9709/May June/33/2017/Q5

A curve has equation y = 2/3 ln(1 + 3 cos² x) for 0 ≤ x ≤ ½

(i) Express dy/dx in terms of tan x.

(ii) Hence find the x-coordinate of the point on the curve where the gradient is −1. Give your answer correct to 3 significant figures.

Solution:

Reference: PYQ - May/Jun 2017 Paper 33 Q5

9709/Oct Nov/2003/3/Q9

Compressed air is escaping from a container. The pressure of the air in the container at time t is P, and the constant atmospheric pressure of the air outside the container is A. The rate of decrease of P is proportional to the square root of the pressure difference (P − A). Thus the differential equation connecting P and t is

where k is a positive constant.
(i) Find, in any form, the general solution of this differential equation.
(ii) Given that P = 5A when t = 0, and that P = 2A when t = 2, show that k =√A.
(iii) Find the value of t when P = A.
(iv) Obtain an expression for P in terms of A and t.

Solution:

Reference: PYQ - Oct/Nov 2003 Paper 3 Q9