OCR C4 (Core Mathematics 4) 2016 June

1 Find the quotient and the remainder when \(4 x ^ { 3 } + 8 x ^ { 2 } - 5 x + 12\) is divided by \(2 x ^ { 2 } + 1\).

2 Use integration to find the exact value of \(\int _ { \frac { 1 } { 16 } \pi } ^ { \frac { 1 } { 8 } \pi } \left( 9 - 6 \cos ^ { 2 } 4 x \right) \mathrm { d } x\).

3 Given that \(y \sin 2 x + \frac { 1 } { x } + y ^ { 2 } = 5\), find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).

4 Find the exact value of \(\int _ { 1 } ^ { 8 } \frac { 1 } { \sqrt [ 3 ] { x } } \ln x \mathrm {~d} x\), giving your answer in the form \(A \ln 2 + B\), where \(A\) and \(B\) are constants to be found.

5 The vector equations of two lines are as follows. $$L : \mathbf { r } = \left( \begin{array} { l } 1
4
5 \end{array} \right) + s \left( \begin{array} { c } 2
- 1
3 \end{array} \right) \quad M : \mathbf { r } = \left( \begin{array} { c } 3
2
- 5 \end{array} \right) + t \left( \begin{array} { c } 5
- 3
1 \end{array} \right)$$

1. Show that the lines \(L\) and \(M\) meet, and find the coordinates of the point of intersection.
2. Show that the line \(L\) can also be represented by the equation \(\mathbf { r } = \left( \begin{array} { c } 7
   1
   14 \end{array} \right) + u \left( \begin{array} { c } - 4
   2
   - 6 \end{array} \right)\).

6 Use the substitution \(u = x ^ { 2 } - 2\) to find \(\int \frac { 6 x ^ { 3 } + 4 x } { \sqrt { x ^ { 2 } - 2 } } \mathrm {~d} x\).

7 Given that the binomial expansion of \(( 1 + k x ) ^ { n }\) is \(1 - 6 x + 30 x ^ { 2 } + \ldots\), find the values of \(n\) and \(k\). State the set of values of \(x\) for which this expansion is valid.

8 The points \(A\) and \(B\) have position vectors relative to the origin \(O\) given by $$\overrightarrow { O A } = \left( \begin{array} { c } 3 \sin \alpha
2 \cos \alpha
- 1 \end{array} \right) \text { and } \overrightarrow { O B } = \left( \begin{array} { c } 2 \cos \alpha
4 \sin \alpha
3 \end{array} \right)$$ where \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). It is given that \(\overrightarrow { O A }\) and \(\overrightarrow { O B }\) are perpendicular.

1. Calculate the two possible values of \(\alpha\).
2. Calculate the area of triangle \(O A B\) for the smaller value of \(\alpha\) from part (i).

9 A curve has parametric equations \(x = 1 - \cos t , y = \sin t \sin 2 t\), for \(0 \leqslant t \leqslant \pi\).

1. Find the coordinates of the points where the curve meets the \(x\)-axis.
2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 \cos 2 t + 2 \cos ^ { 2 } t\). Hence find, in an exact form, the coordinates of the stationary points.
3. Find the cartesian equation of the curve. Give your answer in the form \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x )\) is a polynomial.
4. Sketch the curve.

10

1. Express \(\frac { 16 + 5 x - 2 x ^ { 2 } } { ( x + 1 ) ^ { 2 } ( x + 4 ) }\) in partial fractions.
2. It is given that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { \left( 16 + 5 x - 2 x ^ { 2 } \right) y } { ( x + 1 ) ^ { 2 } ( x + 4 ) }$$ and that \(y = \frac { 1 } { 256 }\) when \(x = 0\). Find the exact value of \(y\) when \(x = 2\). Give your answer in the form \(A \mathrm { e } ^ { n }\).