OCR MEI C1 (Core Mathematics 1) 2013 June

1 Find the equation of the line which is perpendicular to the line \(y = 2 x - 5\) and which passes through the point \(( 4,1 )\). Give your answer in the form \(y = a x + b\).

2 Find the coordinates of the point of intersection of the lines \(y = 3 x - 2\) and \(x + 3 y = 1\).

3

1. Evaluate \(( 0.2 ) ^ { - 2 }\).
2. Simplify \(\left( 16 a ^ { 12 } \right) ^ { \frac { 3 } { 4 } }\).

4 Rearrange the following formula to make \(r\) the subject, where \(r > 0\). $$V = \frac { 1 } { 3 } \pi r ^ { 2 } ( a + b )$$

5 You are given that \(\mathrm { f } ( x ) = x ^ { 5 } + k x - 20\). When \(\mathrm { f } ( x )\) is divided by \(( x - 2 )\), the remainder is 18 . Find the value of \(k\).

6 Find the coefficient of \(x ^ { 3 }\) in the binomial expansion of \(( 2 - 4 x ) ^ { 5 }\).

7

1. Express \(125 \sqrt { 5 }\) in the form \(5 ^ { k }\).
2. Simplify \(10 + 7 \sqrt { 5 } + \frac { 38 } { 1 - 2 \sqrt { 5 } }\), giving your answer in the form \(a + b \sqrt { 5 }\).

8 Express \(3 x ^ { 2 } - 12 x + 5\) in the form \(a ( x - b ) ^ { 2 } - c\). Hence state the minimum value of \(y\) on the curve \(y = 3 x ^ { 2 } - 12 x + 5\).
\(9 \quad n - 1 , n\) and \(n + 1\) are any three consecutive integers.

1. Show that the sum of these integers is always divisible by 3 .
2. Find the sum of the squares of these three consecutive integers and explain how this shows that the sum of the squares of any three consecutive integers is never divisible by 3 .

10 The circle \(( x - 3 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 20\) has centre C.

1. Write down the radius of the circle and the coordinates of C .
2. Find the coordinates of the intersections of the circle with the \(x\) - and \(y\)-axes.
3. Show that the points \(\mathrm { A } ( 1,6 )\) and \(\mathrm { B } ( 7,4 )\) lie on the circle. Find the coordinates of the midpoint of AB . Find also the distance of the chord AB from the centre of the circle.

11 You are given that \(\mathrm { f } ( x ) = ( 2 x - 3 ) ( x + 2 ) ( x + 4 )\).

1. Sketch the graph of \(y = \mathrm { f } ( x )\).
2. State the roots of \(\mathrm { f } ( x - 2 ) = 0\).
3. You are also given that \(\mathrm { g } ( x ) = \mathrm { f } ( x ) + 15\).
   (A) Show that \(\mathrm { g } ( x ) = 2 x ^ { 3 } + 9 x ^ { 2 } - 2 x - 9\).
   (B) Show that \(\mathrm { g } ( 1 ) = 0\) and hence factorise \(\mathrm { g } ( x )\) completely.

12 \begin{figure}[h] \end{figure} Fig. 12 shows the graph of \(y = \frac { 1 } { x - 2 }\).

1. Draw accurately the graph of \(y = 2 x + 3\) on the copy of Fig. 12 and use it to estimate the coordinates of the points of intersection of \(y = \frac { 1 } { x - 2 }\) and \(y = 2 x + 3\).
2. Show algebraically that the \(x\)-coordinates of the points of intersection of \(y = \frac { 1 } { x - 2 }\) and \(y = 2 x + 3\) satisfy the equation \(2 x ^ { 2 } - x - 7 = 0\). Hence find the exact values of the \(x\)-coordinates of the points of intersection.
3. Find the quadratic equation satisfied by the \(x\)-coordinates of the points of intersection of \(y = \frac { 1 } { x - 2 }\) and \(y = - x + k\). Hence find the exact values of \(k\) for which \(y = - x + k\) is a tangent to \(y = \frac { 1 } { x - 2 }\). [4]