OCR C1 (Core Mathematics 1)

1. Solve the equation

$$9 ^ { x } = 3 ^ { x + 2 } .$$

1. The straight line \(l\) has the equation \(x - 5 y = 7\).

The straight line \(m\) is perpendicular to \(l\) and passes through the point \(( - 4,1 )\).
Find an equation for \(m\) in the form \(y = m x + c\).

3.
\includegraphics[max width=\textwidth, alt={}, center]{4fec0924-d727-4d4f-81e6-918e1ccfedbd-1_330_1230_829_386} The diagram shows the rectangles \(A B C D\) and \(E F G H\) which are similar.
Given that \(A B = ( 3 - \sqrt { 5 } ) \mathrm { cm } , A D = \sqrt { 5 } \mathrm {~cm}\) and \(E F = ( 1 + \sqrt { 5 } ) \mathrm { cm }\), find the length \(E H\) in cm, giving your answer in the form \(a + b \sqrt { 5 }\) where \(a\) and \(b\) are integers.

4. (i) Sketch on the same diagram the curves \(y = x ^ { 2 } - 4 x\) and \(y = - \frac { 1 } { x }\).
(ii) State, with a reason, the number of real solutions to the equation $$x ^ { 2 } - 4 x + \frac { 1 } { x } = 0 .$$

1. (i) Solve the inequality

$$x ^ { 2 } + 3 x > 10 .$$ (ii) Find the set of values of \(x\) which satisfy both of the following inequalities: $$\begin{aligned} & 3 x - 2 < x + 3
& x ^ { 2 } + 3 x > 10 \end{aligned}$$

6. $$f ( x ) = 4 x ^ { 2 } + 12 x + 9 .$$

1. Determine the number of real roots that exist for the equation \(\mathrm { f } ( x ) = 0\).
2. Solve the equation \(\mathrm { f } ( x ) = 8\), giving your answers in the form \(a + b \sqrt { 2 }\) where \(a\) and \(b\) are rational.

7. The circle \(C\) has centre \(( - 1,6 )\) and radius \(2 \sqrt { 5 }\).

1. Find an equation for \(C\). The line \(y = 3 x - 1\) intersects \(C\) at the points \(A\) and \(B\).
2. Find the \(x\)-coordinates of \(A\) and \(B\).
3. Show that \(A B = 2 \sqrt { 10 }\).

8. $$f ( x ) = 2 - x + 3 x ^ { \frac { 2 } { 3 } } , \quad x > 0 .$$

1. Find \(f ^ { \prime } ( x )\) and \(f ^ { \prime \prime } ( x )\).
2. Find the coordinates of the turning point of the curve \(y = \mathrm { f } ( x )\).
3. Determine whether the turning point is a maximum or minimum point.

9. (i) Find an equation for the tangent to the curve \(y = x ^ { 2 } + 2\) at the point \(( 1,3 )\) in the form \(y = m x + c\).
(ii) Express \(x ^ { 2 } - 6 x + 11\) in the form \(( x + a ) ^ { 2 } + b\) where \(a\) and \(b\) are integers.
(iii) Describe fully the transformation that maps the graph of \(y = x ^ { 2 } + 2\) onto the graph of \(y = x ^ { 2 } - 6 x + 11\).
(iv) Use your answers to parts (i) and (iii) to deduce an equation for the tangent to the curve \(y = x ^ { 2 } - 6 x + 11\) at the point with \(x\)-coordinate 4.

10. The curve \(C\) has the equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = ( x + 2 ) ^ { 3 }$$

1. Sketch the curve \(C\), showing the coordinates of any points of intersection with the coordinate axes.
2. Find \(\mathrm { f } ^ { \prime } ( x )\). The straight line \(l\) is the tangent to \(C\) at the point \(P ( - 1,1 )\).
3. Find an equation for \(l\). The straight line \(m\) is parallel to \(l\) and is also a tangent to \(C\).
4. Show that \(m\) has the equation \(y = 3 x + 8\).