VTU 1st Year 21MAT21 Advanced Calculus and Numerical Methods set-2 Solved Model Question Paper with answers available on this website.
VTU 1st Year Advanced Calculus and Numerical Methods set-2 Solved Model Question Paper
Module-1
1.A] Evaluate โซ โซ ๐ฅ๐ฆ ๐๐ฅ๐๐ฆ over the region bounded by the x-axis, ordinate x=2a and the curve ๐ฅ2 = 4๐y
1.B] Find the volume bounded by the cylinder ๐ฅ2 + ๐ฆ2 = 4 and the planes ๐ฆ + ๐ง = 4 and ๐ง = 0, by using double integration
1.C] Show that \int_{0}^{\frac{\Pi}{2}}\sqrt{\sin\Theta}d\Theta X\int_{0}^{\frac{\Pi}{2}}\frac{d\Theta}{\sqrt{\sin\Theta}}=\Pi
or
2.A] Change the order of integration and hence evaluate \int_{0}^{\infty }\int_{x}^{\infty }\frac{e^{-y}}{y} dydx.
2.B] Evaluate \int_{0}^{1 }\int_{0}^{\sqrt{1-x^{2}} }\int_{0}^{\sqrt{1-x^{2}-y^{2}}} xyz dxdydz.
2.C] Show that \Gamma\left( \frac{1}{2} \right)=\sqrt{\Pi} .
Module-2
3.A] Find the angle between the surfaces ๐ฅ2 + ๐ฆ2 + ๐ง2 = 9 and ๐ง = ๐ฅ2 + ๐ฆ2 โ 3 at the point (2, โ1, 2)
3.B] If ๐นโ = ๐ฅ2๐ฆ ๐+ ๐ฆ2๐ง ๐+ ๐ง2๐ฅ ๐ , find ๐ถ๐ข๐๐(๐ถ๐ข๐๐๐นโ)
3.C] Show that ๐นโ = (๐ฆ2 โ ๐ง2 + 3๐ฆ๐ง โ 2๐ฅ)๐+ (3๐ฅ๐ง + 2๐ฅ๐ฆ)๐+ (3๐ฅ๐ฆ โ 2๐ฅ๐ง + 2๐ง)๐ is both solenoidal and irrotational.
or
4.A] If ๐นโ = (5๐ฅ๐ฆ โ 6๐ฅ2)๐+ (2๐ฆ โ โ4๐ฅ)๐, evaluate โซc ๐นโ โ ๐๐โ along the curve ๐ถ: ๐ฆ = ๐ฅ3 in the ๐ฅ๐ฆ โplane from the point (1, 1) ๐ก๐ (2, 8)
4.B] Using Greenโs theorem, evaluate โซc (๐ฅ๐ฆ + ๐ฆ2)๐๐ฅ + ๐ฅ2๐๐ฆ, where C is bounded by๐ฆ = ๐ฅ and ๐ฆ = ๐ฅ2
4.C] Using Stokeโs theorem, evaluate โฎc ๐นโ โ ๐๐โ, where ๐นโ = (๐ฅ2 + ๐ฆ2) โ 2๐ฅ๐ฆ, taken around the rectangle whose vertices are (๐, 0), (๐, ๐), (โ๐, ๐), (โ๐, 0).
Module-3
5.A] Form the partial differential equation from the relation ๐ง = ๐(๐ฅ + ๐๐ก) + ๐(๐ฅ โ ๐๐ก).
5.B] Solve \frac{\partial ^{2}z}{\partial x^{2}}+\frac{\partial z}{\partial x}-4z=0 , given that when x = 0, z = 1 and \frac{\partial z}{\partial x}=y
5.C] Derive a one-dimensional wave equation \frac{\partial ^{2}u}{\partial t^{2}}=c^{2}\frac{\partial ^{2}2u}{\partial x^{2}}
or
6.A] Form the partial differential equation from ๐(๐ฅ + ๐ฆ + ๐ง, ๐ฅ2 + ๐ฆ2 + ๐ง2) = 0
6.B] Solve \frac{\partial ^{2}z}{\partial x{\partial y}} = sin ๐ฅ sin ๐ฆ for which \frac{\partial z}{\partial x} = โ2 sin ๐ฆ when x= 0 and z = 0 when y is odd multiple of \frac{\Pi}{2} .
6.C] Solve ๐ฅ(๐ฆ2 โ ๐ง2)๐ + ๐ฆ(๐ง2 โ ๐ฅ2)๐ โ ๐ง(๐ฅ2 โ ๐ฆ2) = 0
Module-4
7.A] Find a real root of ๐ฅ3 โ 9๐ฅ + 1 = 0 ๐๐ (2, 3) by the Regula-Falsi method in four iterations.
7.B] Using Newtonโs forward interpolation find y at x = 5 from the data:-
| x | 4 | 6 | 8 | 10 |
| y | 1 | 3 | 8 | 16 |
7.C] Evaluate \int_{0}^{\frac{\Pi}{2}}\sqrt{\cos\Theta}d\Theta by taking 7 ordinates by Simpsonโs 1/3 rd rule.
or
8.A] Using the Newton-Raphson method, find the root of 3๐ฅ = ๐๐๐ ๐ฅ + 1 correct four decimal places.
8.B] Using Newtonโs divided difference interpolation find ๐(9),Given that :-
| x | 5 | 7 | 11 | 13 | 17 |
| y | 150 | 392 | 1452 | 2366 | 5202 |
8.C] Evaluate \int_{0}^{1}\frac{dx}{1+x^{2}} , using Simpsonโs (3/8)๐กโ rule by taking 7 ordinates.
Module-5
9.A] Use the Taylor series method to find ๐ฆ(0.2) from \frac{dy}{dx}=x^{2}y-1, ๐ค๐๐กโ ๐ฆ(0) = 1.
9.B] By using modified Eulerโs method, find y(0.2), taking h=0.1 from \frac{dy}{dx}=\frac{y-x}{y+x} , with ๐ฆ(0) = 1.
9.C] Applying Milneโs Predictor-Corrector method, find y(0.8), from \frac{dy}{dx}=x^{3}+y , given that ๐ฆ(0) = 2, ๐ฆ(0.2) = 2.073 , ๐ฆ(0.4) = 2.452, ๐ฆ(0.6) = 3.023.
or
10.A] Employ Taylorโs series method to evaluate ๐ฆ(0.2),๐ก๐๐๐๐๐ โ = 0.1 from \frac{dy}{dx}= ๐๐ฅ โ ๐ฆ2, ๐ค๐๐กโ ๐ฆ(0) = 1.
10.B] Using the Runge-Kutta method of order 4, find y at x = 0.1, given that \frac{dy}{dx}= 3๐๐ฅ + 2๐ฆ, ๐ฆ(0) = 1.
10.C] Applying Milneโs Predictor โ Corrector method, to find y(1.4), from \frac{dy}{dx}= x^{2}+\frac{y}{2} given that ๐ฆ(1) = 2, ๐ฆ(1.1) = 2.2156, ๐ฆ(1.2) = 2.4549, ๐ฆ(1.3) = 2.7514.
[โฆ] Advanced Calculus and Numerical Methods โ Set 1Advanced Calculus and Numerical Methods โ Set 2 [โฆ]
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[โฆ] VTU 1st Year Advanced Calculus and Numerical Methods [SET-2] Solved Model Question Paper [โฆ]