OCR MEI C1 (Core Mathematics 1)

2 Fig. 8 shows a right-angled triangle with base \(2 x + 1\), height \(h\) and hypotenuse \(3 x\). \begin{figure}[h] \end{figure} Not to scale

1. Show that \(h ^ { 2 } = 5 x ^ { 2 } - 4 x - 1\).
2. Given that \(h = \sqrt { 7 }\), find the value of \(x\), giving your answer in surd form.

3

1. Find the set of values of \(k\) for which the line \(y = 2 x + k\) intersects the curve \(y = 3 x ^ { 2 } + 12 x + 13\) at two distinct points.
2. Express \(3 x ^ { 2 } + 12 x + 13\) in the form \(a ( x + b ) ^ { 2 } + c\). Hence show that the curve \(y = 3 x ^ { 2 } + 12 x + 13\) lies completely above the \(x\)-axis.
3. Find the value of \(k\) for which the line \(y = 2 x + k\) passes through the minimum point of the curve \(y = 3 x ^ { 2 } + 12 x + 13\).

4 Make \(a\) the subject of \(3 ( a + 4 ) = a c + 5 f\).

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

6 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 \(y = 3 x ^ { 2 } - 12 x + 5\).
\(7 \quad\) Simplify \(\frac { \left( 4 x ^ { 5 } y \right) ^ { 3 } } { \left( 2 x y ^ { 2 } \right) \times \left( 8 x ^ { 10 } y ^ { 4 } \right) }\).

8 You are given that \(\mathrm { f } ( x ) = x ^ { 2 } + k x + c\).
Given also that \(\mathrm { f } ( 2 ) = 0\) and \(\mathrm { f } ( - 3 ) = 35\), find the values of the constants \(k\) and \(c\).

9 Rearrange the equation \(5 c + 9 t = a ( 2 c + t )\) to make \(c\) the subject.

10 Factorise and hence simplify the following expression. $$\frac { x ^ { 2 } - 9 } { x ^ { 2 } + 5 x + 6 }$$

11 Rearrange the following equation to make \(h\) the subject. $$4 h + 5 = 9 a - h a ^ { 2 }$$