Solve your IGNOU Doubts

Question:

Obtain an approximate value of $\int_0^1\frac{dx}{1+x^2}$ using composite Simpson' 's rule with h = 0.25 and h=0.125. Find also the improved value using Romberg integration.

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Question:

The solution of the system of equations $\begin{pmatrix}1&2\\2&1\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}4\\-2\end{pmatrix}$ is attempted by the Gauss Jacobi and Gauss Seidel iteration schemes. Set up the two schemes in matrix form. Will the iteration schemes converge? Justify your answer.

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Question:

Starting with $x^{(0)}=\begin{bmatrix}1&1&1\end{bmatrix}^T,$ find the dominant eigenvalue and corresponding eigenvector for

$A=\begin{bmatrix}4&-1&1\\4&-8&1\\-2&1&5\end{bmatrix}$ the using the power method.

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Question:

Solve the system of equations

$\begin{aligned}8x_1-x_2+2x_3&=4\\-3x_1+11x_2-x_3+3x_4&=23\\-x_2+10x_3-x_4&=-13\\-2x_1+x_2-x_3+8x_4&=13\end{aligned}$

with $x^{(0)}=\begin{bmatrix}0&0&0&0\end{bmatrix}^T,$ , by using the Gauss Jacboi and Gauss Seidel method. The exact solution of the system is $x=\begin{bmatrix}1&2&-1&1\end{bmatrix}^T.$ Perform the required number of iterations so that the same accuracy is obtained by both the methods. What conclusions can you draw from the results obtained?

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Question:

Solve the following linear system Ax=b of equations with partial pivoting

$\begin{aligned}x_1-x_2+3x_3&=3\\2x_1+x_2+4x_3&=7\\3x_1+5x_2-2x_3&=6.\end{aligned}$

Store the multipliers and also write the pivoting vectors.

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Question:

Find the inverse of the matrix $x\begin{bmatrix}3&1&2\\-2&3&-5\\1&2&4\end{bmatrix}$ using Gauss Jordan Method.

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Question:

Solve the system of equations

$\begin{aligned}2x_1+3x_2+4x_3+x_4&=3\\x_1+2x_2+x_4&=2\\2x_1+3x_2+x_3-x_4&=1\\x_1-2x_2-x_3+4x_4&=5\end{aligned}$

using Gauss elimination method with pivoting.

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Question:

Using x₀=0 find an approximation to one of the zeros of x³-4x+1=0 by using Birge- Vieta Method. Perform two iterations.

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Question:

Find an approximate value of the positive real root of xe" -1 using graphical method. Use this value to find the positive real root of xe^x=1 correct to three decimal places by fixed point iteration method.

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Question:

Using Maclaurin's expansion for sin x, find the approximate value of $\sin\frac{\pi}{4}$ with the error 4 bound 10^-5

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Question:

Find an appropriate root of $x^3+2x^2-5=0\quad\text{in}[1,2]\text{with}10^{-5}$ accuracy by

i) Newton Raphson Method

ii) Secant Method

What conclusions can you draw from here about the two methods?

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Question:

The equation x³-x-1=0 has a positive root in the interval ]1, 2[. Write a fixed point iteration method and show that it converges. Starting with initial approximation x₀ = 1.5 find the root of the equation correct to three decimal places.

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Question:

जाँच कीजिए कि अंतराल [2,5] और [7,10] तुल्य हैं या नहीं।

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Question:

दिखाइए कि अनुक्रम (a_n), एकदिष्ट है, जहाँ $a_n\frac{n}{n^2+4}1$ उत्तर की पुष्टि कीजिए। । क्या (a_n) कॉशी अनुक्रम है? अपने

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Question:

समाकलों के लिए बोनट के माध्य मान प्रमेय का कथन दीजिए

$\left|\int_{3}^{5}\frac{\cos x}{x}dx\right|\leq\frac{2}{3}$

दिखाने के लिए इसे लागू कीजिए।

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Question:

जाँच कीजिए कि :

$G'=\left\{\left]\frac{1}{n+2},\frac{1}{n}\right[:n\in\mathbb{N}\right\}$

द्वारा दिया गया संग्रह G, ]0,1[. का विवृत आवरक है या नहीं।

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Question:

जाँच कीजिए :

f (x) = (x-s)³ (x+1)²

द्वारा दिए गए फलन f का स्थानीय उच्चिष्ठ और स्थानीय निम्निष्ठ होता है या नहीं।

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Question:

निरपेक्ष और सप्रतिबंध अभिसरण के लिए श्रेणी

$\sum_{n=1}^{\infty}(-1)^{n-1}\frac{\sin nx}{n\sqrt{n}}$

की जाँच कीजिए।

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Question:

ऐसी श्रेणी $\sum a_n$ का उदाहरण दीजिए जिसके लिए $\sum a_n$ , अभिसारी नहीं है लेकिन अनुक्रम (a_n) 0 की ओर अभिसरण करता है।

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Question:

दिखाइए कि वक्र y = 3x²-7x+6, पर बिन्दुओं को, जिनकी भुज x = 1 और x = 2, हैं, मिलाने वाली जीवा उस बिन्दु पर खींची गई स्पर्श रेखा के समान्तर होती है जिसकी भुजा $x=\frac{3}{2}.$