Download AKTU B-Tech 4th Sem 2017-18 RAS401 Mathematics Iii Question Paper

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Printed Pages: 02

Paper Id: 199421

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Sub Code: RAS401

Roll No.

B. TECH.

(SEM-IV) THEORY EXAMINATION 2017-18

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MATHEMATICS - III

Time: 3 Hours

Total Marks: 70

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

SECTION A

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

1. Discuss Singularity and its types.
2. Write Cauchy-Riemann equation in polar co-ordinates.
3. The life of army shoes is normally distributed with mean 8 months and standard deviation 2 months. If 5000 pairs are insured, how many pairs would be expected to need replacement after 12 months? Given that P (z = 2) = 0.0228.
4. Determine moment generating function of Binomial distribution.

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5. Prove that: E ( Z¹) = µ+ d²
6. Write Newton-Cote's quadrature formula.
7. Find Z-transform of f(k) = u(-k).

SECTION B

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

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1. Determine an analytic function f(z) in terms of u + v = 2 sin 2x + e^2y - 2cos2x.
2. Find the mean variance of Poisson distribution.
3. Find ?⁶₀ e^x / (1+x) dx using (i) Trapezoidal rule, (ii) Simpson's 1/3rd rule and (iii) Simpson's 3/8th rule.
4. A rod is rotating in a plane. The following table gives the angle (in radians) through which the rod has turned for various values of time t (in seconds).
   t: 0 0.2 0.4 0.6 0.8 1.0 1.2
   

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   ?: 0 0.12 0.49 1.12 2.02 3.20 4.67
   Calculate the angular velocity and angular acceleration at t =0.2 and t =1.2 second.
5. Find Fourier cosine transform of 1/(1+x²) hence find Fourier sine transform of 1/(1+x²)

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SECTION C

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

1. Verify Cauchy theorem by integrating e^t along the boundary of the triangle with the vertices at the points 1+i,-1+i and -1-i.
2. Evaluate ?⁸₀ sin mx / x dx, m > 0.

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

1. The following table represents the height of a batch of 100 students. Calculate skewness and kurtosis:
   Height (in cm) 59 61 63 65 67 69 71 73 75
   No. of students 0 2 6 20 40 20 8 2 2
2. Use the method of least squares to fit the curve y = c₀ + c₁vx to the following table of values:
   

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   X 0.1 0.2 0.4 0.5 1 2
   y 21 11 7 6 5 6

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

1. Find the root of the equation xe^x = cosx using Regula-Falsi method correct to four decimal places.
2. Find Newton's divided difference polynomial for the following data:
   

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   X: -3 -1 0 3 5
   f(x): -30 -22 -12 330 3458

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

1. Solve the initial value problem u' =-2tu², u(0)=1 with h=0.2 on the interval [0,0.4]. Use Runge-Kutta fourth order method and compare your result with exact solution.
2. Solve the following system of linear equations by Matrix decomposition method taking l_ii =1 for 1= i =3.
   

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   3x-y+2z=12; x+2y+3z =11; 2x-2y-z = 2

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

1. Using Z-transform, solve the following difference equation: yk+2 +4yk+1+3yk = 3^k, given that y₀ = 0 and y₁ =1.
2. The temperature u in the semi-infinite rod 0=x<8 is determined by the differential equation
   ?u/?t = k ?²u/?x² subject to conditions
   

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   (i) u = 0 when t=0,x=0 (ii) ?u/?x = -µ (a constant) when x = 0 and t > 0, (iii) u (x, t) is bounded.
   Determine the temperature u (x, t).

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