OCR C4 (Core Mathematics 4) 2007 January

1 It is given that $$f ( x ) = \frac { x ^ { 2 } + 2 x - 24 } { x ^ { 2 } - 4 x } \quad \text { for } x \neq 0 , x \neq 4$$ Express \(\mathrm { f } ( x )\) in its simplest form.

3 The points \(A\) and \(B\) have position vectors \(\mathbf { a }\) and \(\mathbf { b }\) relative to an origin \(O\), where \(\mathbf { a } = 4 \mathbf { i } + 3 \mathbf { j } - 2 \mathbf { k }\) and \(\mathbf { b } = - 7 \mathbf { i } + 5 \mathbf { j } + 4 \mathbf { k }\).

1. Find the length of \(A B\).
2. Use a scalar product to find angle \(O A B\).

4 Use the substitution \(u = 2 x - 5\) to show that \(\int _ { \frac { 5 } { 2 } } ^ { 3 } ( 4 x - 8 ) ( 2 x - 5 ) ^ { 7 } \mathrm {~d} x = \frac { 17 } { 72 }\).

5

1. Expand \(( 1 - 3 x ) ^ { - \frac { 1 } { 3 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
2. Hence find the coefficient of \(x ^ { 3 }\) in the expansion of \(\left( 1 - 3 \left( x + x ^ { 3 } \right) \right) ^ { - \frac { 1 } { 3 } }\).

6

1. Express \(\frac { 2 x + 1 } { ( x - 3 ) ^ { 2 } }\) in the form \(\frac { A } { x - 3 } + \frac { B } { ( x - 3 ) ^ { 2 } }\), where \(A\) and \(B\) are constants.
2. Hence find the exact value of \(\int _ { 4 } ^ { 10 } \frac { 2 x + 1 } { ( x - 3 ) ^ { 2 } } \mathrm {~d} x\), giving your answer in the form \(a + b \ln c\), where \(a , b\) and \(c\) are integers.

7 The equation of a curve is \(2 x ^ { 2 } + x y + y ^ { 2 } = 14\). Show that there are two stationary points on the curve and find their coordinates.

8 The parametric equations of a curve are \(x = 2 t ^ { 2 } , y = 4 t\). Two points on the curve are \(P \left( 2 p ^ { 2 } , 4 p \right)\) and \(Q \left( 2 q ^ { 2 } , 4 q \right)\).

1. Show that the gradient of the normal to the curve at \(P\) is \(- p\).
2. Show that the gradient of the chord joining the points \(P\) and \(Q\) is \(\frac { 2 } { p + q }\).
3. The chord \(P Q\) is the normal to the curve at \(P\). Show that \(p ^ { 2 } + p q + 2 = 0\).
4. The normal at the point \(R ( 8,8 )\) meets the curve again at \(S\). The normal at \(S\) meets the curve again at \(T\). Find the coordinates of \(T\).

9

1. Find the general solution of the differential equation $$\frac { \sec ^ { 2 } y } { \cos ^ { 2 } ( 2 x ) } \frac { d y } { d x } = 2$$
2. For the particular solution in which \(y = \frac { 1 } { 4 } \pi\) when \(x = 0\), find the value of \(y\) when \(x = \frac { 1 } { 6 } \pi\).

10 The position vectors of the points \(P\) and \(Q\) with respect to an origin \(O\) are \(5 \mathbf { i } + 2 \mathbf { j } - 9 \mathbf { k }\) and \(4 \mathbf { i } + 4 \mathbf { j } - 6 \mathbf { k }\) respectively.

1. Find a vector equation for the line \(P Q\). The position vector of the point \(T\) is \(\mathbf { i } + 2 \mathbf { j } - \mathbf { k }\).
2. Write down a vector equation for the line \(O T\) and show that \(O T\) is perpendicular to \(P Q\). It is given that \(O T\) intersects \(P Q\).
3. Find the position vector of the point of intersection of \(O T\) and \(P Q\).
4. Hence find the perpendicular distance from \(O\) to \(P Q\), giving your answer in an exact form.