Solve the system of equations:

x + 2y + z = 3

3x − 2y − z4 = −2

2x + 3y − z = −6

using Gauss-Elimination method.

Given the data: 

f(3) = 168,

f(7) = 120

and f(9) = 72 

If  f(k) is estimated as 138 using the Newton’s form of the interpolating polynomial, then find the value of .k

Using Runge-Kutta fourth order method with h = 0.1, find an approximation value of y(0.1) for the initial value problem:

b) Evaluate the integral:

using composite trapezoidal rule with h = ,2.0 compare with the exact value.

Use the Euler’s method to solve numerically the initial value problem: 

with h = 2.0 on the interval [0, 1].

Using the data sin (0.1) = 0.09983 and sin (0.2) = 0.19867, find an approximate value of sin (0.15) by Lagrange’s interpolation. Also obtain a bound on the truncation error.

Use secant method to determine the root of the equation cos x - xe^x = 0. Take the initial approximation as x₀ = 0, x₁ = 1 and perform two iterations of the method.

 Find the inverse of the matrix:

using LU decomposition method with u₁₁ = u₂₂ = u₃₃ = 1.

Calculate the nth divided difference of 1/x, on the nodal points X₀ , X₁ , ......., X_n .

 Using Newton-Rapshson method, find an iterative formula to compute the reciprocal of a natural number N.

 Which of the following statements are true and which are false? Give a short proof or a counterexample in support of your answer:

i) The equation x³ 4x 16 = 0 has a root in the interval [3, 4].

ii) The order of convergence of the secant method is 0.62.

iii) For the system of linear equations:

5x + y + 2z = 34

4y − 3z = 12

10x − 2y + z = −4

the matrix is diagonally dominant.

iv) The numerical method:

is relatively stable.

 v) The method:

converges to 1.5 for any choice of initial approximation.

Which of the following statements are true and which are false? Justify your answer with a short proof or a counter example.

(a) Every subgroup of S₃ is normal.

(b) Every abelian group is cyclic.

(c) In a ring with unity, the sum of any two units is a unit.

(d) If a field has characteristic p, p a prime, the field is finite.

(e) If every element in group has finite order, the group is finite.

 Let R be a ring (not necessarily commutative) and I and J be ideal of R. Show that I ∩ J and I + J = {a + bla ∈ I, b ∈ J} are ideals of R.

Check whether or not (3) is a maximal ideal in Z₉.

Show that (x, 5) is not a principal ideal in Z[x].

 Let S¹ = {z ∈ C | |z| = 1} and Un = {z ∈ C* | z" = 1} for n ∈ N. Check that U_n ⊆ S¹. Further, show that U_n ≤ S¹.

Let F be a field and let f(x) ∈ F[x] be irreducible in F[x]. Show that the ideal (f(x)) is a maximal ideal in F[x]. Use this to deduce that Q[x]/(x² + 6x3 +12) is a field.

Find the order of each of the elements in U(15). Is U (15) cyclic? Justify your answer.

If H and K are normal abelian subgroups of a group, and if H∩ K = {e}, show that HK is abelian. Will the result be still true if we remove the condition that H and K are normal? Justify your answer.

Find the gcd of the polynomials x4 + 3x + 2 and x³ + 3x² + 5x + 3.

 Let   . Check whether Check whether R is a subring of M₂(R). Is R an ideal of M₂(R)? Justify your answer.