The volume of a cube is increasing at the rate of 9 cm³/sec. How fast is the surface area increasing when the length of an edge is 10 cm.
The volume of a sphere is increasing at the rate of 8 cm³/sec. Find the rate at which its surface area is increasing when the radius of the sphere is 12 cm.
For the curve y = 5x – 2x³, if x increases at the rate of 2 units/sec, then find the rate of change of the slope of the curve when x = 3.
Find the equation of tangents to the curve y = x³ + 2x – 4 which are perpendicular to the line x + 14y – 3 = 0.
Find the equations of the normal to the curve y = x³+ 2x + 6, which are parallel to line x + 14y + 4 = 0.
Find the equations of the normal to the curve y = 4x³ – 3x + 5 which are perpendicular to the line 9x – y + 5 = 0..
Find the equations of tangents to the curve y = (x²– 1) (x – 2) at the points, where the curve cuts the X-axis.
Show that the relation S in the set R of real numbers defined as S = {(a, b) : a, b ∈ R and a ≤ b³} is neither reflexive nor symmetric nor transitive.
A and B throw a pair of dice alternatively, till one of them gets a total of 10 and wins the game. Find their respective probabilities of winning, if A starts first.
If C = 0·003x³ + 0·02x² + 6x + 250 gives the amount of carbon pollution in air in an area on the entry of x number of vehicles, then find the marginal carbon pollution in the air, when 3 vehicles have entered in the area.
A bag contains (2n + 1) coins. It is known that (n – 1) of these coins have a head on both sides, whereas the rest of the coins are fair. A coin is picked up at random from the bag and is tossed. If the probability that the toss results in a head is 31/42, determine the value of n.