IGNOU MTE 10 SOLVED ASSIGNMENT HINDI

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IGNOU MTE 10 SOLVED ASSIGNMENT HINDI

~~Rs. 80~~
Rs. 41

IGNOU MTE 10 SOLVED ASSIGNMENT HINDI

~~Rs. 80~~
Rs. 41

Last Date of Submission of IGNOU MTE-010 (BSC) 2025 Assignment is for January 2025 Session: 30th September, 2025 (for December 2025 Term End Exam).
Semester Wise
January 2025 Session: 30th March, 2025 (for June 2025 Term End Exam).
July 2025 Session: 30th September, 2025 (for December 2025 Term End Exam).

Title Name	IGNOU MTE 10 SOLVED ASSIGNMENT HINDI
Type	Soft Copy (E-Assignment) .pdf
University	IGNOU
Degree	BACHELOR DEGREE PROGRAMMES
Course Code	BSC
Course Name	Bachelor in Science
Subject Code	MTE 10
Subject Name	Numerical Analysis
Year	2025
Session
Language	English Medium
Assignment Code	MTE-010/Assignmentt-1//2025
Product Description	Assignment of BSC (Bachelor in Science) 2025. Latest MTE 010 2025 Solved Assignment Solutions
Last Date of IGNOU Assignment Submission	Last Date of Submission of IGNOU MTE-010 (BSC) 2025 Assignment is for January 2025 Session: 30th September, 2025 (for December 2025 Term End Exam). Semester Wise January 2025 Session: 30th March, 2025 (for June 2025 Term End Exam). July 2025 Session: 30th September, 2025 (for December 2025 Term End Exam).

~~Rs. 80~~
Rs. 41

Questions Included in this Help Book

Ques 1.

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.

Ques 2.

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?

Ques 3.

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

Ques 4.

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.

Ques 5.

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

Ques 6.

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.

Ques 7.

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

Ques 8.

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.

Ques 9.

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?

Ques 10.

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.

Ques 11.

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.

Ques 12.

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.

Ques 13.

Find the minimum number of intervals required to evaluate $\int_{0}^{1}e^{-x^2}dx$ dx with an accuracy of $\frac{1}{2}\times 10^{-4},$ by using the Trapezoidal.rule

Ques 14.

From the following table, find the number of students who obtained less than 45 marks.

$\begin{array}{|c|c|}\hline\text{Marks}&\text{No.of Students}\\\hline 30-40&31\\40-50&42\\50-60&51\\60-70&35\\70-80&31\\\hline\end{array}$

Ques 15.

Calculate the third-degree Taylor polynomial about $x_0=0\quad\text{for}f(x)=(1+x)^{1/2}.$

Ques 16.

Use the polynomial in part (a) to approximate $\sqrt[]{1.1}$ and find a bound for the error involved.

Ques 17.

Use the polynomial in part (a) to approximate $\int_{0}^{0.1}(1+x)^{1/2}dx.$

Ques 18.

Using sin(0.1)=0.09983 and sin(0.2)=0.19867, find an approximate value of sin(0.15) by using Lagrange interpolation. Obtain a bound on the truncation error.

Ques 19.

Consider the following data

$\begin{array}{|c|c|c|c|c|c|}\hline x&1.0&1.3&1.6&1.9&2.2\\\hline f(x)&0.7651977&0.6200860&0.4554022&0.2818186&0.1103623\\\hline\end{array}$

Use Stirling's formula to approximate f(1.5) with x_o 1.6.

Ques 20.

Solve the $\text{I.V.P.,}y'=-y+t+1,\quad 0\le t\le 1,\quad y(0)=1$ using R-K method of 0(h⁴) with h = 0.1 and obtain the value of y(0.2). Also find the error at t=0.2, if the exact solution is y(t)-t+e.

Ques 21.

The position f(x) of a particle moving in a line at various times x_k, is given in the following table. Estimate the velocity and acceleration of the particle at x = 1.2

$\begin{array}{|c|c|c|c|c|c|c|c|}\hline x&1.0&1.2&1.4&1.6&1.8&2.0&2.2\\\hline f(x)&2.72&3.32&4.06&4.96&6.05&7.39&9.02\\\hline\end{array}$

Ques 22.

A solid of revolution is formed by rotating about the x-axis the area bounded between x=0, x=1 and the curve given by the table

x	0	0.25	0.5	0.75	1.0
f(x)	1.0	0.9896	0.9587	0.9089	0.8415

Find the volume of the solid so formed using

i) Trapezodial rule

ii) Simpson's rule

Ques 23.

Take 10 figure logarithm to base 10 from x=300 to x=310 by unit increment. Calculate the first derivative of logx when x=310.

Ques 24.

For the table of values of f (x)=xe^x given by

$\begin{array}{|c|c|c|c|c|c|}\hline x&1.8&1.9&2.0&2.1&2.2\\\hline f(x)&10.8894&12.7032&14.7781&17.1489&19.8550\\\hline\end{array}$

Find f"(2.0) using the central difference formula of 0(h²) for h=0.1 and h=0.2. Calculate T.E. and actual erroг.

Ques 25.

Suppose f_n denotes the value of $f(t)\text{at}t=t_n.\text{If}f(t)=t^3$ then find the value of $\frac{(f_{n+1}-2f_n+f_{n-1})}{h^2}.$

Ques 26.

Use Runge-Kutta method of order four to solve y'=x+y. Start with x 1, y=0 and carry to x=1.5 with h = 0.1.

Ques 27.

