21mat21 Model Question Paper with Answers
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VTU Model Question Paper of B.E. / B.Tech 1st and 2nd …
21mat21 Model Question Paper
Visvesvaraya Technological University
Second Semester B.E./B.Tech. Examination, (Year) / (Month)
Advanced Calculus and Numerical Methods
Subject Code: 21MAT21
Time: 3 Hours
Max. Marks: 100
Note: Answer any FIVE full questions, choosing at least ONE question from each module.
Module – 1
a. Find the directional derivative of ϕ = x^2yz + 4xz^2 at the point (1,-2,-1) in the direction of the vector 2i – j – 2k. (06 Marks) b. If F = (x + y + 1)i + j – (x + y)k, show that F.curl F = 0. (07 Marks) c. Verify Green’s theorem for ∫_C [(xy + y^2)dx + x^2dy], where C is the closed curve bounded by y = x and y = x^2. (07 Marks)
a. Find the constants a, b, and c such that the vector F = (axy – z^3)i + (bx^2 + z)j + (bxz^2 + cy)k is irrotational. Also, find the scalar potential function ϕ such that F = ∇ϕ. (06 Marks) b. Using Gauss divergence theorem, evaluate ∬_S F.n dS where F = 4xi – 2y^2j + z^2k and S is the surface bounded by the region x^2 + y^2 = 4, z = 0 and z = 3. (07 Marks) c. Verify Stoke’s theorem for F = (2x – y)i – yz^2j – y^2zk, where S is the upper half surface of the sphere x^2 + y^2 + z^2 = 1 and C is its boundary. (07 Marks)
Module – 2
a. Solve ∂^2z/∂x^2 – ∂^2z/∂x∂y – 2∂^2z/∂y^2 = sin(x + 2y). (06 Marks) b. Solve the equation ∂u/∂t = a^2 ∂^2u/∂x^2 subject to the conditions: i) u is not infinite as t → ∞ ii) u = 0 when x = 0 and x = l for all t iii) u = lx – x^2 when t = 0 and 0 < x < l. (07 Marks) c. Derive one dimensional wave equation in the standard form. (07 Marks)
a. Form the partial differential equation by eliminating the arbitrary functions f and g from z = f(x + ct) + g(x – ct). (06 Marks) b. Solve (∂^2z/∂x∂y) = sin x sin y for which ∂z/∂y = -cos y when x = 0 and z = 0 when y is an odd multiple of π/2. (07 Marks) c. Obtain the solution of two-dimensional Laplace’s equation by the method of separation of variables. (07 Marks)
Module – 3
a. Find the real root of the equation x log_10 x = 1.2 by Regula-Falsi method correct to four decimal places. (06 Marks) b. Apply Gauss-Seidel iteration method to solve the equations: 20x + y – 2z = 17 3x + 20y – z = -18 2x – 3y + 20z = 25. (07 Marks) c. Find the largest eigenvalue and the corresponding eigenvector of the matrix A = [[2,-1,0],[-1,2,-1],[0,-1,2]] by power method, taking the initial eigenvector as [1, 1, 1]^T. (07 Marks)
a. Using Newton-Raphson method, find the real root of the equation 3x = cos x + 1 near x = 0.5 correct to four decimal places. (06 Marks) b. Using Relaxation method, solve the following system of linear equations: 9x – 2y + z = 50 x + 5y – 3z = 18 -2x + 2y + 7z = 19. (07 Marks) c. Using Rayleigh’s power method, find the smallest eigenvalue and the corresponding eigenvector of the matrix A = [[4,1,0],[1,20,1],[0,1,4]] by taking the initial approximation as [1, 0, 0]^T. (07 Marks)
Module – 4
- a. From the following table of values of x and y, obtain dy/dx and d^2y/dx^2 for x = 1.2.
| x | 1.0 | 1.2 | 1.4 | 1.6 | 1.8 | 2.0 | 2.2 |
|---|---|---|---|---|---|---|---|
| y | 2.7183 | 3.3201 | 4.0552 | 4.9530 | 6.0496 | 7.3891 | 9.0250 |
(06 Marks)
b. Evaluate ∫_0^6 dx/(1 + x^2) by using: i) Trapezoidal rule ii) Simpson’s 1/3 rule iii) Simpson’s 3/8 rule. (07 Marks) c. Using Euler’s modified method, solve the initial value problem dy/dx = log_e(x + y), y(0) = 2 to find y(0.2) and y(0.4) by taking h = 0.2. (07 Marks)
- a. The following table gives the values of density of saturated water for various temperatures of saturated steam.
| Temperature (°C) | 100 | 150 | 200 | 250 | 300 |
|---|---|---|---|---|---|
| Density (kg/m^3) | 958 | 917 | 865 | 799 | 712 |
Find by Newton’s divided difference formula, the density of water when the saturated steam temperature is 175°C. (06 Marks) b. Evaluate ∫_0^1 e^x dx by taking six equal strips using Weddle’s rule. (07 Marks) c. Apply Runge-Kutta method of fourth order to find an approximate value of y when x = 0.2 given that dy/dx = x + y and y = 1 when x = 0. (07 Marks)
Module – 5
a. Find by Taylor’s series method the value of y at x = 0.1 to five places of decimals from dy/dx = x^2y – 1, y(0) = 1. (06 Marks) b. Solve numerically using Milne’s predictor-corrector method the differential equation dy/dx = 1/(x + y) with initial conditions y(0) = 2, y(0.2) = 2.0933, y(0.4) = 2.1755, y(0.6) = 2.2493 to compute y(0.8). (07 Marks) c. Solve the equation ∂u/∂t = ∂^2u/∂x^2 subject to the conditions u(x, 0) = sin πx, 0 ≤ x ≤ 1; u(0, t) = u(1, t) = 0 using Bender-Schmidt method, taking h = 1/3 and k = 1/36. (07 Marks)
a. Using modified Euler’s method, find an approximate value of y when x = 0.3 given that dy/dx = x + y and y = 1 when x = 0. (06 Marks)
b. Use Milne’s method to find y(0.8) given that dy/dx = (1 + y^2)/x with y(0.4) = 1, y(0.5) = 1.1114, y(0.6) = 1.25, y(0.7) = 1.4286. (07 Marks)
c. Solve ∂^2u/∂t^2 = ∂^2u/∂x^2, 0 < x < 1, t > 0 given u(x, 0) = 0, ∂u/∂t (x, 0) = 0 for 0 < x < 1, u(0, t) =
