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JNTUH B.Tech I Year MATHEMATICS-I Previous Question Paper (June - 2022 - R18)
Answer any five questions. All questions carry equal marks.
Main
Apply rank test to find whether the following system has any solution other than x = y = z = w = 0:
Answer
The system is a homogeneous system of linear equations AX = 0, where A is the coefficient matrix:
To find if there is a non-trivial solution, we calculate the rank of A. Reducing A to row-echelon form:
- R2 → R2 - 8R1, R3 → R3 - 5R1, R4 → R4 - 8R1: [[1, 2, 3, 4], [0, -11, -23, -28], [0, -4, -7, -19], [0, -13, -17, -30]]
- R2 → R2 - 3R3: [[1, 2, 3, 4], [0, 1, -2, 29], [0, -4, -7, -19], [0, -13, -17, -30]]
- R3 → R3 + 4R2, R4 → R4 + 13R2: [[1, 2, 3, 4], [0, 1, -2, 29], [0, 0, -15, 97], [0, 0, -43, 347]]
Since the determinant of the 4x4 matrix is non-zero (rank = 4), the only solution is the trivial solution x = y = z = w = 0.
By using Gauss’s elimination method solve:
Answer
The augmented matrix [A|B] is:
Applying Gauss elimination:
- Swap R1 and R2: [[1, -3, -1, -30], [5, -1, -2, 142], [2, -1, -3, -5]]
- R2 → R2 - 5R1, R3 → R3 - 2R1: [[1, -3, -1, -30], [0, 14, 3, 292], [0, 5, -1, 55]]
- R2 → R2 - 3R3: [[1, -3, -1, -30], [0, -1, 6, 127], [0, 5, -1, 55]]
- R3 → R3 + 5R2: [[1, -3, -1, -30], [0, -1, 6, 127], [0, 0, 29, 690]]
Solving by back substitution: 29z = 690 gives z ≈ 23.79; -y + 6z = 127 gives y ≈ 15.74; x - 3y - z = -30 gives x ≈ 7.05.
Find the characteristic polynomial of the matrix A and verify Cayley-Hamilton theorem. Hence find A-1, if exists:
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The matrix A = [[a, h], [h, b]] is transformed to the diagonal form D = T-1AT, where T = [[cos θ, sin θ], [-sin θ, cos θ]]. Find the value of θ which gives this diagonal transformation.
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Prove that the following series for conditional convergence:
i) 1 - 1/2 + 1/3 - 1/4 + ...
ii) 1 - 1.3/2 + 1.3.5/2.4 - 1.3.5.7/2.4.6 + ...
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Prove that the series (sin x)/1 - (sin^3 x)/2 + (sin^3 x)/3 - ... converges absolutely.
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Express the log(cos x) as Taylor series about x = π/3.
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Find the volume of solid generated by revolving the ellipse x2/a2 + y2/b2 = 1 (0 < b < a) rotated about minor axis.
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If u = xf(x, y) + yg(x, y), show that:
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Determine whether u and v are functionally dependent, where u = sin x + cos y and v = cos x + sin y.
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Find the greatest value of u = xyz, if x, y and z are positive real numbers for which 4x + 2y + z = 12.
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A function f(x, y) is written in terms of new variables u = ex cos y and v = ex sin y. Show that:
and
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By using techniques involving the Gamma function, find the value of:
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By using techniques involving the Beta function, find the exact value of:
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