Find the solution of the difference equation $y_{k+2}-4y_{k+1}+4y_k=0,\quad k=0,1,\ldots$ Also find the particular solution when y₀. =1 and y₁ =6.

Ques 28.

The iteration method $x_{n+1}=\frac{1}{8}\left[6x_n+\frac{3N}{x_n}-\frac{x_n^3}{N}\right],\quad n=0,1,2$ where N is positive constant, converges to some quantity. Determine this quantity. Also find the rate of convergence of this method.

Ques 29.

Determine the spacing h in a table of equally spaced values for the function

$f(x)=(2+x)^4,\quad 1\le x\le 2,$ so that the quadratic interpolation in this table satisfies | error |≤10^-6

Ques 30.

Determine a unique polynomial f(x) of degree ≤3 such that $f(x_0)=1,f'(x_0)=2,f(x_1)=2,f'(x_1)=3\quad\text{where}x_1-x_0=h.$

Ques 31.

समीकरण x³-x-1=0 का एक धन मूल, अंतराल ]1, 2 [ में है। नियत बिन्दु पुनरावृत्ति विधि लिखिए और यह दिखाइए कि यह अभिसरित होती है। प्रारंभिक सन्निकटन x_o = 1.5 से प्रारंभकरके समीकरण का तीन दशमलव स्थान तक की परिशुद्धता का मूल ज्ञात कीजिए।

ख) i) न्यूटन रैफ्सन विधि

ii) छेदिका विधि

से अंतराल [1, 2] में समीकरण x³+2x²-5=0 का 10^-5 तक की परिशुद्धता वाला एक उपयुक्त मूल ज्ञात कीजिए। यहां आप इन दो विधियों के संबंध में क्या निष्कर्ष निकाल सकते हैं?

Ques 32.

sin x के लिए मैकलारिन प्रसार को लागू करके त्रुटि परिबंध 10^-5 तक $\sin\frac{\pi}{4}$ का सन्निकट मान ज्ञात कीजिए।

Ques 33.

ग्राफीय विधि से xe^x = 1 के धन वास्तविक मूल का सन्निकट मान ज्ञात कीजिए। इस मान का प्रयोग करके नियत बिन्दु पुनरावृत्ति विधि से तीन दशमलव स्थान तक परिशुद्ध xe^x = 1 का धन वास्तविक मूल ज्ञात कीजिए।

Ques 34.

x_o = 0 को एक सन्निकटन मानकर बर्ज-विएटा विधि की दो पुनरावृत्त्तियाँ करके x³- 4x + 1 = 0 के शून्यकों (zeros) में से एक शून्यक का सन्निकटन ज्ञात कीजिए।

Ques 35.

कीलकन के साथ गाउस विलोपन विधि लागू करके निम्नलिखित समीकरण-निकाय को हल कीजिए।

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

Ques 36.

गाउस-जॉर्डन विधि से आव्यूह

$\begin{bmatrix}3&1&2\\-2&3&-5\\1&2&4\end{bmatrix}$

का व्युत्क्रम ज्ञात कीजिए।

Ques 37.

आंशिक कीलकन द्वारा निम्नलिखित रैखिक समीकरण निकाय Ax=b का हल ज्ञात कीजिए।

$x_1-x_2+3x_3=3\\$

$2x_1+x_2+4x_3=7\\$

$3x_1+5x_2-2x_3=6$

गुणकों को संचित कीजिए और कीलकन सदिश भी लिखिए

Ques 38.

x^(o) = [0 0 0 0]^T लेकर गाउस-जैकोबी और गाउस-सीडल विधि से निम्नलिखित समीकरण निकाय का हल ज्ञात कीजिए।

$8x_1-x_2+2x_3=4\\$

$-3x_1+11x_2-x_3+3x_4=23\\$

$-x_2+10x_3-x_4=-13\\$

$-2x_1+x_2-x_3+8x_4=13$

इस निकाय का यथातथ हल x = [1 2-1 1]^T है। अपेक्षित संख्या में पुनरावृत्तियाँ कीजिए जिससे कि दोनों विधियों से समान परिशुद्धता प्राप्त हो। प्राप्त किए गए परिणाम से आप क्या निष्कर्ष निकाल सकते हैं?

Ques 39.

x^(o) = [1 1 1]^T से प्रारंभ करके घात विधि द्वारा निम्नलिखित आव्यूह का प्रमुख आइगनमान और संगत आइगनसदिश ज्ञात कीजिए।

$A=\begin{bmatrix}4&-1&1\\4&-8&1\\-2&1&5\end{bmatrix}$

Ques 40.

गाउस-जैकोबी और गाउस-सीडल पुनरावृत्ति योजनाओं से समीकरण निकाय $\begin{pmatrix}1&2\\2&1\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}4\\-2\end{pmatrix}$ का हल प्राप्त किया गया है। आव्यूह रूप में दोनों योजनाएं स्थापित कीजिए। क्या पुनरावृत्ति योजनाएं अभिसारित होती हैं? तर्क के साथ अपने उत्तर की पुष्टि कीजिए।

Ques 41.

h = 0.25 और h = 0.125 लेकर संयुक्त सिम्प्सन नियम लागू करके $\int_0^1\frac{dx}{1+x^2}$ का सन्निकट मान प्राप्त कीजिए। रॉम्बर्ग समाकलन की सहायता से प्राप्त मान में सुधार कीजिए।

Ques 42.

समलंबी नियम की सहायता से $\frac{1}{2}\times 10^{-4}$ की परिशुद्धता तक $\int_0^1 e^{-x^2}dx$ का मान निकालने के लिए आवश्यक अंतरालों की न्यूनतम संख्या ज्ञात कीजिए।

Ques 43.

