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B.E. IIIRD Semester (New Scheme) Mech. Engg.
Examination, 2021-22
Numerical Analysis
Paper - M - 301
Note :-Attempt any two parts each questions. All questions carry equal marks.
(a) By iterative method find the value (51)1/3 of correct to 3.
(b) Assuming that the following values of y belongs to a polynomial of degree 4, find the next the three consecutive values:
x: 0 1 2 3 4 5 6 7
y: 1 -1 1 -1 1
(c) Using Newton's divided difference formula find y at x = 8 from the following data:
x: -1 0 2 3 7 10
y: -11 1 1 141 561
(a) Evaluate ∫02 (sin x + ex) dx approximately using simpson's 1/3 rule and Weddle's rule correct to four decimal places.
(b) Solve the following system by Crout's method.
x + y + z = 3, 2x - y + 3z = 16, 3x + y - z = -3
(c) Solve by Gauss Seidal method:
20x + y - 2z = 17, 3x + 20y - z = -18, 2x - 3y + 20z = 25.
(a) Solve by Euler's modified method, the equation.
dy/dx = log(x + y); y(0) = 2 at x = 0.2 & 0.4 (Take h = 0.2)
(b) Apply Runge-Kutta method to find an approximate value of y for x = 0.2 in steps of 0.1, if
dy/dx = x2 + y2, y(0) = 1
(c) Apply Milnes method to find a solution of differential equation.
dy/dx = x - y2 in the range 0 ≤ x ≤ 1 with y(0) = 0 (Take h = 0.2)
(c) Find the Fourier Transforms of f(x) = { 1-x2, |x| ≤ 1; 0, |x| > 1 }
and hence evaluate ∫0∞ (x cos x - sin x) / x3 dx
d2y/dt2 - 4 dy/dx + y = 3t + e3t; y(0) = 1 and y'(0) = -1
(a) Find the Laplace Transform ∫0∞ sin t / t dt
(ii) Find the Inverse Laplace Transform of (s2 + 6) / (s3 (s2 + 4))
(b) Solve by using Laplace Transform
(a) Prove that arithmetic mean of binomial distribution is np and variance is npq.
(b) Average rainfall of Bhopal in the month of June is 12 days, according to the past weather records. Determine the probability that during any given week in June.
(i) The first four days will be dry, followed by three days of rain.
(ii) Rain will fall on alternate days
(iii) Rain will fall exactly four days
(c) When the mean of marks was 50% and standard deviation 5%, then 60% of the students failed in an examination. Determine the 'grace' amrks to be awarded in order to show that 70% of the students passed. Assume that the marks are normally distributed. Given that
f(t) = (1/√(2π)) ∫-∞t e-z²/2 dz then f(0.25)=0.1, f(0.52)=0.2, f(0.1)=0.039 and f(0.2)=0.079