MFS Maths Optional 2012 Question Paper (Main)

Q1. Q 1 (a): A company has three operational departments (weaving, processing and packing) with a capacity to produce three different types of cloth namely suiting, shirting and woolens yielding the profit of Rs. 2, Rs. 4 and Rs. 3 per metre respectively. One metre suiting requires 3 minutes in weaving, 2 minutes in processing and 1 minute in packing. One metre of shirting requires 4 minutes in weaving, 1 minute in processing and 3 minutes in packing while one metre of woolen requires 3 minutes in each department. In a week total run time is 60, 40, 80 hours of weaving, processing and packing departments respectively. Formulate the linear programming model to find the product mix to maximize the profit. Solve it using simplex method.

Q1. NOO 2 If
(i) u_k
(x) ∈ c, a ≤ x ≤ b, k = 1, 2, 3, \ldots. 10 (e)
(ii) f
(x) = ∑_k=1^∞ u_k
(x) , uniformly in a ≤ x ≤ b, then prove that ∫₀^b f
(x) dx = ∑_k=1^∞ ∫₀^b u_k
(x) dx SECTION - A

Q2.

(a) Prove the following theorems : Let G = \langle a \rangle be a cyclic group order n and H be a subgroup of G generated 10 i) by a^m, m\leqn. Then, O(H) = n \over g c d (m, n). Any finite group is isomorphic to a permutation group. 10. \rm
(ii) Verify Cayley - Hamilton theorem for the matrix \beginvmatrix 1 & 2 & 3 \\ 2 & 4 & 5 \\ 3 & 5 & 6 \endvmatrix and hence find its

(b) inverse. 20
3. Let f: G → H be a group homomorphism of a group G onto another group H and (a) let ker ( f ) be the kernel of f . Then prove that (G)/(\ker(f)) = H Evaluate Eigen values and Eigen vectors of the matrix \beginbmatrix 1 & 0 & -1 \\ 1 & 2 & 1 \\ 2 & 2 & 3 \endbmatrix, and determine (b) whether the Eigen vectors are orthogonal. SECTION - B (a) If f
(x) is continuous in [a, b], differentiable in (a, b) and f(a) = f(b), then prove 20
4. that there exists at least one c ∈ (a, b) such that f¹

(c) = 0. Plot the graph of the function f
(x) = 1 - x^2/3. Does this function satisfy all conditions above theorem for x ∈ [-1, 1]. Verify the above theorem for the function f
(x) = x(x-2) e^3x/4, 0 ≤ x ≤ 2.

Q2. (a) Prove the following theorems : Let G = \langle a \rangle be a cyclic group order n and H be a subgroup of G generated 10 i) by a^m, m\leqn. Then, O(H) = n \over g c d (m, n). Any finite group is isomorphic to a permutation group. 10. \rm
(ii) Verify Cayley - Hamilton theorem for the matrix \beginvmatrix 1 & 2 & 3 \\ 2 & 4 & 5 \\ 3 & 5 & 6 \endvmatrix and hence find its

(a) Prove the following theorems : Let G = \langle a \rangle be a cyclic group order n and H be a subgroup of G generated 10 i) by a^m, m\leqn. Then, O(H) = n \over g c d (m, n). Any finite group is isomorphic to a permutation group. 10. \rm
(ii) Verify Cayley - Hamilton theorem for the matrix \beginvmatrix 1 & 2 & 3 \\ 2 & 4 & 5 \\ 3 & 5 & 6 \endvmatrix and hence find its

(b) inverse. 20
3. Let f: G → H be a group homomorphism of a group G onto another group H and (a) let ker ( f ) be the kernel of f . Then prove that (G)/(\ker(f)) = H Evaluate Eigen values and Eigen vectors of the matrix \beginbmatrix 1 & 0 & -1 \\ 1 & 2 & 1 \\ 2 & 2 & 3 \endbmatrix, and determine (b) whether the Eigen vectors are orthogonal. SECTION - B (a) If f
(x) is continuous in [a, b], differentiable in (a, b) and f(a) = f(b), then prove 20
4. that there exists at least one c ∈ (a, b) such that f¹

(c) = 0. Plot the graph of the function f
(x) = 1 - x^2/3. Does this function satisfy all conditions above theorem for x ∈ [-1, 1]. Verify the above theorem for the function f
(x) = x(x-2) e^3x/4, 0 ≤ x ≤ 2.

• (a) Prove the following theorems : Let G = \langle a \rangle be a cyclic group order n and H be a subgroup of G generated 10 i) by a^m, m\leqn. Then, O(H) = n \over g c d (m, n). Any finite group is isomorphic to a permutation group. 10. \rm
• (b) inverse. 20

Q3. Q 1
(c) : A particle moves with a central acceleration μ r^{-7} and starts from an apse at a distance a with a velocity equal to the velocity which would be acquired by the particle travelling from rest at an infinity to the apse. Show that the equation of its orbit is r² = a² cos 2θ.

Q4. Q 1 (d): Write a computer program in C for evaluation of the integral ∫ from 2.5 to 7.8 of (x³ + 2x² + 5x + 6)/(x² - 3x + n) dx, using Simpson's rule. Select n = 100.

Q5. Q 2 (a)
(i) : Let G = ⟨a⟩ be a cyclic group of order n and H be the subgroup generated by a^m (m ≤ n). Then |H| = n / gcd(m, n).

Q6. Q 2 (a)
(ii) : Any finite group is isomorphic to a permutation group.

Q7. Q 2 (b): Verify Cayley–Hamilton theorem for the matrix ⎡1 2 3; 2 4 5; 3 5 6⎤ and hence find its inverse.

Q8. Q 3 (a): Let f: G → H be a group homomorphism of a group G onto another group H and let ker(f) be the kernel of f. Then prove that G/ker(f) ≅ H.

Q9. Q 3 (b): Evaluate the eigenvalues and eigenvectors of the matrix ⎡1 0 -1; 1 2 1; 2 2 3⎤, and determine whether the eigenvectors are orthogonal.

Q10. (a) Q 4 : If f
(x) is continuous in [a, b], differentiable in (a, b) and f

(a) Q 4 : If f
(x) is continuous in [a, b], differentiable in (a, b) and f(a) = f

(b) , then prove that there exists at least one c ∈ (a, b) such that f'

(c) = 0.

• (a) Q 4 : If f
• (b) , then prove that there exists at least one c ∈ (a, b) such that f'
• (c) = 0.