नीचे दी गई तालिका से उन छात्रों की संख्या ज्ञात कीजिए जिन्होंने 45 से कम अंक प्राप्त किए हैं।

अंक	छात्रों की संख्या
30-40	31
40-50	42
50-60	51
60-70	35
70-80	31

Ques 44.

sin(0.1) = 0.09983 और sin(0.2) = 0.19867 9867 लेकर लग्राज अंतर्वेशन विधि से sin(0.15) का एक सन्निकट मान ज्ञात कीजिए। रूंडन त्रुटि पर एक परिबंध प्राप्त कीजिए।

Ques 45.

निम्नलिखित आंकड़ों के लिए

x	1.0	1.3	1.6	1.9	2.2
f(x)	0.7651977	0.6200860	0.4554022	0.2818186	0.1103623

स्टर्लिंग सूत्र लागू करके x₀ = 1.6 के लिए f(1.5) का सन्निकट मान ज्ञात कीजिए।

Ques 46.

h = 0.1 के लिए 0(h⁴) की रूंगे कुट्टा विधि लागू करके आदि मान समस्या y' = -y + t + 1, 0 ≤ t ≤1, y(0) = 1 को हल कीजिए और y(0.2) का मान प्राप्त कीजिए। यदि यथातथ हल y(t) = t + e^-t हो तो t = 0.2 पर त्रुटि भी ज्ञात कोजिए।

Ques 47.

विभिन्न समयों x_k पर एक रेखा में गतिमान कण की स्थिति f(x) नीचे की तालिका में दी गई है। x = 1.2 पर कण का वेग और त्वरण आकलित कीजिए।

x	1.0	1.2	1.4	1.6	1.8	2.0	2.2
f(x)	2.72	3.32	4.06	4.96	6.05	7.39	9.02

Ques 48.

x = 0, x = 1 और नीचे की तालिका से प्राप्त वक्र से परिबद्ध क्षेत्र को x-अक्ष के प्रति घूर्णन कराने पर एक परिक्रमण घनाकृति प्राप्त होती है।

x	0	0.25	0.5	0.75	1.0
f(x)	1.0	0.9896	0.9587	0.9089	0.8415

इस प्रकार प्राप्त हुई घनाकृति का आयतन

i) समलंबी नियम और

ii) सिम्प्सन नियम से ज्ञात कीजिए।

Ques 49.

इकाई वृद्धि करके x = 300 से x = 310 तक आधार 10 पर 10 लघुगणक लीजिए। log₁₀ x का प्रथम अवकलज परिकलित कीजिए जबकि x = 310 हो।

Ques 50.

f(x) = xe^x के मानों की निम्नलिखित तालिका से h = 0.1 और h = 0.2 पर 0(h²) का केन्द्रीय अंतर सूत्र लागू करके f" (2.0) ज्ञात कीजिए। रूंडन त्रुटि और वास्तविक त्रुटि परिकलित कीजिए।

x	1.8	1.9	2.0	2.1	2.2
f(x)	10.8894	12.7032	14.7781	17.1489	19.8550

Ques 51.

मान लीजिए f_n, t = t_n पर f (t) के मान को निरूपित करता है। यदि f(t) = t³ हो तो $\frac{(f_{n+1}-2f_n+f_{n-1})}{h^2}$ का मान प्राप्त कीजिए।

Ques 52.

h=0.1 के लिए x = 1, y = 0 से आरंभ करके x = 1.5 तक समीकरण y' = x + y का हल कोटि चार की रूंगे-कुट्टा विधि द्वारा प्राप्त कीजिए।

Ques 53.

अंतर समीकरण y_k+2- 4y_k+1+ 4y_k = 0, k=0,1,... का हल ज्ञात कीजिए। y_o = 1 और y₁ = 6 के लिए विशेष हल भी ज्ञात कीजिए। (2)

Ques 54.

पुनरावृत्ति विधि

$x_{n+1}=\frac{1}{8}\left[6x_n+\frac{3N}{x_n}-\frac{x_n^3}{N}\right],\quad n=0,1,2$

जहां Nएक धन अचर है, एक परिमाण की ओर अभिसरित होती है। यह परिमाण ज्ञात कीजिए। इस विधि की अभिसरण दर भी ज्ञात कीजिए।

Ques 55.

फलन f(x) = (2+x)⁴, 1 ≤ x ≤ 2 के समदूरी मानों की तालिका से एक ऐसा अंतर hज्ञात कीजिए जिससे कि इस तालिका में द्वितीय घात अंतर्वेशन | त्रुटि | ≤10^-6 को संतुष्ट करता हो।

Ques 56.

घात ≤ 3 वाला वह अद्वितीय बहुपद f(x) ज्ञात कीजिए जिसके लिए

$f(x_0)=1,f'(x_0)=2,f(x_1)=2,f'(x_1)=3\text{$ हो जहां $x_1-x_0=h$

Ques 57.

Ques 58.

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?

Ques 59.

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

Ques 60.

Ques 61.

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

Ques 62.

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.

Ques 63.

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

Ques 64.

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.

Ques 65.

Solve the system of equations

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

Ques 66.

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.

Ques 67.

Ques 68.

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.

Ques 69.

Find the minimum number of intervals required to evaluate $\int_{0}^{1}e^{-x^2}dx$ dx with an accuracy of $\frac{1}{2}\times 10^{-4},$ by using the Trapezoidal.rule

Ques 70.

From the following table, find the number of students who obtained less than 45 marks.

