If y = sin(sinx), prove that

Using properties of determinants, prove that = 9(3xyz + xy + yz + zx).

If , find .

OR

If and , find when .

Find the equations of the tangent and the normal, to the curve 16x² + 9y² = 145 at the point (x₁, y₁) where x₁ = 2 and y₁ > 0.

OR

Find the intervals in which the function f(x) = is

(a) Strictly increasing,

(b) Strictly decreasing.

An open tank with a square base and vertical sides is to be constructed from a metal sheet so as to hold a given quantity of water. Show that the cost of material will be least when depth of the tank is half of its width. If the cost is to be least when depth of the tank is half of its width. If the cost is to be borne by nearby settled lower income families, for whom water will be provided, what kind of value is hidden in this question ?