Edexcel C2 (Core Mathematics 2)

\begin{enumerate} \item (a) Show that \(( x + 2 )\) is a factor of \(\mathrm { f } ( x ) = x ^ { 3 } - 19 x - 30\).
(b) Factorise \(\mathrm { f } ( x )\) completely. \item For the binomial expansion, in descending powers of \(x\), of \(\left( x ^ { 3 } - \frac { 1 } { 2 x } \right) ^ { 12 }\),

1. find the first 4 terms, simplifying each term.
2. Find, in its simplest form, the term independent of \(x\) in this expansion.
   [0pt] [P2 June 2004 Question 3] \item The curve \(C\) has equation \(y = \cos \left( x + \frac { \pi } { 4 } \right) , 0 \leq x \leq 2 \pi\).
3. Sketch \(C\).
4. Write down the exact coordinates of the points at which \(C\) meets the coordinate axes.
5. Solve, for \(x\) in the interval \(0 \leq x \leq 2 \pi , \cos \left( x + \frac { \pi } { 4 } \right) = 0.5\), giving your answers in terms of \(\pi\). \item Given that \(\log _ { 2 } x = a\), find, in terms of \(a\), the simplest form of
6. \(\log _ { 2 } ( 16 x )\),
7. \(\log _ { 2 } \left( \frac { x ^ { 4 } } { 2 } \right)\).
8. Hence, or otherwise, solve \(\log _ { 2 } ( 16 x ) - \log _ { 2 } \left( \frac { x ^ { 4 } } { 2 } \right) = \frac { 1 } { 2 }\), giving your answer in its simplest surd form. \item (a) Given that \(3 \sin x = 8 \cos x\), find the value of \(\tan x\).
9. Find, to 1 decimal place, all the solutions of \(3 \sin x - 8 \cos x = 0\) in the interval \(0 \leq x < 360 ^ { \circ }\).
10. Find, to 1 decimal place, all the solutions of \(3 \sin ^ { 2 } y - 8 \cos y = 0\) in the interval \(0 \leq y < 360 ^ { \circ }\). \item \end{enumerate} $$f ( x ) = \frac { \left( x ^ { 2 } - 3 \right) ^ { 2 } } { x ^ { 3 } } , x \neq 0$$
11. Show that \(\mathrm { f } ( x ) \equiv x - 6 x ^ { - 1 } + 9 x ^ { - 3 }\).
12. Hence, or otherwise, differentiate \(\mathrm { f } ( x )\) with respect to \(x\).
13. Verify that the graph of \(y = \mathrm { f } ( x )\) has stationary points at \(x = \pm \sqrt { } 3\).
14. Determine whether the stationary value at \(x = \sqrt { } 3\) is a maximum or a minimum.

7. \begin{figure}[h] \end{figure} Fig. 1 shows part of the curve \(C\) with equation \(y = \frac { 3 } { 2 } x ^ { 2 } - \frac { 1 } { 4 } x ^ { 3 }\).
The curve \(C\) touches the \(x\)-axis at the origin and passes through the point \(A ( p , 0 )\).

1. Show that \(p = 6\).
2. Find an equation of the tangent to \(C\) at \(A\). The curve \(C\) has a maximum at the point \(P\).
3. Find the \(x\)-coordinate of \(P\). The shaded region \(R\), in Fig. 1, is bounded by \(C\) and the \(x\)-axis.
4. Find the area of \(R\).

8. A geometric series is \(a + a r + a r ^ { 2 } + \ldots\)

1. Prove that the sum of the first \(n\) terms of this series is \(S _ { n } = \frac { a \left( 1 - r ^ { n } \right) } { 1 - r }\). The first and second terms of a geometric series \(G\) are 10 and 9 respectively.
2. Find, to 3 significant figures, the sum of the first twenty terms of \(G\).
3. Find the sum to infinity of \(G\). Another geometric series has its first term equal to its common ratio. The sum to infinity of this series is 10.
4. Find the exact value of the common ratio of this series.