VTU 1st Year 21MAT21 Advanced Calculus and Numerical Methods Solved Model Question Paper with answers available on this website.
VTU 1st Year Advanced Calculus and Numerical Methods Solved Model Question Paper
Module-1
1.A] Evaluate \int_{-1}^{1}\int_{0}^{z}\int_{x-z}^{x+z}\text{(π₯ + π¦ + π§)ππ₯ππ¦ππ§).}
1.B] Evaluate \int_{0}^{a}\int_{y}^{a}\frac{x}{x^{2}+y^{2}} by changing the order of Integration.
1.C] Derive the relation between Gamma and Beta functions.
or
2.A] Evaluate \int_{0}^{1}\int_{0}^{\sqrt{1-y^{2}}} (x2+y2 )dxdy by changing into polar coordinates.
2.B] Using double integration find the area between the parabolas π¦2 = 4ππ₯ and π₯2 = 4πy.
2.C] Using beta and gamma functions, evaluate \int_{0}^{\frac{\Pi}{2}}\sqrt{\tan\Theta}d\Theta
Module-2
3.A] Find the directional derivative of β = π₯2π¦π§ + 4π₯π§2 at the point (1, β2,1) in the direction of the vector 2πβπβ2π
3.B] Find πππ£ πΉβ and ππ’ππ πΉβ, where πΉβ = ππππ(π₯3 + π¦3 + π§3 β 3π₯π¦π§).
3.C] Define an irrotational vector. Find the constants π, π and π such that πΉβ = (ππ₯π¦ β π§3)π+ (ππ₯2 + π§)π+(ππ₯π§2+ππ¦)π) is irrotational.
or
4.A] Find the work done in moving a particle in the force field πΉβ = 3π₯2π+ (2π₯π§ β π¦)π+ π§π along the straight line from (0,0,0) to (2,1,3).
4.B] Apply Greenβs theorem to evaluate β«c[(3π₯ β 8π¦2)ππ₯ + (4π¦ β 6π₯π¦)ππ¦] πΆ, where C is the boundary of the region bounded by π₯ = 0, π¦ = 0, π₯ + π¦ = 1
4.C] Apply Stokeβs theorem to evaluate β¬ ππ’ππ πΉβ β πππ , where πΉβ = (π₯2+π¦2)πβ 2π₯π¦π, taken around the rectangle bounded by the lines π₯ = Β±π, π¦ = 0, π¦ = b.
Module-3
5.A] Form the partial differential equation by eliminating the arbitrary function from the relation ππ₯ + ππ¦ + ππ§ = π(π₯2 + π¦2 + π§2).
5.B] Solve \frac{\partial^{2} z}{\partial x^{2}} = π₯π¦ subject to the conditions \frac{\partial z}{\partial x} = log(1 + π¦), when π₯ = 1 and π§ = 0, when π₯ = 0.
5.C] With usual notations derive a one-dimensional heat equation.
or
6.A] Form the partial differential equation by eliminating the arbitrary constants from (π₯ β π)2 + (π¦ β π)2 = 4
Get Answer
6.B] Solve π₯2(π¦ β π§)π + π¦2((π§ β π₯)π = π§2(π₯ β π¦)
6.C] Solve \frac{\partial^{2} z}{\partial x^{2}} = Z given that when y=0, z=ex and \frac{\partial z}{\partial x} = e-x .
Module-4
7.A] Find the root of the equation π₯ππ₯ = πππ π₯ which lies in the interval (0, 1) by Regula-Falsi method correct to four decimal places
7.B] Using Newtonβs backward interpolation formula find the value of y when π₯ = 6 from the given table
| x | 1 | 2 | 3 | 4 | 5 |
| y | 1 | -1 | 1 | -1 | 1 |
7.C] Evaluate \int_{0}^{5}\frac{dx}{4x+5} by Simpsonβs 1/3 rd rule, dividing the interval into 10 equal parts
or
8.A] By Newtonβs-Raphson method find the root of π₯ sin π₯ + cos π₯ = 0 which is near to x = Ο
8.B] Using Lagrangeβs interpolation formula, fit a polynomial which passes through the points (β1, 0), (1, 2), (2, 9) and (3, 8) and hence estimate the value of y when π₯ = 2.2
8.C] Evaluate \int_{4}^{5.2}\log\text{x}dx using Simpsonβs (3/8)th rule by taking 7 ordinates.
Module-5
9.A] Using the Taylor series method find the approximate value of π¦(0.1), from \frac{d y}{dx} , with π¦(0) = 1
9.B] Apply the Runge-Kutta method to find π¦(0.1), if \frac{d y}{dx} = \frac{y^{2}-x^{2}}{y^{2}+x^{2}} , with y(0)=1.
9.C] Using Milneβs Predictor-Corrector method, find π¦(4.5), given \frac{d y}{dx}=\frac{2-y^{2}}{5x} and π¦(4.1) = 1.0049, π¦(4.2) = 1.0097, π¦(4.3) = 1.0143, π¦(4.4) = 1.0187
or
10.A] Using Modified Eulerβs method find y(0.1), taking h = 0.05, given that \frac{d y}{dx}=x^{2}+y , π€ππ‘β π¦(0) = 1.
10.B] Using the Runge-Kutta method of order 4, find \frac{d y}{dx}=3x+\frac{y}{2},y(0)=1
10.C] Given \frac{d y}{dx}=\frac{(1+x^2)y^2}{2} , π¦(0) = 1, π¦(0.1) = 1.06, π¦(0.2) = 1.12, π¦(0.3)=1.21, evaluate π¦(0.4) by using Milneβs predictor-corrector method.
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21MAT21 (Set 2) :
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In 6 a question make RHS side z instead of 4.
Then form pde.
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