Determine a unique polynomial f(x) of degree such that
,
,
,
, where
.
Let (f(x)) be a cubic polynomial: [ f(x) = a_0 + a_1(x-x_0) + a_2(x-x_0)^2 + a_3(x-x_0)^3 ] Then [ f'(x) = a_1 + 2a_2(x-x_0) + 3a_3(x-x_0)^2 ] Using the given conditions: 1. (f(x_0) = 1 \Rightarrow a_0 = 1) 2. (f'(x_0) = 2 \Rightarrow a_1 = 2) Next, at (x = x_1 = x_0 + h): [ f(x_1) = 1 + 2h ______ ____ _____ ________ ______ _________.
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The Gauss-Seidel method is used to solve the system of equations
Determine the rate of convergence of the method.
The method
where N is a positive constant, converges to N1/3. Find the rate of convergence of the method.
Which of the following statements are true, and which are false? Give reasons for your answers.
i) If k is a field, then so is .
ii) If R is an integral domain and I is an ideal of R, then .
iii) In a domain, every prime ideal is a maximal ideal.
iv) If R is a ring with zero divisors, and S is a subring of R, then S has zero divisors.
v) If R is a ring and is of degree
, then f(x) has exactly n roots in R.
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.)
अभिसरण दर ज्ञात कीजिए।
ख) निम्नलिखित आंकड़ों के लिए न्यूटन के विभाजित अंतर सूत्र द्वारा अंतर्वेशन बहुपद ज्ञात कीजिए :
| x | 0 | 1 | 2 | 4 |
| y | 1 | 1 | 2 | 5 |
Divide the polynomial
x5 - 6x4 + 8x3 + 8x2 + 4x - 40
by (x - 3) by the synthetic division method and find the remainder.
Let S be a set, R a ring and f be a 1-1 mapping of S onto R. Define + and on S by:
.
Show that is a ring isomorphic to R.
PART-C (MM: 20 Marks)
(Based on Block 4.)