OCR C1 (Core Mathematics 1) 2012 January

1 Express \(\frac { 15 + \sqrt { 3 } } { 3 - \sqrt { 3 } }\) in the form \(a + b \sqrt { 3 }\), where \(a\) and \(b\) are integers.

2
\includegraphics[max width=\textwidth, alt={}, center]{559e9f1a-340e-4478-adaa-a7361dd70fe8-2_325_479_468_794} The graph of \(y = \mathrm { f } ( x )\) for \(- 2 \leqslant x \leqslant 2\) is shown above.

1. Sketch the graph of \(y = \mathrm { f } ( - x )\) for \(- 2 \leqslant x \leqslant 2\).
2. Sketch the graph of \(y = \mathrm { f } ( x ) + 2\) for \(- 2 \leqslant x \leqslant 2\).

3 Given that $$5 x ^ { 2 } + p x - 8 = q ( x - 1 ) ^ { 2 } + r$$ for all values of \(x\), find the values of the constants \(p , q\) and \(r\).

4 Evaluate

1. \(3 ^ { - 2 }\),
2. \(16 ^ { \frac { 3 } { 4 } }\),
3. \(\frac { \sqrt { 200 } } { \sqrt { 8 } }\).

5 Find the real roots of the equation \(\frac { 3 } { y ^ { 4 } } - \frac { 10 } { y ^ { 2 } } - 8 = 0\).

6 Given that \(\mathrm { f } ( x ) = \frac { 4 } { x } - 3 x + 2\),

1. find \(\mathrm { f } ^ { \prime } ( x )\),
2. find \(\mathrm { f } ^ { \prime \prime } \left( \frac { 1 } { 2 } \right)\).

7 A curve has equation \(y = ( x + 2 ) \left( x ^ { 2 } - 3 x + 5 \right)\).

1. Find the coordinates of the minimum point, justifying that it is a minimum.
2. Calculate the discriminant of \(x ^ { 2 } - 3 x + 5\).
3. Explain why \(( x + 2 ) \left( x ^ { 2 } - 3 x + 5 \right)\) is always positive for \(x > - 2\).

8 The line \(l\) has gradient - 2 and passes through the point \(A ( 3,5 ) . B\) is a point on the line \(l\) such that the distance \(A B\) is \(6 \sqrt { 5 }\). Find the coordinates of each of the possible points \(B\).

9

1. Sketch the curve \(y = 12 - x - x ^ { 2 }\), giving the coordinates of all intercepts with the axes.
2. Solve the inequality \(12 - x - x ^ { 2 } > 0\).
3. Find the coordinates of the points of intersection of the curve \(y = 12 - x - x ^ { 2 }\) and the line \(3 x + y = 4\).

10 A circle has centre \(C ( - 2,4 )\) and radius 5 .

1. Find the equation of the circle, giving your answer in the form \(x ^ { 2 } + y ^ { 2 } + a x + b y + c = 0\).
2. Show that the tangent to the circle at the point \(P ( - 5,8 )\) has equation \(3 x - 4 y + 47 = 0\).
3. Verify that the point \(T ( 3,14 )\) lies on this tangent.
4. Find the area of the triangle \(C P T\). \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE}