If d varies directly as `t^(2)` then `dt^(2)=k` where k is some constant ?

The sum of the perimeter of a circle and square is k, where k is some constant. Prove that the sum of their areas is least when the side of square is double the radius of the circle.

Let h(x) be differentiable for all x and let f(x)=(k x+e^x)h(x) , where k is some constant. If h(0)=5,h^(prime)(0)=-2,a n df^(prime)(0)=18 , then the value of k is

If A and B are the points (3, 4, 5) and (-1, 3,-7) respectively then theset of points Psuch that PA ^(2) + PB ^(2) = K ^(2), where K is a constant lie on a proper sphere if:

If A and B be the points (3, 4, 5) and (-1,3, -7), respectively, find the equation of the set of points P such that PA^2 + PB^2 = k^2 , where k is a constant.

A small piece of mass m moves in such a way the P.E. = -(1)/(2)mkr^(2) . Where k is a constant and r is the distance of the particle from origin. Assuming Bohr's model of quantization of angular momentum and circular orbit, r is directly proportional to :

The temperature T of a cooling object drops at a rate proportional to the difference T-S, where S is the constant temperature of the surrounding medium. Thus, (dT)/(dt)=-k(T-S) , where k(>0) is constant and t is the time. Solve the differntial equation if it is given that T(0)=150.

Let I be the purchase value of an equipment and V(t) be the value after it has been used for t years. The value V(t) depreciates at a rate given by differential equation (d V(t))/(dt) =-k(T-t) , where k"">""0 is a constant and T is the total life in years of the equipment. Then the scrap value V(T) of the equipment is : (1) T^2-1/k (2) I-(k T^2)/2 (3) I-(k(T-t)^2)/2 (4) e^(-k T)

Differentiate a^x w.r.t. x, where a is a positive constant.

In each of the following x and y vary directly with each other . Find the constant of variable . (i) y = 3x (ii) 2x=3y (iii) 9x=y (iv) y = 2x