Solve your IGNOU Doubts

Question:

Give the smallest $n \in \mathbb{N}$ for which A_n is non-abelian. Justify your answer.

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Question:

List two distinct cosets of $\langle r \rangle$ in D₁₀, where r is a reflection in D₁₀.

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Question:

Let $\tau$ be a fixed odd permutation in S₁₀. Show that every odd permutation in S₁₀ is a product of $\tau$ and some permutation in A₁₀.

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Question:

Using Cayley’s theorem, find the permutation group to which a cyclic group of order 12 is isomorphic.

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Question:

Prove that a cyclic group with only one generator can have at most 2 elements.

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Question:

Let G be an infinite group such that for any non-trivial subgroup H of $G, |G : H| < \infty$ . Then prove that
i) $H \le G \Rightarrow H = \{e\}$ or H is infinite;
ii) If $g \in G, g \neq e$ , then o(g) is infinite.

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Question:

Prove that every non-trivial subgroup of a cyclic group has finite index. Hence prove that $(\mathbb{Q}, +)$ is not cyclic.

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Question:

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 $G = \mathbb{Z}$ .
ii) If x and y are elements of a non-abelian group (G, ) such that $x y = y x$ , then $x = e$ or $y = e$ , where e is the identity of G with respect to $$.
iii) There exists a unique non-abelian group of prime order.
iv) If $(a, b) \in A \times A$ , where A is a group, then $o((a, b)) = o(a)o(b)$ .
v) If H and K are normal subgroups of a group G, then $hk = kh \quad \forall \quad h \in H, k \in K$ .

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Question:

Determine a unique polynomial f(x) of degree $\le 3$ such that $f(x_0) = 1$ , $f'(x_0) = 2$ , $f(x_1) = 2$ , $f'(x_1) = 3$ , where $x_1 - x_0 = h$ .

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Question:

Divide the polynomial

x⁵ - 6x⁴ + 8x³ + 8x² + 4x - 40

by (x - 3) by the synthetic division method and find the remainder.

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Question:

Find the inverse of the matrix $\begin{bmatrix} 2 & 1 & 1 \\ 3 & 2 & 1 \\ 2 & 1 & 2 \end{bmatrix}$ using Gauss-Jordan method.

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Question:

The velocity of a vehicle beginning from rest is given in the following table for part of the first four. Using Simpson's $\frac{1}{3}$ rule, find the distance travelled by the vehicle in this hour:

`t=time in min.`	10	20	30	40	50	60
`v=velocity in km/hr.`	80	60	70	75	70	80

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Question:

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).

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Question:

Write the error in linear interpolation. Hence, show that

$$| \text{error} | \leq \frac{h^2}{8} \max | f''(x) |$

where $h = x_1 - x_0, x \in [x_0, x_1].$

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Question:

Set up the Gauss-Seidel iteration scheme in matrix form for solving the system of equations

$$\begin{bmatrix} 1 & 1 & 1 \\ 4 & 3 & -1 \\ 3 & 5 & 3 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 1 \\ 6 \\ 4 \end{bmatrix}.$
Show that the method is convergent and hence find its rate of convergence.

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Question:

Solve the initial value problem using Euler method

$$y' = \frac{1}{x^2 - 3y}, y(3) = 2.$

Find y(3.1) taking $h = 0.1$ .

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Question:

) Evaluate $\int_{0}^{1} \frac{1}{1 + x^2} dx$ by using trapezoidal rule with $h = 0.5$ and $h = 0.25$ . Use Romber's method to find the best value of $\pi$ .

) Estimate the eigenvalues of the matrix

$$\begin{bmatrix} 1 & -1 & 2 \\ -1 & 1 & 2 \\ 2 & 2 & -2 \end{bmatrix}$

using the Gerschgorin bounds.

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Question:

) Evaluate $\int_{0}^{1} \frac{1}{1 + x^2} dx$ by using trapezoidal rule with $h = 0.5$ and $h = 0.25$ . Use Romber's method to find the best value of $\pi$ .

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Question:

Evaluate $\int_{0}^{1} \frac{1}{1 + x^2} dx$ by using trapezoidal rule with $h = 0.5$ and $h = 0.25$ . Use Romber's m

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Question:

The method

$$x_{n+1} = \frac{1}{9} \left[ 5x_n + \frac{5N}{x_n^2} - \frac{N^2}{x_n^5} \right], n = 0, 1, 2, \dots$
where N is a positive constant, converges to N^1/3. Find the rate of convergence of the method.