CAIE P3 (Pure Mathematics 3) 2004 June

1 Sketch the graph of \(y = \sec x\), for \(0 \leqslant x \leqslant 2 \pi\).

2 Solve the inequality \(| 2 x + 1 | < | x |\).

3 Find the gradient of the curve with equation $$2 x ^ { 2 } - 4 x y + 3 y ^ { 2 } = 3$$ at the point \(( 2,1 )\).

4

1. Show that if \(y = 2 ^ { x }\), then the equation $$2 ^ { x } - 2 ^ { - x } = 1$$ can be written as a quadratic equation in \(y\).
2. Hence solve the equation $$2 ^ { x } - 2 ^ { - x } = 1$$

5

1. Prove the identity $$\sin ^ { 2 } \theta \cos ^ { 2 } \theta \equiv \frac { 1 } { 8 } ( 1 - \cos 4 \theta )$$
2. Hence find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \sin ^ { 2 } \theta \cos ^ { 2 } \theta \mathrm {~d} \theta$$

6 Given that \(y = 1\) when \(x = 0\), solve the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y ^ { 3 } + 1 } { y ^ { 2 } }$$ obtaining an expression for \(y\) in terms of \(x\).

7

1. The equation \(x ^ { 3 } + x + 1 = 0\) has one real root. Show by calculation that this root lies between - 1 and 0 .
2. Show that, if a sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { 2 x _ { n } ^ { 3 } - 1 } { 3 x _ { n } ^ { 2 } + 1 }$$ converges, then it converges to the root of the equation given in part (i).
3. Use this iterative formula, with initial value \(x _ { 1 } = - 0.5\), to determine the root correct to 2 decimal places, showing the result of each iteration.

8

1. Find the roots of the equation \(z ^ { 2 } - z + 1 = 0\), giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
2. Obtain the modulus and argument of each root.
3. Show that each root also satisfies the equation \(z ^ { 3 } = - 1\).

9 Let \(\mathrm { f } ( x ) = \frac { x ^ { 2 } + 7 x - 6 } { ( x - 1 ) ( x - 2 ) ( x + 1 ) }\).

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Show that, when \(x\) is sufficiently small for \(x ^ { 4 }\) and higher powers to be neglected, $$f ( x ) = - 3 + 2 x - \frac { 3 } { 2 } x ^ { 2 } + \frac { 11 } { 4 } x ^ { 3 } .$$

10
\includegraphics[max width=\textwidth, alt={}, center]{2718ebbb-29e3-46f7-8d8d-ec7d526483f8-3_458_920_1144_609} The diagram shows the curve \(y = \frac { \ln x } { x ^ { 2 } }\) and its maximum point \(M\). The curve cuts the \(x\)-axis at \(A\).

1. Write down the \(x\)-coordinate of \(A\).
2. Find the exact coordinates of \(M\).
3. Use integration by parts to find the exact area of the shaded region enclosed by the curve, the \(x\)-axis and the line \(x = \mathrm { e }\).

11 With respect to the origin \(O\), the points \(P , Q , R , S\) have position vectors given by $$\overrightarrow { O P } = \mathbf { i } - \mathbf { k } , \quad \overrightarrow { O Q } = - 2 \mathbf { i } + 4 \mathbf { j } , \quad \overrightarrow { O R } = 4 \mathbf { i } + 2 \mathbf { j } + \mathbf { k } , \quad \overrightarrow { O S } = 3 \mathbf { i } + 5 \mathbf { j } - 6 \mathbf { k } .$$

1. Find the equation of the plane containing \(P , Q\) and \(R\), giving your answer in the form \(a x + b y + c z = d\).
2. The point \(N\) is the foot of the perpendicular from \(S\) to this plane. Find the position vector of \(N\) and show that the length of \(S N\) is 7 .