CSS 2022 Applied Mathematics Paper

Find the unit normal vector to the surface x² + y² + z² = 9 at the point (1, 2, 2). Also find the directional derivative of f(x, y, z) = x²yz + 4xz² at (1, -2, -1) in the direction 2i - j - 2k.

A uniform beam of length 2a and weight W rests with its ends on two smooth planes inclined at angles α and β to the horizontal. Find the inclination of the beam to the horizontal in the position of equilibrium.

A particle describes an ellipse under a force to the focus. Find the law of force and the velocity at any point of its path.

A particle is projected with a velocity u at an angle α to the horizontal. Show that the path is a parabola and find the range on the horizontal plane.

Solve the differential equation (D² + 4)y = x sin x using the method of undetermined coefficients.

Evaluate the integral from 0 to 6 of 1/(1+x²) dx using Simpson's 1/3 rule with h = 1.

Define Covariant and Contravariant tensors. Prove that the Kronecker delta is a mixed tensor of rank 2.