OCR C4 (Core Mathematics 4) 2010 June

1 Expand \(( 1 + 3 x ) ^ { - \frac { 5 } { 3 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).

2 Given that \(y = \frac { \cos x } { 1 - \sin x }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), simplifying your answer.

3 Express \(\frac { x ^ { 2 } } { ( x - 1 ) ^ { 2 } ( x - 2 ) }\) in partial fractions.

4 Use the substitution \(u = \sqrt { x + 2 }\) to find the exact value of $$\int _ { - 1 } ^ { 7 } \frac { x ^ { 2 } } { \sqrt { x + 2 } } \mathrm {~d} x$$

5 Find the coordinates of the two stationary points on the curve with equation $$x ^ { 2 } + 4 x y + 2 y ^ { 2 } + 18 = 0$$

6 Lines \(l _ { 1 }\) and \(l _ { 2 }\) have vector equations $$\mathbf { r } = \mathbf { j } + \mathbf { k } + t ( 2 \mathbf { i } + a \mathbf { j } + \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = 3 \mathbf { i } - \mathbf { k } + s ( 2 \mathbf { i } + 2 \mathbf { j } - 6 \mathbf { k } )$$ respectively, where \(t\) and \(s\) are parameters and \(a\) is a constant.

1. Given that \(l _ { 1 }\) and \(l _ { 2 }\) are perpendicular, find the value of \(a\).
2. Given instead that \(l _ { 1 }\) and \(l _ { 2 }\) intersect, find
   (a) the value of \(a\),
   (b) the angle between the lines.

7 The parametric equations of a curve are \(x = \frac { t + 2 } { t + 1 } , y = \frac { 2 } { t + 3 }\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } > 0\).
2. Find the cartesian equation of the curve, giving your answer in a form not involving fractions.

8

1. Find the quotient and the remainder when \(x ^ { 2 } - 5 x + 6\) is divided by \(x - 1\).
2. (a) Find the general solution of the differential equation $$\left( \frac { x - 1 } { x ^ { 2 } - 5 x + 6 } \right) \frac { \mathrm { d } y } { \mathrm {~d} x } = y - 5 .$$ (b) Given that \(y = 7\) when \(x = 8\), find \(y\) when \(x = 6\).

9

1. Find \(\int ( x + \cos 2 x ) ^ { 2 } \mathrm {~d} x\).
2. 
   \includegraphics[max width=\textwidth, alt={}, center]{80f94db1-39be-46f5-896e-277c93cbe4b8-3_538_935_383_646} The diagram shows the part of the curve \(y = x + \cos 2 x\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\). The shaded region bounded by the curve, the axes and the line \(x = \frac { 1 } { 2 } \pi\) is rotated completely about the \(x\)-axis to form a solid of revolution of volume \(V\). Find \(V\), giving your answer in an exact form.