Jawaharlal Nehru Technological University Hyderabad
Jawaharlal Nehru Technological University Hyderabad B.Tech I Year MATHEMATICAL METHODS Previous Question Paper (May - 2018 - R13)
Note: This question paper contains two parts A and B. Part A is compulsory which carries 25 marks. Answer all questions in Part A. Part B consists of 5 Units. Answer any one full question from each unit. Each question carries 10 marks and may have a, b, c as sub questions.
PART - A
If ∑x=15, ∑y=23, ∑x2=25 and ∑xy=55 for n=10, find best fit of straight line y = a + bx.
Answer
The normal equations for the line y = a + bx are:
Σy = na + bΣx
Σxy = aΣx + bΣx2
Substituting the given values (n=10, Σx=15, Σy=23, Σx2=25, Σxy=55):
23 = 10a + 15b
55 = 15a + 25b
Solving these simultaneous equations:
From 10a + 15b = 23, a = (23 - 15b)/10.
Substituting into the second: 15((23 - 15b)/10) + 25b = 55
34.5 - 22.5b + 25b = 55
2.5b = 20.5, so b = 8.2.
Then a = (23 - 15(8.2))/10 = -9.8.
The line is y = -9.8 + 8.2x.
Find the missing value from the given data set.
Answer
Using the method of finite differences for n=4 points, the forward difference operator Δ4f(0) = 0.
(E - 1)4f(0) = 0, which expands to f(4) - 4f(3) + 6f(2) - 4f(1) + f(0) = 0.
With f(0)=2, f(2)=5, f(3)=10, and f(4) unknown, we assume the polynomial degree is 2 (since 4 points are given).
Δ3f(0) = 0 implies f(3) - 3f(2) + 3f(1) - f(0) = 0.
10 - 3(5) + 3f(1) - 2 = 0
10 - 15 + 3f(1) - 2 = 0
3f(1) = 7, so f(1) = 7/3.
Find L and U using LU decomposition method.
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Find the approximate value of 301/3 using Newton's Raphson method.
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Define finite Fourier sine and cosine transforms.
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Find the half range sine series of f(x) = x on (0, l).
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Form the partial differential equation from z = (x+a)(y+b) by eliminating the arbitrary constants a, b.
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Define divergence of a vector point function and what is its geometrical meaning?
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Let F = (x2-yz)i + (y2-xz)j + (z2-xy)k is an irrotational vector, find its scalar potential function.
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PART - B
Define interpolation, and Find the interpolate polynomial from the following data: x: 0, 1, 2, 3, 4; y: 3, 6, 11, 18, 27 and hence find the value of y(0.1), y(2.1) and y(4.5).
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Given points (1, -8), (2, -1) and (3, 18) satisfying the function y = f(x), Determine the values of y(2.5) and y(2.0), using the Cubic spline approximation.
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Find the positive root of the equation 3x = cos x + 1 by iteration method.
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Solve the following system by Gauss-Seidel method: 54x + y + z = 110, 2x + 15y + 6z = 72, -x + 6y + 27z = 85.
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Evaluate ∫e-x2 dx from 0 to 2 using Trapezoidal rule as well as Simpson’s rule, taking step size h=0.2.
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Use Adams-Bashforth Moulton method, find y(0.8) from dy/dx = x+y, y(0)=1. Find the initial values y(0.2), y(0.4) and y(0.6) from Taylor series method.
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Let fs(p) and fc(p) are Fourier sine and cosine transform of f(x), Prove that F{xf(x)} = dfc/dp and F{xf(x)} = -dfs/dp.
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Obtain the Fourier series expansion of f(x) = |x| in (-π, π) and hence deduce that 1/12 + 1/32 + 1/52 + ... = π2/8.
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Find the Fourier transform of f(x) = 1 for |x|<1 and 0 for |x|>1, and hence evaluate ∫ sin(x)/x dx.
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Find the Fourier sine transform of e-ax/x, a > 0.
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Find the solution of the one dimensional heat equation ∂u/∂t = c2(∂2u/∂x2) under the boundary conditions u(0,t) = 0, u(l,t) = 0 and u(x,0) = x(l - x), 0 < x < l.
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Solve (y2-z2)p + (z2-x2)q = z(x2-y2).
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Solve z2(p2+q2) = 1.
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Applying Green's theorem, evaluate ∫(y-sin x)dx + cos x dy, where C is the plane triangle enclosed by y=0, x=π/2 and y=2x/π.
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Use Divergence theorem to evaluate ∫F.n ds over the surface of sphere x2+y2+z2=a2 where F = 3xi + 3yj + 3zk.
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Verify Gauss divergence theorem for F = (x3-yz)i - x2yj + z2k taken over the cube bounded by the planes x=y=z=a.
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