Edexcel C2 (Core Mathematics 2)

1. Given that \(p = \log _ { q } 16\), express in terms of \(p\),
   1. \(\log _ { q } 2\),
   2. \(\log _ { q } ( 8 q )\).
   3. The expansion of \(( 2 - p x ) ^ { 6 }\) in ascending powers of \(x\), as far as the term in \(x ^ { 2 }\), is
   $$64 + A x + 135 x ^ { 2 }$$ Given that \(p > 0\), find the value of \(p\) and the value of \(A\).
   (7)

3. A circle \(C\) has equation $$x ^ { 2 } + y ^ { 2 } - 6 x + 8 y - 75 = 0$$

1. Write down the coordinates of the centre of \(C\), and calculate the radius of \(C\). A second circle has centre at the point \(( 15,12 )\) and radius 10 .
2. Sketch both circles on a single diagram and find the coordinates of the point where they touch.
   (4)

4. (a) Sketch, for \(0 \leq x \leq 360 ^ { \circ }\), the graph of \(y = \sin \left( x + 30 ^ { \circ } \right)\).
(b) Write down the coordinates of the points at which the graph meets the axes.
(c) Solve, for \(0 \leq x < 360 ^ { \circ }\), the equation $$\sin \left( x + 30 ^ { \circ } \right) = - \frac { 1 } { 2 }$$

5. \begin{figure}[h] \end{figure} The shape of a badge is a sector \(A B C\) of a circle with centre \(A\) and radius \(A B\), as shown in Fig 1. The triangle \(A B C\) is equilateral and has a perpendicular height 3 cm .

1. Find, in surd form, the length \(A B\).
2. Find, in terms of \(\pi\), the area of the badge.
3. Prove that the perimeter of the badge is \(\frac { 2 \sqrt { 3 } } { 3 } ( \pi + 6 ) \mathrm { cm }\).

6. \(\mathrm { f } ( x ) = 6 x ^ { 3 } + p x ^ { 2 } + q x + 8\), where \(p\) and \(q\) are constants. Given that \(\mathrm { f } ( x )\) is exactly divisible by ( \(2 x - 1\) ), and also that when \(\mathrm { f } ( x )\) is divided by ( \(x - 1\) ) the remainder is - 7 ,

1. find the value of \(p\) and the value of \(q\).
2. Hence factorise \(\mathrm { f } ( x )\) completely.

7. A geometric series has first term 1200. Its sum to infinity is 960 .

1. Show that the common ratio of the series is \(- \frac { 1 } { 4 }\).
2. Find, to 3 decimal places, the difference between the ninth and tenth terms of the series.
3. Write down an expression for the sum of the first \(n\) terms of the series. Given that \(n\) is odd,
4. prove that the sum of the first \(n\) terms of the series is $$960 \left( 1 + 0.25 ^ { n } \right)$$

1. A circle \(C\) has centre \(( 3,4 )\) and radius \(3 \sqrt { } 2\). A straight line \(l\) has equation \(y = x + 3\).
   1. Write down an equation of the circle \(C\).
   2. Calculate the exact coordinates of the two points where the line \(l\) intersects \(C\), giving your answers in surds.
   3. Find the distance between these two points.
   \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{c85316fe-5c59-4cb3-8cb8-d95a4e97af70-5_730_983_278_404}
   \end{figure} The curve \(C\), shown in Fig. 2, represents the graph of $$y = \frac { x ^ { 2 } } { 25 } , x \geq 0 .$$ The points \(A\) and \(B\) on the curve \(C\) have \(x\)-coordinates 5 and 10 respectively.
2. Write down the \(y\)-coordinates of \(A\) and \(B\).
3. Find an equation of the tangent to \(C\) at \(A\). The finite region \(R\) is enclosed by \(C\), the \(y\)-axis and the lines through \(A\) and \(B\) parallel to the \(x\)-axis.
4. For points \(( x , y )\) on \(C\), express \(x\) in terms of \(y\).
5. Use integration to find the area of \(R\). END