OCR C4 (Core Mathematics 4) 2008 June

1

1. Simplify \(\frac { \left( 2 x ^ { 2 } - 7 x - 4 \right) ( x + 1 ) } { \left( 3 x ^ { 2 } + x - 2 \right) ( x - 4 ) }\).
2. Find the quotient and remainder when \(x ^ { 3 } + 2 x ^ { 2 } - 6 x - 5\) is divided by \(x ^ { 2 } + 4 x + 1\).

2 Find the exact value of \(\int _ { 1 } ^ { \mathrm { e } } x ^ { 4 } \ln x \mathrm {~d} x\).

3 The equation of a curve is \(x ^ { 2 } y - x y ^ { 2 } = 2\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y ^ { 2 } - 2 x y } { x ^ { 2 } - 2 x y }\).
2. (a) Show that, if \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\), then \(y = 2 x\).
   (b) Hence find the coordinates of the point on the curve where the tangent is parallel to the \(x\)-axis.

4 Relative to an origin \(O\), the points \(A\) and \(B\) have position vectors \(3 \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k }\) and \(\mathbf { i } + 3 \mathbf { j } + 4 \mathbf { k }\) respectively.

1. Find a vector equation of the line passing through \(A\) and \(B\).
2. Find the position vector of the point \(P\) on \(A B\) such that \(O P\) is perpendicular to \(A B\).

5

1. Show that \(\sqrt { \frac { 1 - x } { 1 + x } } \approx 1 - x + \frac { 1 } { 2 } x ^ { 2 }\), for \(| x | < 1\).
2. By taking \(x = \frac { 2 } { 7 }\), show that \(\sqrt { 5 } \approx \frac { 111 } { 49 }\).

6 Two lines have equations $$\mathbf { r } = \left( \begin{array} { r } 1
0
- 5 \end{array} \right) + t \left( \begin{array} { l } 2
3
4 \end{array} \right) \quad \text { and } \quad \mathbf { r } = \left( \begin{array} { r } 12
0
5 \end{array} \right) + s \left( \begin{array} { r } 1
- 4
- 2 \end{array} \right) .$$

1. Show that the lines intersect.
2. Find the angle between the lines.

1. Show that, if \(y = \operatorname { cosec } x\), then \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) can be expressed as \(- \operatorname { cosec } x \cot x\).
2. Solve the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = - \sin x \tan x \cot t$$ given that \(x = \frac { 1 } { 6 } \pi\) when \(t = \frac { 1 } { 2 } \pi\).

8

1. Given that \(\frac { 2 t } { ( t + 1 ) ^ { 2 } }\) can be expressed in the form \(\frac { A } { t + 1 } + \frac { B } { ( t + 1 ) ^ { 2 } }\), find the values of the constants \(A\) and \(B\).
2. Show that the substitution \(t = \sqrt { 2 x - 1 }\) transforms \(\int \frac { 1 } { x + \sqrt { 2 x - 1 } } \mathrm {~d} x\) to \(\int \frac { 2 t } { ( t + 1 ) ^ { 2 } } \mathrm {~d} t\).
3. Hence find the exact value of \(\int _ { 1 } ^ { 5 } \frac { 1 } { x + \sqrt { 2 x - 1 } } \mathrm {~d} x\).

12
0
5 \end{array} \right) + s \left( \begin{array} { r } 1
- 4
- 2 \end{array} \right) .$$

1. Show that the lines intersect.
2. Find the angle between the lines.
3. Show that, if \(y = \operatorname { cosec } x\), then \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) can be expressed as \(- \operatorname { cosec } x \cot x\).
4. Solve the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = - \sin x \tan x \cot t$$ given that \(x = \frac { 1 } { 6 } \pi\) when \(t = \frac { 1 } { 2 } \pi\). 8
5. Given that \(\frac { 2 t } { ( t + 1 ) ^ { 2 } }\) can be expressed in the form \(\frac { A } { t + 1 } + \frac { B } { ( t + 1 ) ^ { 2 } }\), find the values of the constants \(A\) and \(B\).
6. Show that the substitution \(t = \sqrt { 2 x - 1 }\) transforms \(\int \frac { 1 } { x + \sqrt { 2 x - 1 } } \mathrm {~d} x\) to \(\int \frac { 2 t } { ( t + 1 ) ^ { 2 } } \mathrm {~d} t\).
7. Hence find the exact value of \(\int _ { 1 } ^ { 5 } \frac { 1 } { x + \sqrt { 2 x - 1 } } \mathrm {~d} x\). 9 The parametric equations of a curve are $$x = 2 \theta + \sin 2 \theta , \quad y = 4 \sin \theta$$ and part of its graph is shown below.
   \includegraphics[max width=\textwidth, alt={}, center]{b8ba126f-c5fa-4828-9439-e5162a03ca5b-3_646_1150_1050_500}
8. Find the value of \(\theta\) at \(A\) and the value of \(\theta\) at \(B\).
9. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \sec \theta\).
10. At the point \(C\) on the curve, the gradient is 2 . Find the coordinates of \(C\), giving your answer in an exact form.