CAIE P3 (Pure Mathematics 3) 2012 November

1 Solve the equation $$\ln ( x + 5 ) = 1 + \ln x$$ giving your answer in terms of e.

2

1. Express \(24 \sin \theta - 7 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give the value of \(\alpha\) correct to 2 decimal places.
2. Hence find the smallest positive value of \(\theta\) satisfying the equation $$24 \sin \theta - 7 \cos \theta = 17$$

3 The parametric equations of a curve are $$x = \frac { 4 t } { 2 t + 3 } , \quad y = 2 \ln ( 2 t + 3 )$$

1. Express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\), simplifying your answer.
2. Find the gradient of the curve at the point for which \(x = 1\).

4 The variables \(x\) and \(y\) are related by the differential equation $$\left( x ^ { 2 } + 4 \right) \frac { d y } { d x } = 6 x y$$ It is given that \(y = 32\) when \(x = 0\). Find an expression for \(y\) in terms of \(x\).

5 The expression \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 3 x \mathrm { e } ^ { - 2 x }\).

1. Find the exact value of \(\mathrm { f } ^ { \prime } \left( - \frac { 1 } { 2 } \right)\).
2. Find the exact value of \(\int _ { - \frac { 1 } { 2 } } ^ { 0 } \mathrm { f } ( x ) \mathrm { d } x\).

6 \includegraphics[max width=\textwidth, alt={}, center]{adbef77f-e2ac-40ce-a56b-cf6776534ec1-3_561_732_255_705} The diagram shows the curve \(y = x ^ { 4 } + 2 x ^ { 3 } + 2 x ^ { 2 } - 4 x - 16\), which crosses the \(x\)-axis at the points \(( \alpha , 0 )\) and \(( \beta , 0 )\) where \(\alpha < \beta\). It is given that \(\alpha\) is an integer.

1. Find the value of \(\alpha\).
2. Show that \(\beta\) satisfies the equation \(x = \sqrt [ 3 ] { } ( 8 - 2 x )\).
3. Use an iteration process based on the equation in part (ii) to find the value of \(\beta\) correct to 2 decimal places. Show the result of each iteration to 4 decimal places.

7 \includegraphics[max width=\textwidth, alt={}, center]{adbef77f-e2ac-40ce-a56b-cf6776534ec1-3_543_1091_1402_529} The diagram shows part of the curve \(y = \sin ^ { 3 } 2 x \cos ^ { 3 } 2 x\). The shaded region shown is bounded by the curve and the \(x\)-axis and its exact area is denoted by \(A\).

1. Use the substitution \(u = \sin 2 x\) in a suitable integral to find the value of \(A\).
2. Given that \(\int _ { 0 } ^ { k \pi } \left| \sin ^ { 3 } 2 x \cos ^ { 3 } 2 x \right| \mathrm { d } x = 40 A\), find the value of the constant \(k\).

8 Two lines have equations $$\mathbf { r } = \left( \begin{array} { r } 5 \\ 1 \\ - 4 \end{array} \right) + s \left( \begin{array} { r } 1 \\ - 1 \\ 3 \end{array} \right) \quad \text { and } \quad \mathbf { r } = \left( \begin{array} { r } p \\ 4 \\ - 2 \end{array} \right) + t \left( \begin{array} { r } 2 \\ 5 \\ - 4 \end{array} \right) ,$$ where \(p\) is a constant. It is given that the lines intersect.

1. Find the value of \(p\) and determine the coordinates of the point of intersection.
2. Find the equation of the plane containing the two lines, giving your answer in the form \(a x + b y + c z = d\), where \(a , b , c\) and \(d\) are integers.

9

1. Express \(\frac { 9 - 7 x + 8 x ^ { 2 } } { ( 3 - x ) \left( 1 + x ^ { 2 } \right) }\) in partial fractions.
2. Hence obtain the expansion of \(\frac { 9 - 7 x + 8 x ^ { 2 } } { ( 3 - x ) \left( 1 + x ^ { 2 } \right) }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).

10

1. Without using a calculator, solve the equation \(\mathrm { i } w ^ { 2 } = ( 2 - 2 \mathrm { i } ) ^ { 2 }\).
2. 1. Sketch an Argand diagram showing the region \(R\) consisting of points representing the complex numbers \(z\) where $$| z - 4 - 4 i | \leqslant 2$$
   2. For the complex numbers represented by points in the region \(R\), it is given that $$p \leqslant | z | \leqslant q \quad \text { and } \quad \alpha \leqslant \arg z \leqslant \beta$$ Find the values of \(p , q , \alpha\) and \(\beta\), giving your answers correct to 3 significant figures.