OCR MEI C3 (Core Mathematics 3) 2006 June

1 Solve the equation \(| 3 x - 2 | = x\).

2 Show that \(\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } x \sin 2 x \mathrm {~d} x = \frac { 3 \sqrt { 3 } - \pi } { 24 }\).

3 Fig. 3 shows the curve defined by the equation \(y = \arcsin ( x - 1 )\), for \(0 \leqslant x \leqslant 2\). \begin{figure}[h] \end{figure}

1. Find \(x\) in terms of \(y\), and show that \(\frac { \mathrm { d } x } { \mathrm {~d} y } = \cos y\).
2. Hence find the exact gradient of the curve at the point where \(x = 1.5\).

4 Fig. 4 is a diagram of a garden pond. \begin{figure}[h] \end{figure} The volume \(V \mathrm {~m} ^ { 3 }\) of water in the pond when the depth is \(h\) metres is given by $$V = \frac { 1 } { 3 } \pi h ^ { 2 } ( 3 - h )$$

1. Find \(\frac { \mathrm { d } V } { \mathrm {~d} h }\). Water is poured into the pond at the rate of \(0.02 \mathrm {~m} ^ { 3 }\) per minute.
2. Find the value of \(\frac { \mathrm { d } h } { \mathrm {~d} t }\) when \(h = 0.4\).

5 Positive integers \(a , b\) and \(c\) are said to form a Pythagorean triple if \(a ^ { 2 } + b ^ { 2 } = c ^ { 2 }\).

1. Given that \(t\) is an integer greater than 1 , show that \(2 t , t ^ { 2 } - 1\) and \(t ^ { 2 } + 1\) form a Pythagorean triple.
2. The two smallest integers of a Pythagorean triple are 20 and 21. Find the third integer. Use this triple to show that not all Pythagorean triples can be expressed in the form \(2 t , t ^ { 2 } - 1\) and \(t ^ { 2 } + 1\).

6 The mass \(M \mathrm {~kg}\) of a radioactive material is modelled by the equation $$M = M _ { 0 } \mathrm { e } ^ { - k t } ,$$ where \(M _ { 0 }\) is the initial mass, \(t\) is the time in years, and \(k\) is a constant which measures the rate of radioactive decay.

1. Sketch the graph of \(M\) against \(t\).
2. For Carbon \(14 , k = 0.000121\). Verify that after 5730 years the mass \(M\) has reduced to approximately half the initial mass. The half-life of a radioactive material is the time taken for its mass to reduce to exactly half the initial mass.
3. Show that, in general, the half-life \(T\) is given by \(T = \frac { \ln 2 } { k }\).
4. Hence find the half-life of Plutonium 239, given that for this material \(k = 2.88 \times 10 ^ { - 5 }\).