JNTU
JNTU B.Tech I MATHEMATICS-I Previous Question Paper (June-2022 - R13)
Answer any five questions. All questions carry equal marks.
Main
Determine the rank of the matrix A =
Answer
To find the rank of matrix A, we reduce it to row-echelon form using elementary row operations:
Apply R2 → R2 - 3R1 and R3 → R3 - R1:
Apply R3 → R3 - (1/3)R2:
The number of non-zero rows in the row-echelon form is 2. Therefore, the rank of matrix A is 2.
Investigate the values of λ and μ so that the equations x + y + z = 2, 3x + 2y + 3z = 5, 7x + 3y + λz = μ have (i) no solution, (ii) a unique solution and (iii) an infinite number of solutions.
Answer
The system of equations is represented by the augmented matrix [A|B]:
Perform row operations: R2 → R2 - 3R1 and R3 → R3 - 7R1:
Perform R3 → R3 - 4R2:
- Unique solution: Occurs when rank(A) = rank(A|B) = 3. This requires λ - 7 ≠ 0, so λ ≠ 7. The value of μ can be any real number.
- No solution: Occurs when rank(A) ≠ rank(A|B). This happens if λ - 7 = 0 (λ = 7) and μ - 10 ≠ 0 (μ ≠ 10).
- Infinite solutions: Occurs when rank(A) = rank(A|B) < 3. This happens if λ - 7 = 0 (λ = 7) and μ - 10 = 0 (μ = 10).
Reduce the quadratic form 3x2 + 5y2 + 3z2 - 2yz + 2zx - 2xy to the canonical form and also write the nature of the quadratic matrix.
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State Rolle’s theorem and verify it for f(x) = e-x sin x in [0, π].
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If x = r cos θ, y = r sin θ then prove that ∂(x, y) / ∂(r, θ) * ∂(r, θ) / ∂(x, y) = 1.
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Prove that β(m, n) = Γ(m)Γ(n) / Γ(m+n).
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Evaluate ∬ e-(x2+y2) dx dy from 0 to ∞ for both integrals by changing to polar coordinates.
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If a substance cools from 370 K to 330 K in 10 min, when the temperature of the surrounding air is 290 K, find the temperature of the substance after 40 min.
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Solve (D2 - 3D + 2)y = ex + cos 3x.
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Solve (D3 - D2 - D + 1)y = 1 + x + cos 2x.
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Solve by method of variation of parameters (d2y/dx2) + 4y = tan 2x.
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Find L{(sin 3t cos t) / t} and L{t2 cos 2t}.
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Find L-1 {s / ((s2+4)(s2+9))} using Convolution theorem.
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(i) Find the Laplace Transform of (1/t) sin3 t and (ii) Find L{t e-t sin 2t}.
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Solve the differential equation y'' + y' + y = e-t, y(0)=0, y'(0)=1 by Laplace transforms method.
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