$\begin{array}{|c|c|}\hline\text{Marks}&\text{No.of Students}\\\hline 30-40&31\\40-50&42\\50-60&51\\60-70&35\\70-80&31\\\hline\end{array}$

Ques 71.

Calculate the third-degree Taylor polynomial about $x_0=0\quad\text{for}f(x)=(1+x)^{1/2}.$

Ques 72.

Use the polynomial in part (a) to approximate $\sqrt[]{1.1}$ and find a bound for the error involved.

Ques 73.

Use the polynomial in part (a) to approximate $\int_{0}^{0.1}(1+x)^{1/2}dx.$

Ques 74.

Using sin(0.1)=0.09983 and sin(0.2)=0.19867, find an approximate value of sin(0.15) by using Lagrange interpolation. Obtain a bound on the truncation error.

Ques 75.

Consider the following data

$\begin{array}{|c|c|c|c|c|c|}\hline x&1.0&1.3&1.6&1.9&2.2\\\hline f(x)&0.7651977&0.6200860&0.4554022&0.2818186&0.1103623\\\hline\end{array}$

Use Stirling's formula to approximate f(1.5) with x_o 1.6.

Ques 76.

Ques 77.

The position f(x) of a particle moving in a line at various times x_k, is given in the following table. Estimate the velocity and acceleration of the particle at x = 1.2

$\begin{array}{|c|c|c|c|c|c|c|c|}\hline x&1.0&1.2&1.4&1.6&1.8&2.0&2.2\\\hline f(x)&2.72&3.32&4.06&4.96&6.05&7.39&9.02\\\hline\end{array}$

Ques 78.

A solid of revolution is formed by rotating about the x-axis the area bounded between x=0, x=1 and the curve given by the table

x	0	0.25	0.5	0.75	1.0
f(x)	1.0	0.9896	0.9587	0.9089	0.8415

Find the volume of the solid so formed using

i) Trapezodial rule

ii) Simpson's rule

Ques 79.

Take 10 figure logarithm to base 10 from x=300 to x=310 by unit increment. Calculate the first derivative of logx when x=310.

Ques 80.

For the table of values of f (x)=xe^x given by

$\begin{array}{|c|c|c|c|c|c|}\hline x&1.8&1.9&2.0&2.1&2.2\\\hline f(x)&10.8894&12.7032&14.7781&17.1489&19.8550\\\hline\end{array}$

Find f"(2.0) using the central difference formula of 0(h²) for h=0.1 and h=0.2. Calculate T.E. and actual erroг.

Ques 81.

Suppose f_n denotes the value of $f(t)\text{at}t=t_n.\text{If}f(t)=t^3$ then find the value of $\frac{(f_{n+1}-2f_n+f_{n-1})}{h^2}.$

Ques 82.

Use Runge-Kutta method of order four to solve y'=x+y. Start with x 1, y=0 and carry to x=1.5 with h = 0.1.

Ques 83.

Find the solution of the difference equation $y_{k+2}-4y_{k+1}+4y_k=0,\quad k=0,1,\ldots$ Also find the particular solution when y₀. =1 and y₁ =6.

Ques 84.

Ques 85.

Determine the spacing h in a table of equally spaced values for the function

$f(x)=(2+x)^4,\quad 1\le x\le 2,$ so that the quadratic interpolation in this table satisfies | error |≤10^-6

Ques 86.

Determine a unique polynomial f(x) of degree ≤3 such that $f(x_0)=1,f'(x_0)=2,f(x_1)=2,f'(x_1)=3\quad\text{where}x_1-x_0=h.$

Ques 87.

ख) i) न्यूटन रैफ्सन विधि

ii) छेदिका विधि

Ques 88.

Ques 89.

Ques 90.

Ques 91.

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

Ques 92.

गाउस-जॉर्डन विधि से आव्यूह

$\begin{bmatrix}3&1&2\\-2&3&-5\\1&2&4\end{bmatrix}$

का व्युत्क्रम ज्ञात कीजिए।

Ques 93.

आंशिक कीलकन द्वारा निम्नलिखित रैखिक समीकरण निकाय Ax=b का हल ज्ञात कीजिए।

$x_1-x_2+3x_3=3\\$

$2x_1+x_2+4x_3=7\\$

$3x_1+5x_2-2x_3=6$

गुणकों को संचित कीजिए और कीलकन सदिश भी लिखिए

Ques 94.

$8x_1-x_2+2x_3=4\\$

$-3x_1+11x_2-x_3+3x_4=23\\$

$-x_2+10x_3-x_4=-13\\$

$-2x_1+x_2-x_3+8x_4=13$

Ques 95.

$A=\begin{bmatrix}4&-1&1\\4&-8&1\\-2&1&5\end{bmatrix}$

Ques 96.

Ques 97.

Ques 98.

Ques 99.

अंक	छात्रों की संख्या
30-40	31
40-50	42
50-60	51
60-70	35
70-80	31

Ques 100.

Ques 101.

निम्नलिखित आंकड़ों के लिए

x	1.0	1.3	1.6	1.9	2.2
f(x)	0.7651977	0.6200860	0.4554022	0.2818186	0.1103623

स्टर्लिंग सूत्र लागू करके x₀ = 1.6 के लिए f(1.5) का सन्निकट मान ज्ञात कीजिए।

Ques 102.

Ques 103.

x	1.0	1.2	1.4	1.6	1.8	2.0	2.2
f(x)	2.72	3.32	4.06	4.96	6.05	7.39	9.02

Ques 104.

x	0	0.25	0.5	0.75	1.0
f(x)	1.0	0.9896	0.9587	0.9089	0.8415

इस प्रकार प्राप्त हुई घनाकृति का आयतन

i) समलंबी नियम और

ii) सिम्प्सन नियम से ज्ञात कीजिए।

Ques 105.

