OCR MEI C1 (Core Mathematics 1)

1 Make \(a\) the subject of the equation \(s = u t + \frac { 1 } { 2 } a t ^ { 2 }\).

2

1. Find the constants \(a\) and \(b\) such that, for all values of \(x\), $$x ^ { 2 } + 4 x + 14 = ( x + a ) ^ { 2 } + b$$
2. Write down the greatest value of \(\frac { 1 } { x ^ { 2 } + 4 x + 14 }\).

3 Find the term independent of \(x\) in the expansion of \(\left( x - \frac { 2 } { x } \right) ^ { 4 }\).

4 The coordinates of the points \(\mathrm { A } , \mathrm { B }\) and C are ( \(- 2,2\) ), ( 1,3 ) and ( \(3 , - 3\) ) respectively.

1. Find the gradients of the lines AB and BC .
2. Show that the triangle ABC is a right-angled triangle.
3. Find the area of the triangle ABC .

5 You are given that \(\mathrm { f } ( x ) = x ^ { 3 } - 7 x + 6\).

1. Show that ( \(x - 2\) ) is a factor of \(\mathrm { f } ( x )\).
2. Solve the equation \(\mathrm { f } ( x ) = 0\).

6 List the integers which satisfy both of the following inequalities: $$2 x - 9 < 0 , \quad 8 - x \leq 6$$

7

1. Express \(( 2 + \sqrt { 3 } ) ^ { 2 }\) in the form \(a + b \sqrt { 3 }\) where \(a\) and \(b\) are integers to be determined.
2. Given that \(x\) and \(y\) are integers, prove that \(\frac { 1 } { x - \sqrt { y } } + \frac { 1 } { x + \sqrt { y } }\) can be written in the form \(\frac { p } { q }\) where \(p\) and \(q\) are both integers.

8 Find the equation of the line that passes through the point \(( 1,2 )\) and is perpendicular to the line \(3 x + 2 y = 5\).

9

1. Show that \(( x - 1 ) ( x - 2 ) ( x - 3 ) - \left( x ^ { 3 } - x ^ { 2 } + 11 x - 12 \right) = 6 - 5 x ^ { 2 }\).
2. Solve the equation \(6 - 5 x ^ { 2 } = 0\).

10

1. A quadratic function is given by \(\mathrm { f } ( x ) = x ^ { 2 } - 6 x + 8\).
   Sketch the graph of \(y = \mathrm { f } ( x )\), giving the coordinates of the points where it crosses the axes. Mark the lowest point on the curve, and give its coordinates.
2. Solve the inequality \(x ^ { 2 } - 6 x + 8 < 0\).
3. On the same graph, sketch \(y = \mathrm { f } ( x + 3 )\).
4. The graph of \(y = \mathrm { f } ( x + 3 ) - 2\) is obtained from the graph of \(y = \mathrm { f } ( x )\) by a transformation. Describe the transformation and sketch the curve on the same axes as in (i) and (iii) above. Label all these curves clearly.

11

1. Show algebraically that the equation \(x ^ { 2 } - 6 x + 10 = 0\) has no real roots.
2. Solve algebraically the simultaneous equations \(y = x ^ { 2 } - 6 x + 10\) and \(y + 2 x = 7\).
3. Plot the graph of the function \(y = x ^ { 2 } - 6 x + 10\) on graph paper, taking \(1 \mathrm {~cm} = 1\) unit on each axis, with the \(x\) axis from 0 to 6 and the \(y\) axis from - 2 to 10 .
   On the same axes plot the line with equation \(y + 2 x = 7\) showing clearly where the line cuts the quadratic curve.
4. Explain why these \(x\) coordinates satisfy the equation \(x ^ { 2 } - 4 x + 3 = 0\). Plot a graph of the function \(y = x ^ { 2 } - 4 x + 3\) on the same axes to illustrate your answer.

12 You are given that the equation of the circle shown in Fig. 12 is $$x ^ { 2 } + y ^ { 2 } - 4 x - 6 y - 12 = 0$$ \begin{figure}[h] \end{figure}

1. Show that the centre, Q , of the circle is \(( 2,3 )\) and find the radius.
2. The circle crosses the \(x\)-axis at B and C . Show that the coordinates of C are \(( 6,0 )\) and find the coordinates of B .
3. Find the gradient of the line QC and hence find the equation of the tangent to the circle at C.
4. Given that M is the mid-point of BC , find the coordinates of the point where QM meets the tangent at C .