सिद्ध कीजिए कि ।
हमारे पास सिद्ध करने के लिए दिया गया है: [ \mu^2 = 1 + \frac{\delta^2}{4}. ] मान लें कि (\mu) और (\delta) किसी द्विघात समीकरण से जुड़े हैं, जैसे कि: [ r^2 - \delta r - 1 = 0. ] इस द्विघात समीकरण के मूल (\mu_1) और (\mu_2) हैं। द्विघात समीकरण का सामान्य हल सूत्र ___ _________ ________ _______ ____ _____ ________ ________ _________ _______.
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Find the interpolating polynomial by Newton’s divided difference formula for the following data:
| x | 0 | 1 | 2 | 4 |
| y | 1 | 1 | 2 | 5 |
Solve the initial value problem using Euler method
Find y(3.1) taking .
Let G be an infinite group such that for any non-trivial subgroup H of . Then prove that
i) or H is infinite;
ii) If , then o(g) is infinite.
Check whether or not is a field.
निम्नलिखित आंकड़ों के लिए, गाउस पश्चांतर विधि का प्रयोग करके को अंतर्वेशी करने वाला बहुपद प्राप्त कीजिए :
| x | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 |
| f (x) | 1.40 | 1.56 | 1.76 | 2.00 | 2.28 |
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.)
List two distinct cosets of in D10, where r is a reflection in D10.