OCR C4 (Core Mathematics 4) 2010 January

1 Find the quotient and the remainder when \(x ^ { 4 } + 11 x ^ { 3 } + 28 x ^ { 2 } + 3 x + 1\) is divided by \(x ^ { 2 } + 5 x + 2\).

2 Points \(A , B\) and \(C\) have position vectors \(- 5 \mathbf { i } - 10 \mathbf { j } + 12 \mathbf { k } , \mathbf { i } + 2 \mathbf { j } - 3 \mathbf { k }\) and \(3 \mathbf { i } + 6 \mathbf { j } + p \mathbf { k }\) respectively, where \(p\) is a constant.

1. Given that angle \(A B C = 90 ^ { \circ }\), find the value of \(p\).
2. Given instead that \(A B C\) is a straight line, find the value of \(p\).

3 By expressing \(\cos 2 x\) in terms of \(\cos x\), find the exact value of \(\int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 1 } { 3 } \pi } \frac { \cos 2 x } { \cos ^ { 2 } x } \mathrm {~d} x\).

4 Use the substitution \(u = 2 + \ln t\) to find the exact value of $$\int _ { 1 } ^ { \mathrm { e } } \frac { 1 } { t ( 2 + \ln t ) ^ { 2 } } \mathrm {~d} t$$

5

1. Expand \(( 1 + x ) ^ { \frac { 1 } { 3 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).
2. (a) Hence, or otherwise, expand \(( 8 + 16 x ) ^ { \frac { 1 } { 3 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).
   (b) State the set of values of \(x\) for which the expansion in part (ii) (a) is valid.

6 A curve has parametric equations $$x = 9 t - \ln ( 9 t ) , \quad y = t ^ { 3 } - \ln \left( t ^ { 3 } \right)$$ Show that there is only one value of \(t\) for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3\) and state that value.

7 Find the equation of the normal to the curve \(x ^ { 3 } + 2 x ^ { 2 } y = y ^ { 3 } + 15\) at the point \(( 2,1 )\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

8

1. State the derivative of \(\mathrm { e } ^ { \cos x }\).
2. Hence use integration by parts to find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \cos x \sin x \mathrm { e } ^ { \cos x } \mathrm {~d} x$$

9 The equation of a straight line \(l\) is \(\mathbf { r } = \left( \begin{array} { l } 3 \\ 1 \\ 1 \end{array} \right) + t \left( \begin{array} { r } 1 \\ - 1 \\ 2 \end{array} \right) . O\) is the origin.

1. The point \(P\) on \(l\) is given by \(t = 1\). Calculate the acute angle between \(O P\) and \(l\).
2. Find the position vector of the point \(Q\) on \(l\) such that \(O Q\) is perpendicular to \(l\).
3. Find the length of \(O Q\).

10

1. Express \(\frac { 1 } { ( 3 - x ) ( 6 - x ) }\) in partial fractions.
2. In a chemical reaction, the amount \(x\) grams of a substance at time \(t\) seconds is related to the rate at which \(x\) is changing by the equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = k ( 3 - x ) ( 6 - x )$$ where \(k\) is a constant. When \(t = 0 , x = 0\) and when \(t = 1 , x = 1\).
   1. Show that \(k = \frac { 1 } { 3 } \ln \frac { 5 } { 4 }\).
   2. Find the value of \(x\) when \(t = 2\).