OCR C4 (Core Mathematics 4) 2007 June

1 The equation of a curve is \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = \frac { 3 x + 1 } { ( x + 2 ) ( x - 3 ) }\).

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Hence find \(\mathrm { f } ^ { \prime } ( x )\) and deduce that the gradient of the curve is negative at all points on the curve.

2 Find the exact value of \(\int _ { 0 } ^ { 1 } x ^ { 2 } \mathrm { e } ^ { x } \mathrm {~d} x\).

3 Find the exact volume generated when the region enclosed between the \(x\)-axis and the portion of the curve \(y = \sin x\) between \(x = 0\) and \(x = \pi\) is rotated completely about the \(x\)-axis.

4

1. Expand \(( 2 + x ) ^ { - 2 }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), and state the set of values of \(x\) for which the expansion is valid.
2. Hence find the coefficient of \(x ^ { 3 }\) in the expansion of \(\frac { 1 + x ^ { 2 } } { ( 2 + x ) ^ { 2 } }\).

5 A curve \(C\) has parametric equations $$x = \cos t , \quad y = 3 + 2 \cos 2 t , \quad \text { where } 0 \leqslant t \leqslant \pi$$

1. Express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\) and hence show that the gradient at any point on \(C\) cannot exceed 8 .
2. Show that all points on \(C\) satisfy the cartesian equation \(y = 4 x ^ { 2 } + 1\).
3. Sketch the curve \(y = 4 x ^ { 2 } + 1\) and indicate on your sketch the part which represents \(C\).

6 The equation of a curve is \(x ^ { 2 } + 3 x y + 4 y ^ { 2 } = 58\). Find the equation of the normal at the point \(( 2,3 )\) on the curve, giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

7

1. Find the quotient and the remainder when \(2 x ^ { 3 } + 3 x ^ { 2 } + 9 x + 12\) is divided by \(x ^ { 2 } + 4\).
2. Hence express \(\frac { 2 x ^ { 3 } + 3 x ^ { 2 } + 9 x + 12 } { x ^ { 2 } + 4 }\) in the form \(A x + B + \frac { C x + D } { x ^ { 2 } + 4 }\), where the values of the constants \(A , B , C\) and \(D\) are to be stated.
3. Use the result of part (ii) to find the exact value of \(\int _ { 1 } ^ { 3 } \frac { 2 x ^ { 3 } + 3 x ^ { 2 } + 9 x + 12 } { x ^ { 2 } + 4 } \mathrm {~d} x\).

8 The height, \(h\) metres, of a shrub \(t\) years after planting is given by the differential equation $$\frac { \mathrm { d } h } { \mathrm {~d} t } = \frac { 6 - h } { 20 }$$ A shrub is planted when its height is 1 m .

1. Show by integration that \(t = 20 \ln \left( \frac { 5 } { 6 - h } \right)\).
2. How long after planting will the shrub reach a height of 2 m ?
3. Find the height of the shrub 10 years after planting.
4. State the maximum possible height of the shrub.

9 Lines \(L _ { 1 } , L _ { 2 }\) and \(L _ { 3 }\) have vector equations $$\begin{aligned} & L _ { 1 } : \mathbf { r } = ( 5 \mathbf { i } - \mathbf { j } - 2 \mathbf { k } ) + s ( - 6 \mathbf { i } + 8 \mathbf { j } - 2 \mathbf { k } ) , \\ & L _ { 2 } : \mathbf { r } = ( 3 \mathbf { i } - 8 \mathbf { j } ) + t ( \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k } ) , \\ & L _ { 3 } : \mathbf { r } = ( 2 \mathbf { i } + \mathbf { j } + 3 \mathbf { k } ) + u ( 3 \mathbf { i } + c \mathbf { j } + \mathbf { k } ) . \end{aligned}$$

1. Calculate the acute angle between \(L _ { 1 }\) and \(L _ { 2 }\).
2. Given that \(L _ { 1 }\) and \(L _ { 3 }\) are parallel, find the value of \(c\).
3. Given instead that \(L _ { 2 }\) and \(L _ { 3 }\) intersect, find the value of \(c\). 4