Ques 106.

x	1.8	1.9	2.0	2.1	2.2
f(x)	10.8894	12.7032	14.7781	17.1489	19.8550

Ques 107.

Ques 108.

Ques 109.

Ques 110.

पुनरावृत्ति विधि

$x_{n+1}=\frac{1}{8}\left[6x_n+\frac{3N}{x_n}-\frac{x_n^3}{N}\right],\quad n=0,1,2$

Ques 111.

Ques 112.

घात ≤ 3 वाला वह अद्वितीय बहुपद f(x) ज्ञात कीजिए जिसके लिए

$f(x_0)=1,f'(x_0)=2,f(x_1)=2,f'(x_1)=3\text{$ हो जहां $x_1-x_0=h$

Ques 113.

Ques 114.

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?

Ques 115.

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

Ques 116.

Ques 117.

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

Ques 118.

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.

Ques 119.

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

Ques 120.

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.

Ques 121.

Solve the system of equations

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

Ques 122.

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.

Ques 123.

Ques 124.

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.

Ques 125.

Find the minimum number of intervals required to evaluate $\int_{0}^{1}e^{-x^2}dx$ dx with an accuracy of $\frac{1}{2}\times 10^{-4},$ by using the Trapezoidal.rule

Ques 126.

From the following table, find the number of students who obtained less than 45 marks.

$\begin{array}{|c|c|}\hline\text{Marks}&\text{No.of Students}\\\hline 30-40&31\\40-50&42\\50-60&51\\60-70&35\\70-80&31\\\hline\end{array}$

Ques 127.

Calculate the third-degree Taylor polynomial about $x_0=0\quad\text{for}f(x)=(1+x)^{1/2}.$

Ques 128.

Use the polynomial in part (a) to approximate $\sqrt[]{1.1}$ and find a bound for the error involved.

Ques 129.

Use the polynomial in part (a) to approximate $\int_{0}^{0.1}(1+x)^{1/2}dx.$

Ques 130.

Using sin(0.1)=0.09983 and sin(0.2)=0.19867, find an approximate value of sin(0.15) by using Lagrange interpolation. Obtain a bound on the truncation error.

Ques 131.

Consider the following data

$\begin{array}{|c|c|c|c|c|c|}\hline x&1.0&1.3&1.6&1.9&2.2\\\hline f(x)&0.7651977&0.6200860&0.4554022&0.2818186&0.1103623\\\hline\end{array}$

Use Stirling's formula to approximate f(1.5) with x_o 1.6.

Ques 132.

Ques 133.

The position f(x) of a particle moving in a line at various times x_k, is given in the following table. Estimate the velocity and acceleration of the particle at x = 1.2

$\begin{array}{|c|c|c|c|c|c|c|c|}\hline x&1.0&1.2&1.4&1.6&1.8&2.0&2.2\\\hline f(x)&2.72&3.32&4.06&4.96&6.05&7.39&9.02\\\hline\end{array}$

Ques 134.

A solid of revolution is formed by rotating about the x-axis the area bounded between x=0, x=1 and the curve given by the table

x	0	0.25	0.5	0.75	1.0
f(x)	1.0	0.9896	0.9587	0.9089	0.8415

Find the volume of the solid so formed using

i) Trapezodial rule

ii) Simpson's rule

Ques 135.

Take 10 figure logarithm to base 10 from x=300 to x=310 by unit increment. Calculate the first derivative of logx when x=310.

Ques 136.

For the table of values of f (x)=xe^x given by

$\begin{array}{|c|c|c|c|c|c|}\hline x&1.8&1.9&2.0&2.1&2.2\\\hline f(x)&10.8894&12.7032&14.7781&17.1489&19.8550\\\hline\end{array}$

Find f"(2.0) using the central difference formula of 0(h²) for h=0.1 and h=0.2. Calculate T.E. and actual erroг.

Ques 137.

Suppose f_n denotes the value of $f(t)\text{at}t=t_n.\text{If}f(t)=t^3$ then find the value of $\frac{(f_{n+1}-2f_n+f_{n-1})}{h^2}.$

Ques 138.

Use Runge-Kutta method of order four to solve y'=x+y. Start with x 1, y=0 and carry to x=1.5 with h = 0.1.

Ques 139.

Find the solution of the difference equation $y_{k+2}-4y_{k+1}+4y_k=0,\quad k=0,1,\ldots$ Also find the particular solution when y₀. =1 and y₁ =6.

Ques 140.

Ques 141.

Determine the spacing h in a table of equally spaced values for the function

$f(x)=(2+x)^4,\quad 1\le x\le 2,$ so that the quadratic interpolation in this table satisfies | error |≤10^-6

Ques 142.

Determine a unique polynomial f(x) of degree ≤3 such that $f(x_0)=1,f'(x_0)=2,f(x_1)=2,f'(x_1)=3\quad\text{where}x_1-x_0=h.$

Ques 143.

ख) i) न्यूटन रैफ्सन विधि

ii) छेदिका विधि

Ques 144.

Ques 145.

Ques 146.

Ques 147.

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

Ques 148.

गाउस-जॉर्डन विधि से आव्यूह

$\begin{bmatrix}3&1&2\\-2&3&-5\\1&2&4\end{bmatrix}$

का व्युत्क्रम ज्ञात कीजिए।

Ques 149.

