A spherical iron ball of radius 10 cm is coated with a layer of ice of uniform thickness that melts at a rate of 50 cm³ /min. When the thickness of the ice is 5 cm, then the rate at which the thickness (in cm/min) of the ice decreases, is :

If z and w are two complex numbers such that |zw| = 1 and arg(z) – arg(w) = $${\pi \over 2}$$ , then :

Let f(x) = log_e(sin x), (0 < x < $$\pi $$) and g(x) = sin^–1 (e^–x ), (x $$ \ge $$ 0). If $$\alpha $$ is a positive real number such that a = (fog)'($$\alpha $$) and b = (fog)($$\alpha $$), then :

If both the mean and the standard deviation of 50 observations x₁, x₂,..., x₅₀ are equal to 16, then the mean of (x₁ – 4)² , (x₂ – 4)² ,....., (x₅₀ – 4)² is :