CAIE P3 (Pure Mathematics 3) 2013 June

1 Solve the equation \(| x - 2 | = \left| \frac { 1 } { 3 } x \right|\).

2 The sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { x _ { n } \left( x _ { n } ^ { 3 } + 100 \right) } { 2 \left( x _ { n } ^ { 3 } + 25 \right) }$$ with initial value \(x _ { 1 } = 3.5\), converges to \(\alpha\).

1. Use this formula to calculate \(\alpha\) correct to 4 decimal places, showing the result of each iteration to 6 decimal places.
2. State an equation satisfied by \(\alpha\) and hence find the exact value of \(\alpha\).

3
\includegraphics[max width=\textwidth, alt={}, center]{7c125770-1ded-4763-8453-b07ef43e83e9-2_392_727_927_708} The variables \(x\) and \(y\) satisfy the equation \(y = A e ^ { - k x ^ { 2 } }\), where \(A\) and \(k\) are constants. The graph of \(\ln y\) against \(x ^ { 2 }\) is a straight line passing through the points \(( 0.64,0.76 )\) and \(( 1.69,0.32 )\), as shown in the diagram. Find the values of \(A\) and \(k\) correct to 2 decimal places.

4 The polynomial \(a x ^ { 3 } - 20 x ^ { 2 } + x + 3\), where \(a\) is a constant, is denoted by \(\mathrm { p } ( x )\). It is given that \(( 3 x + 1 )\) is a factor of \(\mathrm { p } ( x )\).

1. Find the value of \(a\).
2. When \(a\) has this value, factorise \(\mathrm { p } ( x )\) completely.

5
\includegraphics[max width=\textwidth, alt={}, center]{7c125770-1ded-4763-8453-b07ef43e83e9-2_446_601_1969_772} The diagram shows the curve with equation $$x ^ { 3 } + x y ^ { 2 } + a y ^ { 2 } - 3 a x ^ { 2 } = 0$$ where \(a\) is a positive constant. The maximum point on the curve is \(M\). Find the \(x\)-coordinate of \(M\) in terms of \(a\).

1. By differentiating \(\frac { 1 } { \cos x }\), show that the derivative of \(\sec x\) is \(\sec x \tan x\). Hence show that if \(y = \ln ( \sec x + \tan x )\) then \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \sec x\).
2. Using the substitution \(x = ( \sqrt { } 3 ) \tan \theta\), find the exact value of $$\int _ { 1 } ^ { 3 } \frac { 1 } { \sqrt { \left( 3 + x ^ { 2 } \right) } } \mathrm { d } x$$ expressing your answer as a single logarithm.

1. By first expanding \(\cos \left( x + 45 ^ { \circ } \right)\), express \(\cos \left( x + 45 ^ { \circ } \right) - ( \sqrt { } 2 ) \sin x\) in the form \(R \cos ( x + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give the value of \(R\) correct to 4 significant figures and the value of \(\alpha\) correct to 2 decimal places.
2. Hence solve the equation $$\cos \left( x + 45 ^ { \circ } \right) - ( \sqrt { } 2 ) \sin x = 2$$ for \(0 ^ { \circ } < x < 360 ^ { \circ }\).
3. Express \(\frac { 1 } { x ^ { 2 } ( 2 x + 1 ) }\) in the form \(\frac { A } { x ^ { 2 } } + \frac { B } { x } + \frac { C } { 2 x + 1 }\).
4. The variables \(x\) and \(y\) satisfy the differential equation $$y = x ^ { 2 } ( 2 x + 1 ) \frac { \mathrm { d } y } { \mathrm {~d} x }$$ and \(y = 1\) when \(x = 1\). Solve the differential equation and find the exact value of \(y\) when \(x = 2\). Give your value of \(y\) in a form not involving logarithms.
   (a) The complex number \(w\) is such that \(\operatorname { Re } w > 0\) and \(w + 3 w ^ { * } = \mathrm { i } w ^ { 2 }\), where \(w ^ { * }\) denotes the complex conjugate of \(w\). Find \(w\), giving your answer in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
   (b) On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) which satisfy both the inequalities \(| z - 2 i | \leqslant 2\) and \(0 \leqslant \arg ( z + 2 ) \leqslant \frac { 1 } { 4 } \pi\). Calculate the greatest value of \(| z |\) for points in this region, giving your answer correct to 2 decimal places.

10 The points \(A\) and \(B\) have position vectors \(2 \mathbf { i } - 3 \mathbf { j } + 2 \mathbf { k }\) and \(5 \mathbf { i } - 2 \mathbf { j } + \mathbf { k }\) respectively. The plane \(p\) has equation \(x + y = 5\).

1. Find the position vector of the point of intersection of the line through \(A\) and \(B\) and the plane \(p\).
2. A second plane \(q\) has an equation of the form \(x + b y + c z = d\), where \(b , c\) and \(d\) are constants. The plane \(q\) contains the line \(A B\), and the acute angle between the planes \(p\) and \(q\) is \(60 ^ { \circ }\). Find the equation of \(q\).