आंशिक कीलकन द्वारा निम्नलिखित रैखिक समीकरण निकाय Ax=b का हल ज्ञात कीजिए।

$x_1-x_2+3x_3=3\\$

$2x_1+x_2+4x_3=7\\$

$3x_1+5x_2-2x_3=6$

गुणकों को संचित कीजिए और कीलकन सदिश भी लिखिए

Ques 150.

$8x_1-x_2+2x_3=4\\$

$-3x_1+11x_2-x_3+3x_4=23\\$

$-x_2+10x_3-x_4=-13\\$

$-2x_1+x_2-x_3+8x_4=13$

Ques 151.

$A=\begin{bmatrix}4&-1&1\\4&-8&1\\-2&1&5\end{bmatrix}$

Ques 152.

Ques 153.

Ques 154.

Ques 155.

अंक	छात्रों की संख्या
30-40	31
40-50	42
50-60	51
60-70	35
70-80	31

Ques 156.

Ques 157.

निम्नलिखित आंकड़ों के लिए

x	1.0	1.3	1.6	1.9	2.2
f(x)	0.7651977	0.6200860	0.4554022	0.2818186	0.1103623

स्टर्लिंग सूत्र लागू करके x₀ = 1.6 के लिए f(1.5) का सन्निकट मान ज्ञात कीजिए।

Ques 158.

Ques 159.

x	1.0	1.2	1.4	1.6	1.8	2.0	2.2
f(x)	2.72	3.32	4.06	4.96	6.05	7.39	9.02

Ques 160.

x	0	0.25	0.5	0.75	1.0
f(x)	1.0	0.9896	0.9587	0.9089	0.8415

इस प्रकार प्राप्त हुई घनाकृति का आयतन

i) समलंबी नियम और

ii) सिम्प्सन नियम से ज्ञात कीजिए।

Ques 161.

Ques 162.

x	1.8	1.9	2.0	2.1	2.2
f(x)	10.8894	12.7032	14.7781	17.1489	19.8550

Ques 163.

Ques 164.

Ques 165.

Ques 166.

पुनरावृत्ति विधि

$x_{n+1}=\frac{1}{8}\left[6x_n+\frac{3N}{x_n}-\frac{x_n^3}{N}\right],\quad n=0,1,2$

Ques 167.

Ques 168.

घात ≤ 3 वाला वह अद्वितीय बहुपद f(x) ज्ञात कीजिए जिसके लिए

$f(x_0)=1,f'(x_0)=2,f(x_1)=2,f'(x_1)=3\text{$ हो जहां $x_1-x_0=h$

Ques 169.

Ques 170.

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?

Ques 171.

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

Ques 172.

Ques 173.

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

Ques 174.

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.

Ques 175.

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

Ques 176.

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.

Ques 177.

Solve the system of equations

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

Ques 178.

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.

Ques 179.

Ques 180.

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.

Ques 181.

Find the minimum number of intervals required to evaluate $\int_{0}^{1}e^{-x^2}dx$ dx with an accuracy of $\frac{1}{2}\times 10^{-4},$ by using the Trapezoidal.rule

Ques 182.

From the following table, find the number of students who obtained less than 45 marks.

$\begin{array}{|c|c|}\hline\text{Marks}&\text{No.of Students}\\\hline 30-40&31\\40-50&42\\50-60&51\\60-70&35\\70-80&31\\\hline\end{array}$

Ques 183.

Calculate the third-degree Taylor polynomial about $x_0=0\quad\text{for}f(x)=(1+x)^{1/2}.$

Ques 184.

Use the polynomial in part (a) to approximate $\sqrt[]{1.1}$ and find a bound for the error involved.

Ques 185.

Use the polynomial in part (a) to approximate $\int_{0}^{0.1}(1+x)^{1/2}dx.$

Ques 186.

Using sin(0.1)=0.09983 and sin(0.2)=0.19867, find an approximate value of sin(0.15) by using Lagrange interpolation. Obtain a bound on the truncation error.

Ques 187.

Consider the following data

$\begin{array}{|c|c|c|c|c|c|}\hline x&1.0&1.3&1.6&1.9&2.2\\\hline f(x)&0.7651977&0.6200860&0.4554022&0.2818186&0.1103623\\\hline\end{array}$

Use Stirling's formula to approximate f(1.5) with x_o 1.6.

Ques 188.

Ques 189.

The position f(x) of a particle moving in a line at various times x_k, is given in the following table. Estimate the velocity and acceleration of the particle at x = 1.2

$\begin{array}{|c|c|c|c|c|c|c|c|}\hline x&1.0&1.2&1.4&1.6&1.8&2.0&2.2\\\hline f(x)&2.72&3.32&4.06&4.96&6.05&7.39&9.02\\\hline\end{array}$

Ques 190.

A solid of revolution is formed by rotating about the x-axis the area bounded between x=0, x=1 and the curve given by the table

x	0	0.25	0.5	0.75	1.0
f(x)	1.0	0.9896	0.9587	0.9089	0.8415

Find the volume of the solid so formed using

i) Trapezodial rule

ii) Simpson's rule

Ques 191.

Take 10 figure logarithm to base 10 from x=300 to x=310 by unit increment. Calculate the first derivative of logx when x=310.

Ques 192.

For the table of values of f (x)=xe^x given by

$\begin{array}{|c|c|c|c|c|c|}\hline x&1.8&1.9&2.0&2.1&2.2\\\hline f(x)&10.8894&12.7032&14.7781&17.1489&19.8550\\\hline\end{array}$

Find f"(2.0) using the central difference formula of 0(h²) for h=0.1 and h=0.2. Calculate T.E. and actual erroг.

Ques 193.

