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}}} (x^{2}+y^{2} )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₯^{2}π¦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₯^{2}π+ (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=e^{x} and \frac{\partial z}{\partial x} = e^{-x} .

**Module-4**

7.A] Find the root of the equation π₯π^{π₯} = 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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