LAC Assignment-2 PDF

CHAITANYA BHARATHI INSTITUTE OF TECHNOLOGY (A) ASSIGNMENT-II SUB: LAC CODE: 22MTC01 SEMESTER-I CSE&IT BRANCH- CSE & IT SECTION: C2,C4&H2...

CHAITANYA BHARATHI INSTITUTE OF TECHNOLOGY (A) ASSIGNMENT-II SUB: LAC CODE: 22MTC01 SEMESTER-I CSE&IT BRANCH- CSE & IT SECTION: C2,C4&H2 C3 & H1 Date: Marks:10 S. No M CO BT 1 Make use of Green’s Theorem for the verification of ∫𝑐 [(3𝑥 2 − 1 3 L6 8𝑦 2 )𝑑𝑥 + (4𝑦 − 6𝑥𝑦)𝑑𝑦] where ‘c’ is the region bounded by 𝑦 = √𝑥 and 𝑦 = 𝑥 2 2 Verify Gauss Divergence Theorem for2𝑥 2 𝑦𝑖 − 𝑦 2 𝑗 + 4𝑥𝑧 2 𝑘 taken over the cylinder 𝑦 2 + 𝑧 2 = 9 and 𝑥 = 2 1 3 L6 3 Verify Stoke’s theorem for 𝐹̅ = 𝑦 2 𝑖 − 2𝑥𝑦𝑗 taken around the rectangle L6 bounded by 𝑥 = ±𝑏, 𝑦 = 0, 𝑦 = 𝑎 1 3 4 Examine the vectors are linearly independent or not (3,1,1) , (2,0,1), 1 4 L5 (4,2,1) 5 Express (2,3,4,−1) as linear combination of the vectors 1 4 (1,1,1,2), (1, −1,0,0), (0,0,1,1), (0,1,0,0). L4 6 Let 𝑇 be the linear transformation on 𝑅3 defined by (𝑥, 𝑦, 𝑧) = (2𝑦 + 𝑧, 𝑥 − 4𝑦, 3𝑥). Find the matrix representation of T relative to the 1 4 L5 basis 𝑆 = {𝑢1, 𝑢2, 𝑢3} = {(1,1,1), (1,1,0), (1,0,0)}. 7 Verify rank-nullity theorem for the linear transformation 𝑇: 𝑅3 → L6 𝑅3 where 𝑇(𝑥, 𝑦, 𝑧) = (𝑥 − 𝑦, 2𝑦 + 𝑧, 𝑥 + 𝑦 + 𝑧). 1 4 8 Prove that the set 𝑆 = {(1,1,1), (1,1,0), (1,0,0)} is a basis of 𝑅3. 1 4 L6 9 2 2 0 Find the row and column space of matrix A = [1 4 1 ] 1 4 L3 4 1 −1 10 Solve the following system of equations 1 5 2x + t – z + 3 w = 8, x + y + z – w = -2 L5 3x + 2y – z = 6, 4y + 3z + 2w = -8 11 Find the values of a and b for which the equations 1 5 𝑥 + 𝑦 + 𝑧 = 3, 𝑥 + 2𝑦 + 2𝑧 = 6; 𝑥 + 9𝑦 + 𝑎𝑧 = 𝑏 have L6 (i) No solution (ii) Unique solution (iii) Infinite solutions. 12 Determine the values of λ for which the system of equations may possess non trivial solution and solve in each case 1 5 L5 3𝑥 + 𝑦 − 𝜆𝑧 = 0, 4𝑥 − 2𝑦 − 3𝑧 = 0, 2𝜆𝑥 + 4𝑦 + 𝜆𝑧 = 0 13 2 −1 1 Verify Canley – Hamilton theorem for A = [−1 2 −1] and find 1 5 1 −1 2 L4 A-1 and A4 14 Reduce the quadratic form 7𝑥 2 + 6𝑦 2 + 5𝑧 2 − 4𝑥𝑦 − 4𝑦𝑧 to canonical form and also find Nature, index, sign 1 5 L5 15 Reduce the quadratic form 𝑥2 + 𝑦 2 + 2𝑧 2 − 2𝑥𝑦 + 4𝑥𝑧 + 4𝑦𝑧 to 1 5 L5 canonical form and also find Nature, index, sign Paper Setter Name: Dr. Omeshwar Reddy V