OCR C4 (Core Mathematics 4) 2006 June

1 Find the gradient of the curve \(4 x ^ { 2 } + 2 x y + y ^ { 2 } = 12\) at the point \(( 1,2 )\).

2

1. Expand \(( 1 - 3 x ) ^ { - 2 }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).
2. Find the coefficient of \(x ^ { 2 }\) in the expansion of \(\frac { ( 1 + 2 x ) ^ { 2 } } { ( 1 - 3 x ) ^ { 2 } }\) in ascending powers of \(x\).

3

1. Express \(\frac { 3 - 2 x } { x ( 3 - x ) }\) in partial fractions.
2. Show that \(\int _ { 1 } ^ { 2 } \frac { 3 - 2 x } { x ( 3 - x ) } \mathrm { d } x = 0\).
3. What does the result of part (ii) indicate about the graph of \(y = \frac { 3 - 2 x } { x ( 3 - x ) }\) between \(x = 1\) and \(x = 2\) ?

4 The position vectors of three points \(A , B\) and \(C\) relative to an origin \(O\) are given respectively by and $$\begin{aligned} & \overrightarrow { O A } = 7 \mathbf { i } + 3 \mathbf { j } - 3 \mathbf { k }
& \overrightarrow { O B } = 4 \mathbf { i } + 2 \mathbf { j } - 4 \mathbf { k }
& \overrightarrow { O C } = 5 \mathbf { i } + 4 \mathbf { j } - 5 \mathbf { k } \end{aligned}$$

1. Find the angle between \(A B\) and \(A C\).
2. Find the area of triangle \(A B C\).

5 A forest is burning so that, \(t\) hours after the start of the fire, the area burnt is \(A\) hectares. It is given that, at any instant, the rate at which this area is increasing is proportional to \(A ^ { 2 }\).

1. Write down a differential equation which models this situation.
2. After 1 hour, 1000 hectares have been burnt; after 2 hours, 2000 hectares have been burnt. Find after how many hours 3000 hectares have been burnt.
3. Show that the substitution \(u = \mathrm { e } ^ { x } + 1\) transforms \(\int \frac { \mathrm { e } ^ { 2 x } } { \mathrm { e } ^ { x } + 1 } \mathrm {~d} x\) to \(\int \frac { u - 1 } { u } \mathrm {~d} u\).
4. Hence show that \(\int _ { 0 } ^ { 1 } \frac { \mathrm { e } ^ { 2 x } } { \mathrm { e } ^ { x } + 1 } \mathrm {~d} x = \mathrm { e } - 1 - \ln \left( \frac { \mathrm { e } + 1 } { 2 } \right)\).

7 Two lines have vector equations $$\mathbf { r } = \mathbf { i } - 2 \mathbf { j } + 4 \mathbf { k } + \lambda ( 3 \mathbf { i } + \mathbf { j } + a \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = - 8 \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } + \mu ( \mathbf { i } - 2 \mathbf { j } - \mathbf { k } ) ,$$ where \(a\) is a constant.

1. Given that the lines are skew, find the value that \(a\) cannot take.
2. Given instead that the lines intersect, find the point of intersection.

8

1. Show that \(\int \cos ^ { 2 } 6 x \mathrm {~d} x = \frac { 1 } { 2 } x + \frac { 1 } { 24 } \sin 12 x + c\).
2. Hence find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 12 } \pi } x \cos ^ { 2 } 6 x \mathrm {~d} x\).

9 A curve is given parametrically by the equations $$x = 4 \cos t , \quad y = 3 \sin t$$ where \(0 \leqslant t \leqslant \frac { 1 } { 2 } \pi\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
2. Show that the equation of the tangent at the point \(P\), where \(t = p\), is $$3 x \cos p + 4 y \sin p = 12$$
3. The tangent at \(P\) meets the \(x\)-axis at \(R\) and the \(y\)-axis at \(S . O\) is the origin. Show that the area of triangle \(O R S\) is \(\frac { 12 } { \sin 2 p }\).
4. Write down the least possible value of the area of triangle \(O R S\), and give the corresponding value of \(p\).