Prove that every non-trivial subgroup of a cyclic group has finite index. Hence prove that is not cyclic.
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 .
Divide the polynomial
x5 - 6x4 + 8x3 + 8x2 + 4x - 40
by (x - 3) by the synthetic division method and find the remainder.
See Answer → Find the inverse of the matrix using Gauss-Jordan method.
The velocity of a vehicle beginning from rest is given in the following table for part of the first four. Using Simpson's rule, find the distance travelled by the vehicle in this hour:
|
| 10 | 20 | 30 | 40 | 50 | 60 |
|
| 80 | 60 | 70 | 75 | 70 | 80 |
For the following data, use Gauss backward difference method to obtain the interpolating polynomial f(x):
| x | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 |
| f(x) | 1.40 | 1.56 | 1.76 | 2.00 | 2.28 |
Hence, find the value of f(0.45).
See Answer →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.
) Evaluate by using trapezoidal rule with
and
. Use Romber's method to find the best value of
.
) Estimate the eigenvalues of the matrix
using the Gerschgorin bounds.
) Evaluate by using trapezoidal rule with
and
. Use Romber's method to find the best value of
.
Evaluate by using trapezoidal rule with
and
. Use Romber's m
The method
where N is a positive constant, converges to N1/3. Find the rate of convergence of the method.
Determine the largest eigenvalue in magnitude and the corresponding eigenvector of the matrix using the power method. Take (1, 0, 0)T as the initial approximation and perform 4 iterations.
Determine the order of convergence of the iterative method
for finding a simple root of the equation .
Prove that .
Obtain the interpolating polynomial in simplest form which fits the following data:
| x | -1 | 0 | 1 | 2 |
| f(x) | 3 | -4 | 5 | -6 |
Using synthetic division method, show that 2 is a simple root of the equation
.
Find the interpolating polynomial by Newton’s divided difference formula for the following data:
| x | 0 | 1 | 2 | 4 |
| y | 1 | 1 | 2 | 5 |