Calculate the third-degree Taylor polynomial about 

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

Find the minimum number of intervals required to evaluate   with an accuracy of

by using the Trapezoidal rule.

Obtain an approximate value of  using composite Simpson’s rule with h = 0.25 and

h = 0.125  Find also the improved value using Romberg integration.

The solution of the system of equations 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.

Starting with  find the dominant eigenvalue and corresponding eigenvector for the matrix using the power method.

Solve the system of equations

 by using the Gauss Jacboi and Gauss Seidel method. The exact solution of the system is   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?

Solve the following linear system Ax = b of equations with partial pivoting

Store the multipliers and also write the pivoting vectors.

Find the inverse of the matrix   using Gauss Jordan Method.

Solve the system of equations

using Gauss elimination method with pivoting.

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

Find an approximate value of the positive real root of xe ^x= 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.

Using Maclaurin’s expansion for sin x , find the approximate value of 4 sin  with the error bound 5 10 ^-5

Find an appropriate root of   with 10^-5 accuracy by

i) Newton Raphson Method

ii) Secant Method

What conclusions can you draw from here about the two methods?

The equation   has a positive root in the interval ]1, 2[. Write a fixed point iteration method and show that it converges. Starting with initial approximation x = 1.5 find the root of the equation correct to three decimal places.

Find the Boolean expression in the DNF form for the function defined in tabular form below:

x	y	z
1	0	1	1
0	1	0	0
0	0	1	1
1	1	1	1
1	0	0	0
0	1	1	1
1	1	0	1
0	0	0	0

Is there a 4-regular graph on 7 vertices? Justify your answer.

Solve the recurrence relation:

using generating function technique. Also find 5 a using your answer.

How many numbers from 0 to 759 are not divisible by either 3 or 7?

Define vertex connectivity and cut vertex set of any graph G. Find the vertex connectivity and cut vertex set for the following graph: