JNTU
JNTU B.Tech I Year MATHEMATICAL METHODS Previous Question Paper (March/April - 2023 - R15)
Note: i) Question paper consists of Part A, Part B. ii) Part A is compulsory, which carries 25 marks. In Part A, Answer all questions. iii) In Part B, Answer any one question from each unit. Each question carries 10 marks and may have a, b as sub questions.
Part A
Show that E - 1 = (1 - ∇)-1 - 1.
Answer
The shift operator E is defined by E f(x) = f(x + h) and the backward difference operator ∇ is defined by ∇ f(x) = f(x) - f(x - h). From the definition of ∇, we have ∇ = 1 - E-1. Rearranging this gives E-1 = 1 - ∇, which implies E = (1 - ∇)-1. Therefore, E - 1 = (1 - ∇)-1 - 1.
If f(x) = sin(px + q), find Δ2f(x).
Answer
Given the function f(x) = sin(px + q), the forward difference operator Δ is defined as Δf(x) = f(x + h) - f(x), where h is the interval of differencing. Assuming h = 1 for standard finite difference notation:
Using the trigonometric identity sin(A) - sin(B) = 2 cos((A + B)/2) sin((A - B)/2):
Applying the operator again to find the second difference Δ2f(x) = Δ(Δf(x)):
Using the identity cos(A) - cos(B) = -2 sin((A + B)/2) sin((A - B)/2):
If the first two approximations for the root of x3 - x2 + 1 = 0 are -1 and 0, find the next two approximations using Regula falsi method.
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The values of x and f(x) are given below. Find ∫0π/2 f(x)dx.
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If f(x) = |sinx| in (-L, L), then find the Fourier coefficient bn.
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Find the finite Fourier cosine transform of f(x) = x2, 0 < x < 4.
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Find the general solution of pqz = p2(qx + p2) + q2(py + q2).
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Solve px + qy = 1 using the method of separation of variables.
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If Φ = ax2 + by2 + cz2 satisfies Laplacian equation, then find the value of a + b + c.
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If F = axi + byj + czk where a, b, c are constants evaluate ∬ F.n ds where S is the surface of the unit sphere.
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Part B
Find f(2.4) from the following data using appropriate interpolation method.
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Fit a second degree polynomial to the following data by the method of least squares.
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Using Lagrange’s interpolation formula find y(8) from the table.
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Fit y = abx by the method of least squares to the data provided.
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Solve the system 10x-2y-z-w=3; -2x+10y-z-w=15; -x-y+10z-2w=15; -x-y-2z+10w= -9 using Gauss Seidel method.
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Find the root of the equation x log10(x) = 1.2 using False position method.
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Evaluate ∫-20 e-x2 dx using Simpson’s rule taking h = 0.25.
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Using Milne’s method, find y(4.5), given 5xy' + y2 - 2 = 0, y(4)=1, y(4.1)= 1.0049, y(4.2)= 1.0097, y(4.3) = 1.0143 and y(4.4)= 1.0187.
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Find the Fourier series of f(x) = (π-x)2/2 in the interval 0 < x < 2π.
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Find the finite Fourier sine transform of f(x) = x, 0 < x < π.
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Obtain the Fourier cosine series for f(x) = x sin x, 0 < x < π.
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Find the Fourier transform of f(x) = 1/|x|.
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Find the partial differential equation arising from Φ(x2+y2+z2, x+y+z) = 0.
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Solve p(z-y) = (z-x)(y+2)q.
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Solve 4zx - zy = 3z, z(0, y) = ey by the method of separation of variables.
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Solve uxx + uyy = 0 subject to the boundary conditions.
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Find the angle of intersection of the spheres x2 + y2 + z2 = 39 and x2 + y2 + z2 + 4x - 6y - 8z + 52 = 0 at the point (4, -3, 2).
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Evaluate ∫C F.dr where F = 3xyi - y2j and C is the curve y = 2x2 in xy-plane from (0, 0) to (1, 2).
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Find the directional derivative of xyz + xz2 at (1, 1, 1) in a direction of the normal to the surface xy + y = z3 at (0, 1, 1).
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Evaluate ∬S F.n ds, if F = yz2i + 2j + xz2k and S is the surface of the cylinder x2 + y2 = 9 contained in the first octant between the planes z = 0 and z = 2.
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