OCR C4 (Core Mathematics 4) 2012 January

1 When the polynomial \(\mathrm { f } ( x )\) is divided by \(x ^ { 2 } + 1\), the quotient is \(x ^ { 2 } + 4 x + 2\) and the remainder is \(x - 1\). Find \(\mathrm { f } ( x )\), simplifying your answer.

2

1. Find, in the form \(\mathbf { r } = \mathbf { a } + t \mathbf { b }\), an equation of the line \(l\) through the points ( \(4,2,7\) ) and ( \(5 , - 4 , - 1\) ).
2. Find the acute angle between the line \(l\) and a line in the direction of the vector \(\left( \begin{array} { l } 1
   2
   3 \end{array} \right)\).

3 The equation of a curve \(C\) is \(( x + 3 ) ( y + 4 ) = x ^ { 2 } + y ^ { 2 }\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
2. The line \(2 y = x + 3\) meets \(C\) at two points. What can be said about the tangents to \(C\) at these points? Justify your answer.
3. Find the equation of the tangent at the point ( 6,0 ), giving your answer in the form \(a x + b y = c\), where \(a , b\) and \(c\) are integers.

4

1. Expand \(( 1 - 4 x ) ^ { \frac { 1 } { 4 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
2. The term of lowest degree in the expansion of $$( 1 + a x ) \left( 1 + b x ^ { 2 } \right) ^ { 7 } - ( 1 - 4 x ) ^ { \frac { 1 } { 4 } }$$ in ascending powers of \(x\) is the term in \(x ^ { 3 }\). Find the values of the constants \(a\) and \(b\).

5 Use the substitution \(u = \cos x\) to find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \sin ^ { 3 } x \cos ^ { 2 } x d x$$

6
\includegraphics[max width=\textwidth, alt={}, center]{cf154c94-6248-4dda-91e8-61349cc10482-3_606_846_251_614} The diagram shows the curves \(y = \cos x\) and \(y = \sin x\), for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\). The region \(R\) is bounded by the curves and the \(x\)-axis. Find the volume of the solid of revolution formed when \(R\) is rotated completely about the \(x\)-axis, giving your answer in terms of \(\pi\).

7 The equation of a straight line \(l\) is $$\mathbf { r } = \left( \begin{array} { l } 1
0
2 \end{array} \right) + t \left( \begin{array} { r } 1
- 1
0 \end{array} \right) .$$ \(O\) is the origin.

1. Find the position vector of the point \(P\) on \(l\) such that \(O P\) is perpendicular to \(l\).
2. A point \(Q\) on \(l\) is such that the length of \(O Q\) is 3 units. Find the two possible position vectors of \(Q\). [3]

8 A curve is defined by the parametric equations $$x = \sin ^ { 2 } \theta , \quad y = 4 \sin \theta - \sin ^ { 3 } \theta ,$$ where \(- \frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 4 - 3 \sin ^ { 2 } \theta } { 2 \sin \theta }\).
2. Find the coordinates of the point on the curve at which the gradient is 2 .
3. Show that the curve has no stationary points.
4. Find a cartesian equation of the curve, giving your answer in the form \(y ^ { 2 } = \mathrm { f } ( x )\).

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

10

1. Write down the derivative of \(\sqrt { y ^ { 2 } + 1 }\) with respect to \(y\).
2. Given that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { ( x - 1 ) \sqrt { y ^ { 2 } + 1 } } { x y }\) and that \(y = \sqrt { \mathrm { e } ^ { 2 } - 2 \mathrm { e } }\) when \(x = \mathrm { e }\),
   find a relationship between \(x\) and \(y\).