Suppose f_n denotes the value of $f(t)\text{at}t=t_n.\text{If}f(t)=t^3$ then find the value of $\frac{(f_{n+1}-2f_n+f_{n-1})}{h^2}.$

Ques 194.

Use Runge-Kutta method of order four to solve y'=x+y. Start with x 1, y=0 and carry to x=1.5 with h = 0.1.

Ques 195.

Find the solution of the difference equation $y_{k+2}-4y_{k+1}+4y_k=0,\quad k=0,1,\ldots$ Also find the particular solution when y₀. =1 and y₁ =6.

Ques 196.

Ques 197.

Determine the spacing h in a table of equally spaced values for the function

$f(x)=(2+x)^4,\quad 1\le x\le 2,$ so that the quadratic interpolation in this table satisfies | error |≤10^-6

Ques 198.

Determine a unique polynomial f(x) of degree ≤3 such that $f(x_0)=1,f'(x_0)=2,f(x_1)=2,f'(x_1)=3\quad\text{where}x_1-x_0=h.$

Ques 199.

ख) i) न्यूटन रैफ्सन विधि

ii) छेदिका विधि

Ques 200.

Ques 201.

Ques 202.

Ques 203.

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

Ques 204.

गाउस-जॉर्डन विधि से आव्यूह

$\begin{bmatrix}3&1&2\\-2&3&-5\\1&2&4\end{bmatrix}$

का व्युत्क्रम ज्ञात कीजिए।

Ques 205.

आंशिक कीलकन द्वारा निम्नलिखित रैखिक समीकरण निकाय Ax=b का हल ज्ञात कीजिए।

$x_1-x_2+3x_3=3\\$

$2x_1+x_2+4x_3=7\\$

$3x_1+5x_2-2x_3=6$

गुणकों को संचित कीजिए और कीलकन सदिश भी लिखिए

Ques 206.

$8x_1-x_2+2x_3=4\\$

$-3x_1+11x_2-x_3+3x_4=23\\$

$-x_2+10x_3-x_4=-13\\$

$-2x_1+x_2-x_3+8x_4=13$

Ques 207.

$A=\begin{bmatrix}4&-1&1\\4&-8&1\\-2&1&5\end{bmatrix}$

Ques 208.

Ques 209.

Ques 210.

Ques 211.

अंक	छात्रों की संख्या
30-40	31
40-50	42
50-60	51
60-70	35
70-80	31

Ques 212.

Ques 213.

निम्नलिखित आंकड़ों के लिए

x	1.0	1.3	1.6	1.9	2.2
f(x)	0.7651977	0.6200860	0.4554022	0.2818186	0.1103623

स्टर्लिंग सूत्र लागू करके x₀ = 1.6 के लिए f(1.5) का सन्निकट मान ज्ञात कीजिए।

Ques 214.

Ques 215.

x	1.0	1.2	1.4	1.6	1.8	2.0	2.2
f(x)	2.72	3.32	4.06	4.96	6.05	7.39	9.02

Ques 216.

x	0	0.25	0.5	0.75	1.0
f(x)	1.0	0.9896	0.9587	0.9089	0.8415

इस प्रकार प्राप्त हुई घनाकृति का आयतन

i) समलंबी नियम और

ii) सिम्प्सन नियम से ज्ञात कीजिए।

Ques 217.

Ques 218.

x	1.8	1.9	2.0	2.1	2.2
f(x)	10.8894	12.7032	14.7781	17.1489	19.8550

Ques 219.

Ques 220.

Ques 221.

Ques 222.

पुनरावृत्ति विधि

$x_{n+1}=\frac{1}{8}\left[6x_n+\frac{3N}{x_n}-\frac{x_n^3}{N}\right],\quad n=0,1,2$

Ques 223.

Ques 224.

घात ≤ 3 वाला वह अद्वितीय बहुपद f(x) ज्ञात कीजिए जिसके लिए

$f(x_0)=1,f'(x_0)=2,f(x_1)=2,f'(x_1)=3\text{$ हो जहां $x_1-x_0=h$

Ques 225.

Ques 226.

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?

Ques 227.

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

Ques 228.

Ques 229.

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

Ques 230.

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.

Ques 231.

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

Ques 232.

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.

Ques 233.

Solve the system of equations

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

Ques 234.

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.

Ques 235.

Ques 236.

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.

Ques 237.

Find the minimum number of intervals required to evaluate $\int_{0}^{1}e^{-x^2}dx$ dx with an accuracy of $\frac{1}{2}\times 10^{-4},$ by using the Trapezoidal.rule

Ques 238.

From the following table, find the number of students who obtained less than 45 marks.

$\begin{array}{|c|c|}\hline\text{Marks}&\text{No.of Students}\\\hline 30-40&31\\40-50&42\\50-60&51\\60-70&35\\70-80&31\\\hline\end{array}$

Ques 239.

Calculate the third-degree Taylor polynomial about $x_0=0\quad\text{for}f(x)=(1+x)^{1/2}.$

Ques 240.

Use the polynomial in part (a) to approximate $\sqrt[]{1.1}$ and find a bound for the error involved.

Ques 241.

Use the polynomial in part (a) to approximate $\int_{0}^{0.1}(1+x)^{1/2}dx.$

Ques 242.

Using sin(0.1)=0.09983 and sin(0.2)=0.19867, find an approximate value of sin(0.15) by using Lagrange interpolation. Obtain a bound on the truncation error.

Ques 243.

