9709/May June/2011/42/Q3

The velocity-time graph shown models the motion of a parachutist falling vertically. There are four
stages in the motion:
• falling freely with the parachute closed,
• decelerating at a constant rate with the parachute open,
• falling with constant speed with the parachute open,
• coming to rest instantaneously on hitting the ground.

(i) Show that the total distance fallen is 1048m.
The weight of the parachutist is 850N.
(ii) Find the upward force on the parachutist due to the parachute, during the second stage.

Solution:

Reference: PYQ - May/Jun 2011 Paper 42 Q3

9709/May June/2011/42/Q2

An object of mass 8 kg slides down a line of greatest slope of an inclined plane. Its initial speed at the top of the plane is 3ms−1 and its speed at the bottom of the plane is 8ms−1. The work done against the resistance to motion of the object is 120 J. Find the height of the top of the plane above the level of the bottom.

Solution:

Reference: PYQ - May/Jun 2011 Paper 42 Q2

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/May Jun/2008/3/Q10

The points A and B have position vectors, relative to the origin O, given by

OA = i + 2j + 3k  and  OB = 2i + j + 3k.

The line l has vector equation

r = (1 − 2t)i + (5 + t)j + (2 − t)k.

(i) Show that l does not intersect the line passing through A and B.

(ii) The point P lies on l and is such that angle PAB is equal to 60^o. Given that the position vector

of P is (1 − 2t)i + (5 + t)j + (2 − t)k, show that 3t² + 7t + 2 = 0. Hence find the only possible

position vector of P.

Solution:

Reference: PYQ - May/Jun 2008 Paper 3 Q10

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

9709/Oct Nov/2009/32/Q6

6i) Use the substitution x = 2 tan q to show that

6ii) Hence find the exact value of

Solution:

Reference: PYQ - Oct/Nov 2009 Paper 32 Q6

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9709/Oct Nov/2003/4/Q6

One end of a light inextensible string is attached to a fixed point A of a fixed vertical wire. The other end of the string is attached to a small ring B, of mass 0.2 kg, through which the wire passes.

A horizontal force of magnitude 5N is applied to the mid-point M of the string. The system is in equilibrium with the string taut, with B below A, and with angles ABM and BAM equal to 30◦ (see diagram).

(i) Show that the tension in BM is 5N.

(ii) The ring is on the point of sliding up the wire. Find the coefficient of friction between the ring and the wire.

(iii) A particle of mass m kg is attached to the ring. The ring is now on the point of sliding down the wire. Given that the coefficient of friction between the ring and the wire is unchanged, find the value of m.

Solution:

Reference: PYQ - Oct/Nov 2003 Paper 4 Q6