All India - 2013

Write the value of

For what value of x, is the matrix a skew symmetric matrix?

If matrix and A² = kA, then write the value of k.

Write the differential equation representing the family of curves y = mx, where m is an arbitrary constant.

If A_ij is the cofactor of the element a_ij of the determinant then write the value of a₃₂. A₃₂.

P and Q are two points with position vectors and respectively. Write the position vector of a point R which divides the line segment PQ in the ratio 2:1 externally.

Find if for a unit vector

Find the length of the perpendicular drawn from the origin to the plane 2x – 3y + 6z + 21 = 0.

The money to be spent for the welfare of the employees of a firm is proportional to the rate of change of its total revenue (marginal revenue). If the total revenue (in rupees) received from the sale of x units of a product is given by R(x) =3x² +36x + 5, find the marginal revenue when x = 5, and write which value does the question indicate.