Exercise 1 from note (Question 3)

Find the area of the shaded region for each of the following graphs.

Solution:

Vector: The intersection of two lines (Exercise 2, Question 3)

In this question the origin is taken to be at a harbour and the unit vectors i and j to have lengths of 1 km in the directions E and N.

A cargo vessel leaves the harbour and its position vector t hours later is given by r1 = 12ti + 16tj.

A fishing boat is trawling nearby and its position at time t is given by r2 = (10 - 3t)i + (8 + 4t)j.

Solution:

 

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/Oct Nov/2002/4/Q3

A light inextensible string has its ends attached to two fixed points A and B, with A vertically above B. A smooth ring R, of mass 0.8 kg, is threaded on the string and is pulled by a horizontal force of magnitude X newtons. The sections AR and BR of the string make angles of 50^o and 20^o  respectively with the horizontal, as shown in the diagram. The ring rests in equilibrium with the string taut. Find

(i) the tension in the string,

(ii) the value of X.

Solution:

Reference: PYQ - Oct/Nov 2002 Paper 4 Q3

9709/MayJune/2002/3/Q3

The polynomial x^4 + 4^2 + x + a is denoted by p(x). It is given that (x^2 + x + 2) is a factor of p(x). Find the value of a and the other quadratic factor of p(x).

Solution:

Reference: PYQ - May/June 2002 Paper 3 Q3

9709/OctNov/2008/2/Q3

The variables x and y satisfy the equation y = A(b^−x), where A and b are constants. The graph of ln y against x is a straight line passing through the points (0, 1.3) and (1.6, 0.9), as shown in the diagram. Find the values of A and b, correct to 2 decimal places.

Solution:

Reference: PYQ - Oct/Nov 2008 Paper 2 Q3

9709/MayJune/13/2010/Q3

The function f : x  → a + b cos x is defined for 0 ≤ x ≤ 2pi. Given that f(0) = 10 and that f(2/3 pi) = 1, find
(i) the values of a and b, [2]
(ii) the range of f, [1]
(iii) the exact value of f(5/6  pi).  [2]

Solution:
i)

ii)

iii)

Reference: PYQ - May/June 2010 Paper 13 Q3