CAIE P3 (Pure Mathematics 3) 2011 June

1 Use logarithms to solve the equation \(5 ^ { 2 x - 1 } = 2 \left( 3 ^ { x } \right)\), giving your answer correct to 3 significant figures.

2 The curve \(y = \frac { \ln x } { x ^ { 3 } }\) has one stationary point. Find the \(x\)-coordinate of this point.

3 Show that \(\int _ { 0 } ^ { 1 } ( 1 - x ) \mathrm { e } ^ { - \frac { 1 } { 2 } x } \mathrm {~d} x = 4 \mathrm { e } ^ { - \frac { 1 } { 2 } } - 2\).

4

1. Show that the equation $$\tan \left( 60 ^ { \circ } + \theta \right) + \tan \left( 60 ^ { \circ } - \theta \right) = k$$ can be written in the form $$( 2 \sqrt { } 3 ) \left( 1 + \tan ^ { 2 } \theta \right) = k \left( 1 - 3 \tan ^ { 2 } \theta \right)$$
2. Hence solve the equation $$\tan \left( 60 ^ { \circ } + \theta \right) + \tan \left( 60 ^ { \circ } - \theta \right) = 3 \sqrt { } 3$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).

5 The polynomial \(a x ^ { 3 } + b x ^ { 2 } + 5 x - 2\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( 2 x - 1 )\) is a factor of \(\mathrm { p } ( x )\) and that when \(\mathrm { p } ( x )\) is divided by \(( x - 2 )\) the remainder is 12 .

1. Find the values of \(a\) and \(b\).
2. When \(a\) and \(b\) have these values, find the quadratic factor of \(\mathrm { p } ( x )\).

6

1. By sketching a suitable pair of graphs, show that the equation $$\cot x = 1 + x ^ { 2 }$$ where \(x\) is in radians, has only one root in the interval \(0 < x < \frac { 1 } { 2 } \pi\).
2. Verify by calculation that this root lies between 0.5 and 0.8.
3. Use the iterative formula $$x _ { n + 1 } = \tan ^ { - 1 } \left( \frac { 1 } { 1 + x _ { n } ^ { 2 } } \right)$$ to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

7

1. Find the roots of the equation $$z ^ { 2 } + ( 2 \sqrt { } 3 ) z + 4 = 0$$ giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
2. State the modulus and argument of each root.
3. Showing all your working, verify that each root also satisfies the equation $$z ^ { 6 } = - 64$$

8
\includegraphics[max width=\textwidth, alt={}, center]{5b219e1c-e5a0-4f75-910d-fca9761e5088-3_435_895_799_625} The diagram shows the curve \(y = 5 \sin ^ { 3 } x \cos ^ { 2 } x\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\), and its maximum point \(M\).

1. Find the \(x\)-coordinate of \(M\).
2. Using the substitution \(u = \cos x\), find by integration the area of the shaded region bounded by the curve and the \(x\)-axis.

9 In a chemical reaction, a compound \(X\) is formed from two compounds \(Y\) and \(Z\). The masses in grams of \(X , Y\) and \(Z\) present at time \(t\) seconds after the start of the reaction are \(x , 10 - x\) and \(20 - x\) respectively. At any time the rate of formation of \(X\) is proportional to the product of the masses of \(Y\) and \(Z\) present at the time. When \(t = 0 , x = 0\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 2\).

1. Show that \(x\) and \(t\) satisfy the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = 0.01 ( 10 - x ) ( 20 - x )$$
2. Solve this differential equation and obtain an expression for \(x\) in terms of \(t\).
3. State what happens to the value of \(x\) when \(t\) becomes large.

10 With respect to the origin \(O\), the lines \(l\) and \(m\) have vector equations \(\mathbf { r } = 2 \mathbf { i } + \mathbf { k } + \lambda ( \mathbf { i } - \mathbf { j } + 2 \mathbf { k } )\) and \(\mathbf { r } = 2 \mathbf { j } + 6 \mathbf { k } + \mu ( \mathbf { i } + 2 \mathbf { j } - 2 \mathbf { k } )\) respectively.

1. Prove that \(l\) and \(m\) do not intersect.
2. Calculate the acute angle between the directions of \(l\) and \(m\).
3. Find the equation of the plane which is parallel to \(l\) and contains \(m\), giving your answer in the form \(a x + b y + c z = d\).