JNTU B.Tech I MATRICES AND CALCULUS Previous Question Paper (September - 2023 - R22)

Define rank of a matrix.

What do you mean by indirect methods?

What is characteristic equation of a matrix?

Write the symmetric matrix in the matrix notation of the quadratic form: 2xy + 2yz + 2zx.

State Lagrange’s mean value theorem.

Define improper integral.

State Euler’s theorem.

What are stationary points of z = f(x, y)?

Define volume integral of a vector point function F over a region V.

For finding the area of a plane region R, which double integral is to be evaluated.

Determine the rank of the matrix by reducing it into normal form:
1 2 1 4
2 4 3 4
1 2 3 4
1 2 6 7

Apply Gauss elimination method to solve the equations: x + 4y - z = -5; x + y - 6z = -12; 3x - y - z = 4.

Apply Gauss -Jordon method to solve the equations: 2x + 3y - z = -1, x + 4y + 5z = 25, 3x - 4y + z = 2.

Apply Gauss -Siedel method to solve the equations: 2x + y + 6z = 9, 8x + 3y + 2z = 13, x + 5y + z = 7.

Find the Eigen values and Eigen vectors of the matrix:
8 -6 2
-6 7 -4
1 -4 3

Using Cayley-Hamilton theorem, compute A^-1 for A = [[2, 1, 1], [0, 1, 0], [1, 1, 2]]. Also find A^8 - 5A^7 + 7A^6 - 3A^5 + A^4 - 5A^3 + 8A^2 - 2A + I.

Using Cayley Hamilton theorem for the matrix [[1, 1, 2], [0, -2, 0], [0, 0, 3]], find its inverse.

Reduce the quadratic form 8x^2 + 7y^2 + 3z^2 - 12xy - 8yz + 4zx into a sum of squares by an orthogonal transformation and discuss its nature.

Verify Rolle's theorem for the function f(x) = log(x) in [1, e].

Expand f(x, y) = sin(xy) in powers of (x-1) and (y - pi) upto second degree terms.

Find the volume of the revolution of a loop of the curve y^2 = x^2(3-x)/(1+x) about the x-axis.

Prove that Γ(m)Γ(m + 1/2) = (sqrt(pi)/2^(2m-1))Γ(2m).

If u = cos^-1((x+y)/(sqrt(x)+sqrt(y))), prove that x*du/dx + y*du/dy = -1/4 * tan(u).

If u = x+y and v = tan^-1(x)+tan^-1(y), find ∂(u, v)/∂(x, y). Are u and v functionally related? If so find the relation.

Find the maximum and minimum values of x^3 * y^2 * (1 - x - y).

A rectangular box open at top is to have a volume of 32 cubic feet. Find the dimensions of the box requiring least material for its construction.

Evaluate ∬ xy dxdy over the positive quadrant of the circle x^2 + y^2 = a^2.

Evaluate the integral ∬ (e^-y/y) dy dx from y=x to infinity and x=0 to infinity by changing the order of integration.

Find by double integration the area of the lemniscate r^2 = a^2 * cos(2θ).

Find the volume bounded by the cylinder x^2 + y^2 = 4 and the planes y+z = 4 and z = 0.