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

9709/Oct Nov/2005/3/Q9

(i)  Express (3x^2 + x) / (x + 2)(x^2 + 1) in partial fractions.
(ii) Hence obtain the expansion of (3x^2 + x) / (x + 2)(x^2 + 1) in ascending powers of x, up to and including the term in x^3.

Solution:

Reference: PYQ - Oct/Nov 2005 Paper 3 Q9

Mechanics 1 Modelling with Friction Question 9

In each of the following situations a brick is about to slide down a rough inclined plane. Find the unknown quantity.
(i) The plane is inclined at 30° to the horizontal and the brick has mass 2 kg: find μ.
(ii) The brick has mass 4 kg and the coefficient of friction is 0.7: find the angle of the slope.
(iii) The plane is at 65° to the horizontal and the brick has mass 5 kg: find μ.
(iv) The brick has mass 6 kg and μ is 1.2: find the angle of slope.

Solution:

9709/MayJune/2006/3/Q9

(i) Express 10/(2-x)(1+x^2) in partial fractions. [5]

(ii) Hence, given that |x| < 1, obtain the expansion of 10/(2-x)(1+x^2) in ascending powers of x, up to and including the term in x^3, simplifying the coefficients. [5]

Solution:

Reference: PYQ - May/June 2006 Paper 3 Q9