Obtain the interpolating polynomial in simplest form which fits the following data:
| x | -1 | 0 | 1 | 2 |
| f(x) | 3 | -4 | 5 | -6 |
We are asked to find the interpolating polynomial for the data (x = -1, 0, 1, 2) with (f(x) = 3, -4, 5, -6) using the Newton forward difference method (or equivalently constructing a polynomial manually). First, construct the divided differences table: | (x) | (f(x)) | (\Delta f) | (\Delta^2 f) | (\Delta^3 f) | | --- | ------ | ---------- | ------------ | ------------ _______ ____ ___ ______ _____ ________.
_____ ____ ___ ___ __________ ______ _____ __________ _______ _____.
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___ __________ _____ ___ _____ _______ _________ ___ _______ _____ ____.
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________ _____ ________ _________ _________ ________ _________ __________ ________ _______ _______.
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_______ __________ ____ _______ _________ _______ ____ ______ ___ ________ __________.
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सांशलेषिक विभाजन विधि का प्रयोग करके यह दर्शाइए कि 2, समीकरण का एक सरल मूल है।
Solve the initial value problem using Euler method
Find y(3.1) taking .
Which of the following statements are true? Give reasons for your answers.
i) If a group G is isomorphic to one of its proper subgroups, then .
ii) If x and y are elements of a non-abelian group (G, ) such that , then
or
, where e is the identity of G with respect to $$.
iii) There exists a unique non-abelian group of prime order.
iv) If , where A is a group, then
.
v) If H and K are normal subgroups of a group G, then .
घात विधि द्वारा निम्नलिखित आव्यूह का परिमाण में अधिकतम आइगेनमान व संगत आइगेनसदिश ज्ञात कीजिए :$
प्रारम्भिक सन्निकटन (1, 0, 0)T लेकर 4 पुनरावृत्तियाँ कीजिए।
Set up the Gauss-Seidel iteration scheme in matrix form for solving the system of equations
Show that the method is convergent and hence find its rate of convergence.
Use the Fundamental Theorem of Homomorphism for Groups to prove the following theorem, which is called the Zassenhaus (Butterfly) Lemma:
Let H and K be subgroups of a group G and H' and K' be normal subgroups of H and K, respectively. Then
i)
ii)
iii)
The situation can be represented by the subgroup diagram below, which explains the name ‘butterfly’.
PART-B (MM: 30 Marks)
(Based on Block 3.)
विश्रामावस्था से आरंभ कर रही एक गाड़ी का वेग पहले घंटे के लिए निम्नलिखित तालिका में दिया गया है। सिम्पसन का नियम लागू करके, इस घंटे में गाड़ी द्वारा तय की गई दूरी ज्ञात कीजिए :