Edexcel C2 (Core Mathematics 2)

1. \(\quad \mathrm { f } ( x ) = 3 x ^ { 3 } - 2 x ^ { 2 } + k x + 9\).

Given that when \(\mathrm { f } ( x )\) is divided by ( \(x + 2\) ) there is a remainder of - 35 ,

1. find the value of the constant \(k\),
2. find the remainder when \(\mathrm { f } ( x )\) is divided by ( \(3 x - 2\) ).

2. \begin{figure}[h] \end{figure} Figure 1 shows the curve with equation \(y = 2 ^ { x }\).
Use the trapezium rule with four intervals of equal width to estimate the area of the shaded region bounded by the curve, the \(x\)-axis and the lines \(x = - 2\) and \(x = 2\).

3. Giving your answers in terms of \(\pi\), solve the equation $$3 \tan ^ { 2 } \theta - 1 = 0 ,$$ for \(\theta\) in the interval \(- \pi \leq \theta \leq \pi\).

4. (a) Expand \(( 1 + 3 x ) ^ { 8 }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\). You should simplify each coefficient in your expansion.
(b) Use your series, together with a suitable value of \(x\) which you should state, to estimate the value of (1.003) \({ } ^ { 8 }\), giving your answer to 8 significant figures.

5. (a) Given that \(t = \log _ { 3 } x\), find expressions in terms of \(t\) for

1. \(\log _ { 3 } x ^ { 2 }\),
2. \(\log _ { 9 } x\).
   (b) Hence, or otherwise, find to 3 significant figures the value of \(x\) such that $$\log _ { 3 } x ^ { 2 } - \log _ { 9 } x = 4 .$$

1. The circle \(C\) has centre \(( - 3,2 )\) and passes through the point \(( 2,1 )\).
   1. Find an equation for \(C\).
   2. Show that the point with coordinates \(( - 4,7 )\) lies on \(C\).
   3. Find an equation for the tangent to \(C\) at the point ( - 4 , 7). Give your answer in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers.
   \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c7c8cf84-06ac-4059-b8f0-d68b6d1d8dcc-3_664_1016_1276_376} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure} Figure 2 shows the curve \(y = 2 x ^ { 2 } + 6 x + 7\) and the straight line \(y = 2 x + 13\).

1. Find the coordinates of the points where the curve and line intersect.
2. Find the area of the shaded region bounded by the curve and line.

8. A geometric series has first term \(a\) and common ratio \(r\) where \(r > 1\). The sum of the first \(n\) terms of the series is denoted by \(S _ { n }\). Given that \(S _ { 4 } = 10 \times S _ { 2 }\),

1. find the value of \(r\). Given also that \(S _ { 3 } = 26\),
2. find the value of \(a\),
3. show that \(S _ { 6 } = 728\).

9. \begin{figure}[h] \end{figure} Figure 3 shows a design consisting of two rectangles measuring \(x \mathrm {~cm}\) by \(y \mathrm {~cm}\) joined to a circular sector of radius \(x \mathrm {~cm}\) and angle 0.5 radians. Given that the area of the design is \(50 \mathrm {~cm} ^ { 2 }\),

1. show that the perimeter, \(P\) cm, of the design is given by $$P = 2 x + \frac { 100 } { x }$$
2. Find the value of \(x\) for which \(P\) is a minimum.
3. Show that \(P\) is a minimum for this value of \(x\).
4. Find the minimum value of \(P\) in the form \(k \sqrt { 2 }\).