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/2003/4/Q5

Particles A and B, of masses 0.4 kg and 0.1 kg respectively, are attached to the ends of a light inextensible string. Particle A is held at rest on a horizontal table with the string passing over a smooth pulley at the edge of the table. Particle B hangs vertically below the pulley (see diagram). The system is released from rest. In the subsequent motion a constant frictional force of magnitude 0.6N acts on A. Find

(i) the tension in the string, 

(ii) the speed of B 1.5 s after it starts to move.

Solution:

 
Reference: PYQ - Oct/Nov 2003 Paper 4 Q5

9709/May Jun/2003/4/Q5

S₁ and S₂ are light inextensible strings, and A and B are particles each of mass 0.2 kg. Particle A is suspended from a fixed point O by the string S₁, and particle B is suspended from A by the string S₂.

The particles hang in equilibrium as shown in the diagram.

(i) Find the tensions in S₁ and S₂.

The string S₁ is cut and the particles fall. The air resistance acting on A is 0.4N and the air resistance acting on B is 0.2N.

(ii) Find the acceleration of the particles and the tension in S₂.

Solution:

Reference: PYQ - May/June 2003 Paper 4 Q5

9709/MayJune/1/2005/Q5

The diagram shows a rhombus ABCD. The points B and D have coordinates (2, 10) and (6, 2) respectively, and A lies on the x-axis. The mid-point of BD isM. Find, by calculation, the coordinates of each of M, A and C.

Solution:

Reference: PYQ - May/June 2005 Paper 1 Q5