radius R is of the volume of the sphere. . Show that the right circular cone of least curved surface and given volume has an altitude equal to time the radius of the base. .
Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is tan . Show that semi-vertical angle of right circular cone of given surface area and maximum volume is sin − . Choose the correct answer in Questions and . .
The point on the curve x = y which is nearest to the point ( , ) is (A) ( , ) (B) ( , ) (C) ( , ) (D) ( , ) . For all real values of x , the minimum value of is (A) (B) (C) (D) . The maximum value of [ ( ) ] x x − , ≤ ≤ is (A) (B) (C) (D) Miscellaneous Examples Example A car starts from a point P at time t = seconds and stops at point Q. The distance x , in metres, covered by it, in t seconds is given by t t Find the time taken by it to reach Q and also find distance between P and Q.
Solution Let v be the velocity of the car at t seconds. Now x = t t Therefore v = dx dt = t – t = t ( – t ) Thus, v = gives t = and/or t = . Now v = at P as well as at Q and at P, t = . So, at Q, t = .
Thus, the car will reach the point Q after seconds. Also the distance travelled in seconds is given by x ] t = = m Example A water tank has the shape of