Solve your IGNOU Doubts

Question:

Check that f(x) is well defined and 1 - 1.
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Question:

Consider the function $f: \mathbf{R} \setminus \{-1\} \to \mathbf{R}$ defined by $f(x) = \frac{2x+1}{x+1}$ .

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Question:

) Which of the following statements are true and which are false? Justify your answer with a short proof or a counterexample.
i) The function $f: \mathbf{R} \to \mathbf{R}$ defined by $f(x) = \cos x$ is 1-1.
ii) The operation $ defined byx y = \log(xy)is a binary operation onS, whereSis the set\{x \in \mathbf{R} \mid x > 0\}$.
iii) The set $\{(x_1, x_2, \dots, x_n) \mid x_1, x_2, \dots, x_n \in \mathbf{R}, x_1 = 2x_2 + 3\}$ is a subspace of $\mathbf{R}^n$ .
iv) There is no $7 \times 5$ matrix of rank 6.
v) If V and V' are vector spaces and $T: V \to V'$ is a linear transformation, then whenever $u_1, u_2, \dots, u_k$ are linearly independent, $Tu_1, Tu_2, \dots, Tu_k$ are also linearly independent.
vi) If V is a vector space and $T: V \to V$ is a linear operator with $\det(T) = 0$ , then T is not diagonalisable.
vii) The degree of the minimal polynomial of a $3 \times 3$ matrix is at most 2.
viii) For any $2 \times 2$ matrix A, $\text{Adj}(A^t) = (\text{Adj}(A))^t$ .
ix) The only matrix which is both symmetric and skew-symmetric is the zero matrix.
x) There is no co-ordinate transformation that transforms the quadratic form x² + y² + z² to the quadratic form xz + yz.

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Question:

Show that there are infinitely many values of $\alpha$ for which $x^7 + 15x^2 - 30x + \alpha$ is irreducible in $\mathbb{Q}[x]$ .

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Question:

Show that R/M is a field, and give two distinct non-zero elements of this field.

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Question:

Hence show that if N is an ideal of R properly containing M, then $N = R$ .

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Question:

Show that if $5 \nmid a$ or $5 \nmid b$ , then $5 \nmid (a^2 - 2b^2)$ , for $a, b \in \mathbb{Z}$ .

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Question:

Show that M is an ideal of R.

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Question:

Let $R = \mathbb{Z}[\sqrt{2}]$ and $M = \{a + b\sqrt{2} \in R \mid 5 \mid a \text{ and } 5 \mid b\}$

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Question:

Let $D = \{f(x, y) + g(x, y)i \mid f, g \in \mathbb{Z}[x, y]\} \subseteq \mathbb{C}[x, y]$ . Check whether D is a UFD or not.

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Question:

Find all the units of $\mathbb{Z}_{12}$ . Let R be a commutative ring with unity and $r \in R$ . Prove that $\frac{R[x]}{\langle x - r \rangle} \simeq R$ using the Fundamental Theorem of Homomorphism. Hence show that $\frac{R[x, y]}{\langle y - r \rangle} \simeq R[x]$ .

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Question:

Find all the units of $\mathbb{Z}_{12}$ .

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Question:

Prove that $\frac{\mathbb{R}[x]}{\langle x^2 + 1 \rangle} \simeq \mathbb{C}$ as rings.

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Question:

heck whether $I = \left\{ \begin{bmatrix} m & 0 \\ n & 0 \end{bmatrix} \middle| m, n \in \mathbb{Z} \right\}$ is a subring of the ring $M_2(\mathbb{Z})$ or not. If it is, check whether or not it is an ideal of the ring also. If I is not a subring of the ring, then provide a subring of the ring.

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Question:

Let G be a group such that $\text{Aut } G$ is cyclic. Prove that G is abelian.

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Question:

Find a group G, and a homomorphism $\phi$ of G, so that $\phi(G) \simeq S_3$ and $\text{Ker } \phi \simeq A_4$ . Is G abelian? Give reasons for your answer.

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Question:

Let G be a group of order 56. What are all its Sylow p-subgroups? Show that G is not simple, i.e., G must have a proper normal non-trivial subgroup.

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Question:

Explicitly give the elements and structure of the group S_n / A_n, $n \geq 5$ .

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Question:

Consider the map $f_{ab} : \mathbb{R} \rightarrow \mathbb{R} : f_{ab}(x) = ax + b$ . Let $B = \{f_{ab} \mid a, b \in \mathbb{R}, a \neq 0\}$ . Then B is a group with respect to the composition of functions. Check whether or not $A = \{f_{ab} \mid a \in \mathbb{Q}^+, b \in \mathbb{R}\}$ is a normal subgroup of B.

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Question:

Let G be a group of order $n \geq 2$ , with only two subgroups — $\{e\}$ and itself. Find a minimal generating set for G. Also, find out whether n is a prime or a composite number, or can be either.