OCR C1 (Core Mathematics 1)

1. Evaluate \(49 ^ { \frac { 1 } { 2 } } + 8 ^ { \frac { 2 } { 3 } }\).
2. Solve the equation

$$3 x - \frac { 5 } { x } = 2 .$$

1. Find the set of values of \(x\) for which
   1. \(6 x - 11 > x + 4\),
   2. \(x ^ { 2 } - 6 x - 16 < 0\).
   3. (i) Sketch on the same diagram the graphs of \(y = ( x - 1 ) ^ { 2 } ( x - 5 )\) and \(y = 8 - 2 x\).
   Label on your diagram the coordinates of any points where each graph meets the coordinate axes.
2. Explain how your diagram shows that there is only one solution, \(\alpha\), to the equation $$( x - 1 ) ^ { 2 } ( x - 5 ) = 8 - 2 x$$
3. State the integer, \(n\), such that $$n < \alpha < n + 1 .$$

5. $$f ( x ) = x ^ { 2 } - 10 x + 17$$

1. Express \(\mathrm { f } ( x )\) in the form \(a ( x + b ) ^ { 2 } + c\).
2. State the coordinates of the minimum point of the curve \(y = \mathrm { f } ( x )\).
3. Deduce the coordinates of the minimum point of each of the following curves:
   1. \(y = \mathrm { f } ( x ) + 4\),
   2. \(y = \mathrm { f } ( 2 x )\).

6. The points \(P , Q\) and \(R\) have coordinates (-5, 2), (-3, 8) and (9, 4) respectively.

1. Show that \(\angle P Q R = 90 ^ { \circ }\). Given that \(P , Q\) and \(R\) all lie on a circle,
2. find the coordinates of the centre of the circle,
3. show that the equation of the circle can be written in the form $$x ^ { 2 } + y ^ { 2 } - 4 x - 6 y = k$$ where \(k\) is an integer to be found.

7. The straight line \(l _ { 1 }\) has gradient \(\frac { 3 } { 2 }\) and passes through the point \(A ( 5,3 )\).

1. Find an equation for \(l _ { 1 }\) in the form \(y = m x + c\). The straight line \(l _ { 2 }\) has the equation \(3 x - 4 y + 3 = 0\) and intersects \(l _ { 1 }\) at the point \(B\).
2. Find the coordinates of \(B\).
3. Find the coordinates of the mid-point of \(A B\).
4. Show that the straight line parallel to \(l _ { 2 }\) which passes through the mid-point of \(A B\) also passes through the origin.

8.
\includegraphics[max width=\textwidth, alt={}, center]{b7078372-d0d3-4563-818d-637260be5efc-3_592_727_251_493} The diagram shows the curve with equation \(y = 2 + 3 x - x ^ { 2 }\) and the straight lines \(l\) and \(m\). The line \(l\) is the tangent to the curve at the point \(A\) where the curve crosses the \(y\)-axis.

1. Find an equation for \(l\). The line \(m\) is the normal to the curve at the point \(B\).
   Given that \(l\) and \(m\) are parallel,
2. find the coordinates of \(B\).

9. The curve \(C\) has the equation $$y = 3 - x ^ { \frac { 1 } { 2 } } - 2 x ^ { - \frac { 1 } { 2 } } , \quad x > 0 .$$

1. Find the coordinates of the points where \(C\) crosses the \(x\)-axis.
2. Find the exact coordinates of the stationary point of \(C\).
3. Determine the nature of the stationary point.
4. Sketch the curve \(C\).