OCR C1 (Core Mathematics 1)

1. Solve the equation

$$x ^ { 2 } - 4 x - 8 = 0$$ giving your answers in the form \(a + b \sqrt { 3 }\) where \(a\) and \(b\) are integers.

2. The curve \(C\) has the equation $$y = x ^ { 2 } + a x + b$$ where \(a\) and \(b\) are constants.
Given that the minimum point of \(C\) has coordinates \(( - 2,5 )\), find the values of \(a\) and \(b\).

3. (i) Solve the simultaneous equations $$\begin{aligned} & y = x ^ { 2 } - 6 x + 7
& y = 2 x - 9 \end{aligned}$$ (ii) Hence, describe the geometrical relationship between the curve \(y = x ^ { 2 } - 6 x + 7\) and the straight line \(y = 2 x - 9\).

4. (i) Evaluate $$\left( 36 ^ { \frac { 1 } { 2 } } + 16 ^ { \frac { 1 } { 4 } } \right) ^ { \frac { 1 } { 3 } }$$ (ii) Solve the equation $$3 x ^ { - \frac { 1 } { 2 } } - 4 = 0 .$$

1. (i) Sketch on the same diagram the curve with equation \(y = ( x - 2 ) ^ { 2 }\) and the straight line with equation \(y = 2 x - 1\).

Label on your sketch the coordinates of any points where each graph meets the coordinate axes.
(ii) Find the set of values of \(x\) for which $$( x - 2 ) ^ { 2 } > 2 x - 1$$

1. (i) Given that \(y = x ^ { \frac { 1 } { 3 } }\), show that the equation

$$2 x ^ { \frac { 1 } { 3 } } + 3 x ^ { - \frac { 1 } { 3 } } = 7$$ can be rewritten as $$2 y ^ { 2 } - 7 y + 3 = 0 .$$ (ii) Hence, solve the equation $$2 x ^ { \frac { 1 } { 3 } } + 3 x ^ { - \frac { 1 } { 3 } } = 7$$

1. Given that

$$y = \sqrt { x } - \frac { 4 } { \sqrt { x } }$$

1. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
2. find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\),
3. show that $$4 x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 x \frac { \mathrm {~d} y } { \mathrm {~d} x } - y = 0 .$$

\begin{enumerate} \setcounter{enumi}{7} \item \(f ( x ) = 2 + 6 x ^ { 2 } - x ^ { 3 }\).

1. Find the coordinates of the stationary points of the curve \(y = \mathrm { f } ( x )\).
2. Determine whether each stationary point is a maximum or minimum point.
3. Sketch the curve \(y = \mathrm { f } ( x )\).
4. State the set of values of \(k\) for which the equation \(\mathrm { f } ( x ) = k\) has three solutions. \item The points \(P\) and \(Q\) have coordinates \(( 7,4 )\) and \(( 9,7 )\) respectively.

1. Find an equation for the straight line \(l\) which passes through \(P\) and \(Q\). Give your answer in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers. \end{enumerate} The straight line \(m\) has gradient 8 and passes through the origin, \(O\).
2. Write down an equation for \(m\). The lines \(l\) and \(m\) intersect at the point \(R\).
3. Show that \(O P = O R\).

10.
\includegraphics[max width=\textwidth, alt={}, center]{af6fdbed-fcab-4db8-9cdf-fd049ce720fd-3_668_787_918_431} The diagram shows the circle \(C\) and the straight line \(l\).
The centre of \(C\) lies on the \(x\)-axis and \(l\) intersects \(C\) at the points \(A ( 2,4 )\) and \(B ( 8 , - 8 )\).

1. Find the gradient of 1 .
2. Find the coordinates of the mid-point of \(A B\).
3. Find the coordinates of the centre of \(C\).
4. Show that \(C\) has the equation $$x ^ { 2 } + y ^ { 2 } - 18 x + 16 = 0$$