Consider the following data

$\begin{array}{|c|c|c|c|c|c|}\hline x&1.0&1.3&1.6&1.9&2.2\\\hline f(x)&0.7651977&0.6200860&0.4554022&0.2818186&0.1103623\\\hline\end{array}$

Use Stirling's formula to approximate f(1.5) with x_o 1.6.

Ques 244.

Ques 245.

The position f(x) of a particle moving in a line at various times x_k, is given in the following table. Estimate the velocity and acceleration of the particle at x = 1.2

$\begin{array}{|c|c|c|c|c|c|c|c|}\hline x&1.0&1.2&1.4&1.6&1.8&2.0&2.2\\\hline f(x)&2.72&3.32&4.06&4.96&6.05&7.39&9.02\\\hline\end{array}$

Ques 246.

A solid of revolution is formed by rotating about the x-axis the area bounded between x=0, x=1 and the curve given by the table

x	0	0.25	0.5	0.75	1.0
f(x)	1.0	0.9896	0.9587	0.9089	0.8415

Find the volume of the solid so formed using

i) Trapezodial rule

ii) Simpson's rule

Ques 247.

Take 10 figure logarithm to base 10 from x=300 to x=310 by unit increment. Calculate the first derivative of logx when x=310.

Ques 248.

For the table of values of f (x)=xe^x given by

$\begin{array}{|c|c|c|c|c|c|}\hline x&1.8&1.9&2.0&2.1&2.2\\\hline f(x)&10.8894&12.7032&14.7781&17.1489&19.8550\\\hline\end{array}$

Find f"(2.0) using the central difference formula of 0(h²) for h=0.1 and h=0.2. Calculate T.E. and actual erroг.

Ques 249.

Suppose f_n denotes the value of $f(t)\text{at}t=t_n.\text{If}f(t)=t^3$ then find the value of $\frac{(f_{n+1}-2f_n+f_{n-1})}{h^2}.$

Ques 250.

Use Runge-Kutta method of order four to solve y'=x+y. Start with x 1, y=0 and carry to x=1.5 with h = 0.1.

Ques 251.

Find the solution of the difference equation $y_{k+2}-4y_{k+1}+4y_k=0,\quad k=0,1,\ldots$ Also find the particular solution when y₀. =1 and y₁ =6.

Ques 252.

Ques 253.

Determine the spacing h in a table of equally spaced values for the function

$f(x)=(2+x)^4,\quad 1\le x\le 2,$ so that the quadratic interpolation in this table satisfies | error |≤10^-6

Ques 254.

Determine a unique polynomial f(x) of degree ≤3 such that $f(x_0)=1,f'(x_0)=2,f(x_1)=2,f'(x_1)=3\quad\text{where}x_1-x_0=h.$

Ques 255.

ख) i) न्यूटन रैफ्सन विधि

ii) छेदिका विधि

Ques 256.

Ques 257.

Ques 258.

Ques 259.

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

Ques 260.

गाउस-जॉर्डन विधि से आव्यूह

$\begin{bmatrix}3&1&2\\-2&3&-5\\1&2&4\end{bmatrix}$

का व्युत्क्रम ज्ञात कीजिए।

Ques 261.

आंशिक कीलकन द्वारा निम्नलिखित रैखिक समीकरण निकाय Ax=b का हल ज्ञात कीजिए।

$x_1-x_2+3x_3=3\\$

$2x_1+x_2+4x_3=7\\$

$3x_1+5x_2-2x_3=6$

गुणकों को संचित कीजिए और कीलकन सदिश भी लिखिए

Ques 262.

$8x_1-x_2+2x_3=4\\$

$-3x_1+11x_2-x_3+3x_4=23\\$

$-x_2+10x_3-x_4=-13\\$

$-2x_1+x_2-x_3+8x_4=13$

Ques 263.

$A=\begin{bmatrix}4&-1&1\\4&-8&1\\-2&1&5\end{bmatrix}$

Ques 264.

Ques 265.

Ques 266.

Ques 267.

अंक	छात्रों की संख्या
30-40	31
40-50	42
50-60	51
60-70	35
70-80	31

Ques 268.

Ques 269.

निम्नलिखित आंकड़ों के लिए

x	1.0	1.3	1.6	1.9	2.2
f(x)	0.7651977	0.6200860	0.4554022	0.2818186	0.1103623

स्टर्लिंग सूत्र लागू करके x₀ = 1.6 के लिए f(1.5) का सन्निकट मान ज्ञात कीजिए।

Ques 270.

Ques 271.

x	1.0	1.2	1.4	1.6	1.8	2.0	2.2
f(x)	2.72	3.32	4.06	4.96	6.05	7.39	9.02

Ques 272.

x	0	0.25	0.5	0.75	1.0
f(x)	1.0	0.9896	0.9587	0.9089	0.8415

इस प्रकार प्राप्त हुई घनाकृति का आयतन

i) समलंबी नियम और

ii) सिम्प्सन नियम से ज्ञात कीजिए।

Ques 273.

Ques 274.

x	1.8	1.9	2.0	2.1	2.2
f(x)	10.8894	12.7032	14.7781	17.1489	19.8550

Ques 275.

Ques 276.

Ques 277.

Ques 278.

पुनरावृत्ति विधि

$x_{n+1}=\frac{1}{8}\left[6x_n+\frac{3N}{x_n}-\frac{x_n^3}{N}\right],\quad n=0,1,2$

Ques 279.

Ques 280.

घात ≤ 3 वाला वह अद्वितीय बहुपद f(x) ज्ञात कीजिए जिसके लिए

$f(x_0)=1,f'(x_0)=2,f(x_1)=2,f'(x_1)=3\text{$ हो जहां $x_1-x_0=h$

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