21mat21 Model Question Paper with Answers

21mat21 Model Question Paper with Answers

If you are looking for “21mat21 Model Question Paper” then these are the top 10 pages, where you can get your previous years Question Paper “with Answers” at ease.

Model Question Paper-I with effect from 2021

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Model Question Paper-I with effect from 2021. USN. Second Semester B.E Degree Examination. Advanced Calculus and Numerical Methods (21MAT21). TIME: 03 Hours.

VTU Advanced Calculus and Numerical Methods Question …

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Download VTU Advanced Calculus and Numerical Methods of 2nd semester Physics Cycle with subject code 21MAT21 2021 scheme Question Papers.

model question paper ug & pg

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Model Question paper B.E/B.Tech/B.Arch Program 1st & 2nd semester (2021 scheme ). 1. 21MAT11 Calculus and Linear Algebra.

VTU Model Question Paper of B.E. / B.Tech 1st and 2nd …

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25 Jan 2022 — VTU Model Question Paper of B.E. / B.Tech 1st … 21MAT21 Advanced Calculus and Numerical Methods … pdf. 210.7 KB · Views: 3,752. 21ELN24set2.pdf.

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

  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)

  2. 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

  1. 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)

  2. 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

  1. 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)

  2. 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

  1. 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)

  1. 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

  1. 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)

  2. 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) =

 


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21MAT21 Advanced Calculus and Numerical Methods

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21MAT21 – Solution of Model Question Paper – II

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21MAT21– solution of model question paper -II (1) – Free download as PDF File (.pdf) or read online for free. Question paper.