JNTU
JNTU B.Tech I Year MATHEMATICS-I Previous Question Paper (January/February - 2024 - R13)
i) Question paper consists of Part A, Part B.
ii) Part A is compulsory, which carries 25 marks. In Part A, answer all questions.
iii) In Part B, Answer any one question from each unit. Each question carries 10 marks and may have a, b as sub questions.
Part A
Define the rank of a matrix.
Answer
The rank of a matrix is defined as the maximum number of linearly independent rows or columns in the matrix. Alternatively, it is the order of the largest square sub-matrix with a non-zero determinant.
Consider the system of linear equations:
Answer
To determine the nature of the solutions, we analyze the augmented matrix [A|B]:
Performing row operations R2 → R2 - R1 and R3 → R3 - R1:
Performing R3 → R3 - 2R2:
The system is consistent if the rank of the coefficient matrix equals the rank of the augmented matrix.
- Unique solution: Occurs when the determinant of A is non-zero, i.e., 1+a ≠ 0. Thus, the system has a unique solution for a ≠ -1, regardless of the value of b.
- Infinite solutions: Occurs when the rank of the augmented matrix is less than the number of variables (3) and equal to the rank of A. This requires 1+a = 0 and 12-2b = 0. Thus, a = -1 and b = 6.
- No solution: Occurs when the rank of the augmented matrix is greater than the rank of A. This happens when 1+a = 0 but 12-2b ≠ 0. Thus, a = -1 and b ≠ 6.
If x = r cos θ, y = r sin θ, find J(x, y) / (r, θ).
Answer
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State Rolle’s Mean value theorem and give an example of function for which Rolle’s theorem can’t be applied.
Answer
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Apply Beta or Gamma Function to evaluate:
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Apply Beta or Gamma Function to evaluate:
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Explain Law of natural growth/decay.
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The general form of a differential equation is given by:
Answer
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Determine L { (2 sinh 2t)/t }.
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Determine L { e^(-2t) ∫[0 to t] e^(2τ) cos(3τ) dτ }.
Answer
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Part B
Find the inverse of the matrix:
Answer
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Given A = [[1, 1, -1, 3], [2, -2, 6, 8], [3, 5, -7, 8]]. Then, find the rank(AA^T).
Answer
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Use Gauss Elimination process to solve the system:
Answer
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Apply Lagrange’s Multipliers method to find the shortest and longest distance between the line y = 10 - 2x and the ellipse (x^2/4) + (y^2/9) = 1.
Answer
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If u = x + y^2, v = y + z^2, w = z + x^2, then apply Jacobian concept to check the functional dependency of u, v, w.
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Apply triple integral to find the volume enclosed by the solid:
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Find the area of that part of the surface of the paraboloid y^2 + z^2 = 2ax, which lies between the cylinder y^2 = ax and the plane x = a.
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Solve (D^3 + 1)y = cos^2(x/2) + e^(-x).
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Solve (D^2 + 5D + 6)y = e^(-2x) sec^2 x (1 + 2 tan x).
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Determine the solution of the linear differential equations
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Determine L^(-1){F(s)}, where F(s) = (s^3 - 5s^2 + 6s + 7) / (s(s-1)(s^3 + 6s^2 + 11s + 6)).
Answer
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