CAIE P3 (Pure Mathematics 3) Specimen

1 Solve the inequality \(| 2 x - 5 | > 3 | 2 x + 1 |\).

2 Using the substitution \(u = 3 ^ { x }\), solve the equation \(3 ^ { x } + 3 ^ { 2 x } = 3 ^ { 3 x }\) giving your answer correct to 3 significant figures.

3 The angles \(\theta\) and \(\phi\) lie between \(0 ^ { \circ }\) and \(180 ^ { \circ }\), and are such that $$\tan ( \theta - \phi ) = 3 \quad \text { and } \quad \tan \theta + \tan \phi = 1$$ Find the possible values of \(\theta\) and \(\phi\).

4 The equation \(x ^ { 3 } - x ^ { 2 } - 6 = 0\) has one real root, denoted by \(\alpha\).

1. Find by calculation the pair of consecutive integers between which \(\alpha\) lies.
2. Show that, if a sequence of values given by the iterative formula $$x _ { n + 1 } = \sqrt { } \left( x _ { n } + \frac { 6 } { x _ { n } } \right)$$ converges, then it converges to \(\alpha\).
3. Use this iterative formula to determine \(\alpha\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.

5 The equation of a curve is \(y = \mathrm { e } ^ { - 2 x } \tan x\), for \(0 \leqslant x < \frac { 1 } { 2 } \pi\).

1. Obtain an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and show that it can be written in the form \(\mathrm { e } ^ { - 2 x } ( a + b \tan x ) ^ { 2 }\), where \(a\) and \(b\) are constants.
2. Explain why the gradient of the curve is never negative.
3. Find the value of \(x\) for which the gradient is least.

6 The polynomial \(8 x ^ { 3 } + a x ^ { 2 } + b x - 1\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( x + 1 )\) is a factor of \(\mathrm { p } ( x )\) and that when \(\mathrm { p } ( x )\) is divided by ( \(2 x + 1\) ) the remainder is 1 .

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

7 The points \(A , B\) and \(C\) have position vectors, relative to the origin \(O\), given by $$\overrightarrow { O A } = \left( \begin{array} { l } 1
2
0 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { l } 3
0
1 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { l } 1
1
4 \end{array} \right)$$ The plane \(m\) is perpendicular to \(A B\) and contains the point \(C\).

1. Find a vector equation for the line passing through \(A\) and \(B\).
2. Obtain the equation of the plane \(m\), giving your answer in the form \(a x + b y + c z = d\).
3. The line through \(A\) and \(B\) intersects the plane \(m\) at the point \(N\). Find the position vector of \(N\) and show that \(C N = \sqrt { } ( 13 )\).

8 The variables \(x\) and \(\theta\) satisfy the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} \theta } = ( x + 2 ) \sin ^ { 2 } 2 \theta$$ and it is given that \(x = 0\) when \(\theta = 0\). Solve the differential equation and calculate the value of \(x\) when \(\theta = \frac { 1 } { 4 } \pi\), giving your answer correct to 3 significant figures.

9 The complex number \(3 - \mathrm { i }\) is denoted by \(u\). Its complex conjugate is denoted by \(u ^ { * }\).

1. On an Argand diagram with origin \(O\), show the points \(A , B\) and \(C\) representing the complex numbers \(u , u ^ { * }\) and \(u ^ { * } - u\) respectively. What type of quadrilateral is \(O A B C\) ?
2. Showing your working and without using a calculator, express \(\frac { u ^ { * } } { u }\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
3. By considering the argument of \(\frac { u ^ { * } } { u }\), prove that $$\tan ^ { - 1 } \left( \frac { 3 } { 4 } \right) = 2 \tan ^ { - 1 } \left( \frac { 1 } { 3 } \right) .$$ \includegraphics[max width=\textwidth, alt={}, center]{d4a7604c-9e2c-47ef-a496-8697bc88fdd4-18_360_758_260_689} The diagram shows the curve \(y = \frac { x ^ { 2 } } { 1 + x ^ { 3 } }\) for \(x \geqslant 0\), and its maximum point \(M\). The shaded region \(R\) is enclosed by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = p\).
4. Find the exact value of the \(x\)-coordinate of \(M\).
5. Calculate the value of \(p\) for which the area of \(R\) is equal to 1 . Give your answer correct to 3 significant figures.