JNTU
JNTU B.Tech I MATHEMATICS - I Previous Question Paper (March/April - 2023 - R18)
Note: 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
Find the rank of the matrix by reducing to Echelon form A =
Answer
To find the rank of matrix A, we reduce it to row echelon form using elementary row operations:
Apply row operations:
- R2 → R2 - 2R1: [[1, 4, 2, 1], [0, -5, 0, 1], [3, 3, 6, 12]]
- R3 → R3 - 3R1: [[1, 4, 2, 1], [0, -5, 0, 1], [0, -9, 0, 9]]
- R3 → R3 - (9/5)R2: [[1, 4, 2, 1], [0, -5, 0, 1], [0, 0, 0, 7.2]]
The matrix is now in row echelon form. Since there are 3 non-zero rows, the rank of the matrix is 3.
Check whether the matrix is Skew-Hermitian or not.
Answer
A square matrix A is defined as Skew-Hermitian if it satisfies the condition Aθ = -A, where Aθ is the conjugate transpose of the matrix (the transpose of the complex conjugate of A). To check if a given matrix is Skew-Hermitian, follow these steps:
- Find the complex conjugate of each element in the matrix (replace i with -i).
- Transpose the resulting matrix to obtain Aθ.
- Compare Aθ with -A (the matrix obtained by multiplying every element of the original matrix by -1).
- If Aθ = -A, the matrix is Skew-Hermitian; otherwise, it is not.
Note that for a matrix to be Skew-Hermitian, all its principal diagonal elements must be either zero or purely imaginary numbers.
If A =
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If A =
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Test for convergence of the series
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Test for convergence of the series
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Evaluate
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Find the value of 'c' using Cauchy's mean value theorem for the function f(x) = x2 and g(x) = x3 in [1, 2].
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If x3 + y3 - 3axy = 0 then find the value of dy/dx.
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Using Euler’s theorem find the value of x(∂u/∂x) + y(∂u/∂y), if u = (x3+y3)/(x-y).
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PART - B
Find rank of matrix A by reducing it to normal form.
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Solve the system of equations x – y + 2z = 4, 3x + y + 4z = 6, x + y + z = 1 by using Gauss elimination method.
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Find the inverse of the matrix by using Gauss-Jordan method.
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Solve the system of equations x+y+54z=110; 27x + 6y – z = 85; 6x+15y + 2z = 72 using Gauss Seidel method.
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Find the Eigen values and Eigen vectors of matrix A.
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Verify Cayley Hamilton theorem for the Matrix A and hence find A-1.
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Reduce the quadratic form to canonical form using orthogonal transformation. Also find signature and rank of the quadratic form.
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Examine the convergence of the series.
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Examine the convergence of the series.
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Test for convergence of the given series.
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Test for convergence of the given series.
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If f(x) = 1/x and g(x) = 1/x2 prove that ‘c’ of the Cauchy’s generalized mean value theorem is the geometric mean of ‘a’ and ‘b’ for a > 0, b > 0.
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Find Maclaurin’s series expansion of f(x, y) = sin(x/y).
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Find the area bounded by pair of curves.
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Evaluate the given integral.
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Find using Euler’s theorem for the function.
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If u=x/yz, v=y/xz, w=z/xy find the Jacobian.
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Determine whether the functions are dependent. If dependent find the relationship between them.
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Find the highest temperature on the surface of the unit sphere.
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