Maharashtra Math 2008 Optional Paper – Page1-2

Q1. Show that every group is isomorphic to a subgroup of a permutation group A(S).
(a) for some appropriate S.

Q1. \overline2 MNS SECTION - A Show that every group is isomorphic to a subgroup of a permutation group A(S)

Q2.

(a) for some appropriate S. 18 State the name of this theorem. 2 If \v₁, v₂, \ldots, v_n\ is a basis of a vector space V and if \w₁, w₂, \ldots, w_m\ is linearly

(b) independent in V, then show that m ≤ n. 20
3. Define
(i) Euclidean Domain. 3 (a)
(ii) Principal Ideal Domain (PID). 3 Show that every Euclidean Domain is PID. 14 (b) State Cayley Hamilton theorem and using it find inverse of the matrix A if it exists. 2 A = \beginbmatrix 1 & 2 & 3 \\ 2 & 2 & 4 \\ 3 & 4 & 8 \endbmatrix 18 SECTION - B If f ' is a real-valued continuous function on a compact metric space X, then
4. (a) show that f
(X) , range of f' is compact and f' attains a maximum and minimum at points of X. 10 + Define absolute convergence and conditional convergence for improper integrals (b) of the type ∫ f
(x) dx for continuous function f
(x) . 2 + 2 Show that ∫₀^∞ (sin x)/(x) dx is convergent but not absolutely. 8 + 8
5. Define differentiability of a function of two variables at a point. (a) 2 Let f: E → \mathbbR be defined on a neighbourhood E of (a,b) ∈ \mathbbR × \mathbbR such that (\partial f)/(\partial x), (\partial f)/(\partial y) are continuous at (a,b). Show that 'f' is differentiable at (a, b). Is the converse of this is true? Justify your answer.

Q2. (a) for some appropriate S. 18 State the name of this theorem. 2 If \v₁, v₂, \ldots, v_n\ is a basis of a vector space V and if \w₁, w₂, \ldots, w_m\ is linearly

(a) for some appropriate S. 18 State the name of this theorem. 2 If \v₁, v₂, \ldots, v_n\ is a basis of a vector space V and if \w₁, w₂, \ldots, w_m\ is linearly

(b) independent in V, then show that m ≤ n. 20
3. Define
(i) Euclidean Domain. 3 (a)
(ii) Principal Ideal Domain (PID). 3 Show that every Euclidean Domain is PID. 14 (b) State Cayley Hamilton theorem and using it find inverse of the matrix A if it exists. 2 A = \beginbmatrix 1 & 2 & 3 \\ 2 & 2 & 4 \\ 3 & 4 & 8 \endbmatrix 18 SECTION - B If f ' is a real-valued continuous function on a compact metric space X, then
4. (a) show that f
(X) , range of f' is compact and f' attains a maximum and minimum at points of X. 10 + Define absolute convergence and conditional convergence for improper integrals (b) of the type ∫ f
(x) dx for continuous function f
(x) . 2 + 2 Show that ∫₀^∞ (sin x)/(x) dx is convergent but not absolutely. 8 + 8
5. Define differentiability of a function of two variables at a point. (a) 2 Let f: E → \mathbbR be defined on a neighbourhood E of (a,b) ∈ \mathbbR × \mathbbR such that (\partial f)/(\partial x), (\partial f)/(\partial y) are continuous at (a,b). Show that 'f' is differentiable at (a, b). Is the converse of this is true? Justify your answer.

• (a) for some appropriate S. 18 State the name of this theorem. 2 If \v₁, v₂, \ldots, v_n\ is a basis of a vector space V and if \w₁, w₂, \ldots, w_m\ is linearly
• (b) independent in V, then show that m ≤ n. 20

Q3. (a) 4. show that f
(X) , range of f' is compact and f' attains a maximum and minimum at points of X.

(a) 4. show that f
(X) , range of f' is compact and f' attains a maximum and minimum at points of X.

(b) Define absolute convergence and conditional convergence for improper integrals (b) of the type ∫ f
(x) dx for continuous function f
(x) .
Show that ∫0^∞ sin x/x dx is convergent but not absolutely.

• (a) 4. show that f
• (b) Define absolute convergence and conditional convergence for improper integrals (b) of the type ∫ f

Q4. 5. Define differentiability of a function of two variables at a point. (a) Let f: E → R be defined on a neighbourhood E of (a,b) ∈ R×R such that ∂f/∂x, ∂f/∂y are continuous at (a,b). Show that 'f' is differentiable at (a, b). Is the converse of this true? Justify your answer.

Q5. 8.