JNTU B.Tech I MATHEMATICS-I Previous Question Paper (August - 2018 - R15)

A body is heated to 1100°C and placed in air at 100°C. After one hour its temperature is 600°C. How much additional time is required for it to cool to 300°C?

Find the integrating factor for the following equation:

Find the rank of the matrix:

Find the value of x such that the Rank of the following matrix is 2:

Find the sum and product of Eigen values of A =

If 1, 3, -2 are eigen values of a matrix A, then find the eigen values of 2A+3I.

Write the Taylor’s series expansion for f(x,y) about the point (a,b).

Find the value of ∂^3u / ∂x∂y∂z, where u = x^3 y^3 z^3

Form the partial differential equation of z = ax^2 + by^2 where a and b are arbitrary constants.

Find the order and degree of the partial differential equation:

Solve the differential equation (D^2 - 1)y = x^2e^x.

Find the orthogonal trajectories of the lines y = mx, m is arbitrary constant.

Solve the differential equation (D^2 - 5D + 6)y = Sin 4x.

In a culture of yeast, the active ferment doubles itself in 3 hours. Determine the number of times it multiplies itself in 15 hours.

Find the eigen values and eigen vectors of the matrix:

Find by LU decomposition method solve the linear system:

Find the value of k for which the system of equations (k+1)x + 8y = 4k; kx + (k-3)y = 3k-1 has infinitely many solutions.

Solve the system of equations by the Gauss Seidel method: 10x + y + z = 12, 2x + 10y + z = 13, 2x + 2y + 10z = 14

If a matrix A = [[-1, 0, 0], [2, -3, 0], [1, 4, 2]], then find the Eigen values of A^2?

If A = [[1, 2, -3], [0, 3, 2], [0, 0, -2]], then find eigenvalues of 3A^3 + 5A^2 - 6A - 2I.

Reduce the quadratic form 6x1^2 + 3x2^2 + 3x3^2 - 4x1x2 - 2x3x2 + 4x3x1 to canonical form.

If λ is an eigenvalue of A, then prove that λ^m is an eigenvalue of A^m (m is any positive integer).

If u = x log(xy) where x^3 + y^3 + 3xy = 1, find the value of du/dx.

Find the Taylor’s series expansion of f(x, y) = Sin(xy) about the point (0, π).

Find the maximum and minimum values of the function f(x, y, z) = 2x + 3y + z, under the conditions x^2 + y^2 = 5 and x + y = 1.

The sum of three positive numbers is constant. For what values of x, y and z is their product a maximum?

If p = ∂z/∂x and q = ∂z/∂y, solve: p - q = log(x + y).

If p = ∂z/∂x and q = ∂z/∂y, solve: y^2p - xyq = x(z - 2y).

If p = ∂z/∂x and q = ∂z/∂y, solve: p = log(px - y).

If p = ∂z/∂x and q = ∂z/∂y, solve: p - x^2 = q + y^2.