Find the eigen values of the matrix 4 -2 2 -5 3 2 -2 4 1
8. (h) Write the condition for AX = B is consistent.
9. (i) Apply Euler's method to solve y' = x + y, y(0) = 1, find y(1).

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10. (j) Discuss the Netwon-Raphson method for convergence.

Part - B

(Answer all five units, 05 X 10 = 50 Marks)

Unit - I

1. 1. Using Cayley-Hamilton theorem, find the inverse of A = 1 2 1 2 1 1 1 -1 1 and also find A^-3.
   

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   OR
2. 2. Reduce the quadratic form 6x² + 3y² + 3z² - 4xy - 4xz - 2yz to the sum of squares form and find the index and signature.

Unit - II

1. 3. Using Newton's forward interpolation formula, find the polynomial y = tan x satisfying the following data. Hence evaluate tan(0.12) and tan(0.28)
   

   X	0.10	0.15	0.20	0.25	0.30
   y	0.1003	0.1511	0.2027	0.2533	0.3093

   
   

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   OR
2. 4. Dividing the range into 10 equal parts find the value of ?₀^p/2 sin x dx. Using (i) Trapezoidal rule. (ii) Simpson's 1/3 rd rule.

Unit - III

1. 5. Using Taylors series method with first five terms in the expansion find y(0.1) correct to three decimal places, given that dy/dx = e^x – y², y(0) = 1.
   OR

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2. 6. Given f(x) = - x + 1 2 for - p = x = 0 x + 1 mn>2 for 0 = x = p
   Is the function even or odd? Find the Fourier series for f(x) and deduce the value of 1/1² + 1/3² + 1/5² + ...

Unit - IV

1. 7. Find the Fourier sine transform of f(x) = e^-ax/x
   OR

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2. 8. Find: (i) Z[sin(3k + 5)]. (ii) Z^-1 (4z / (z-a)²) if |z| < a.

Unit - V

1. 9. Form partial differential equations by eliminating arbitrary constants.
   (i) Z = f(x + at) + g(x – at). (ii) z = f₁(x) + f₂(y).
   OR

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2. 10. Using the method of separation of variables, solve ?u/?x = 2?u/?t + u where U(x, 0)= 6e^-3x.