All India - 2015

If and , then find the projection of .

If a line makes angles 90°, 60° and θ with x, y and z-axis respectively, where θ is acute, then find θ.

Write the element a₂₃ of a 3 × 3 matrices A = (a_ij) whose elements a_ij are given by a_ij = .

Find the differential equation representing the family of curves v = where A and B are arbitrary constants.

Find the integrating factor of the differential equation

If A = find A² – 5A + 4I and hence find a matrix X such that A² – 5A + 4I + X = 0.

OR

If A = , find (A’)^-1

If f(x) = ,using properties of determinants find the value of f(2x) – f(x).

Find:

OR

Integrate w.r.t x

Evaluate: