7
7
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1.
$$f ( x ) = 4 x ^ { 3 } + 3 x ^ { 2 } - 2 x - 6$$
Find the remainder when \(\mathrm { f } ( x )\) is divided by ( \(2 x + 1\) ).
2. The point \(A\) has coordinates \(( 2,5 )\) and the point \(B\) has coordinates \(( - 2,8 )\).
Find, in cartesian form, an equation of the circle with diameter \(A B\).
3.
$$f ( x ) = x ^ { 3 } - 19 x - 30$$
- Show that \(( x + 2 )\) is a factor of \(\mathrm { f } ( x )\).
- Factorise \(\mathrm { f } ( x )\) completely.
4. Express \(\frac { 3 } { x ^ { 2 } + 2 x } + \frac { x - 4 } { x ^ { 2 } - 4 }\) as a single fraction in its simplest form.
5. Find, in degrees, the value of \(\theta\) in the interval \(0 \leq \theta < 360 ^ { \circ }\) for which
$$2 \cos ^ { 2 } \theta - \cos \theta - 1 = \sin ^ { 2 } \theta$$
Give your answers to 1 decimal place where appropriate.
6. A geometric series is \(a + a r + a r ^ { 2 } + \ldots\) - Prove that the sum of the first \(n\) terms of this series is given by
$$S _ { n } = \frac { a \left( 1 - r ^ { n } \right) } { 1 - r }$$
The second and fourth terms of the series are 3 and 1.08 respectively.
Given that all terms in the series are positive, find - the value of \(r\) and the value of \(a\),
- the sum to infinity of the series.
7. .
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 2}
\includegraphics[alt={},max width=\textwidth]{8d60bedd-6496-4fc9-abc8-16fc5ac52e01-3_1141_1297_280_360}
\end{figure}
A rectangular sheet of metal measures 50 cm by 40 cm . Squares of side \(x \mathrm {~cm}\) are cut from each corner of the sheet and the remainder is folded along the dotted lines to make an open tray, as shown in Fig. 2. - Show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of the tray is given by
$$V = 4 x \left( x ^ { 2 } - 45 x + 500 \right) .$$
- State the range of possible values of \(x\).
- Find the value of \(x\) for which \(V\) is a maximum.
- Hence find the maximum value of \(V\).
- Justify that the value of \(V\) you found in part (d) is a maximum.
\section*{8.}
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 1}
\includegraphics[alt={},max width=\textwidth]{8d60bedd-6496-4fc9-abc8-16fc5ac52e01-4_556_554_276_840}
\end{figure}
Figure 1 shows the sector \(A O B\) of a circle, with centre \(O\) and radius 6.5 cm , and \(\angle A O B = 0.8\) radians. - Calculate, in \(\mathrm { cm } ^ { 2 }\), the area of the sector \(A O B\).
- Show that the length of the chord \(A B\) is 5.06 cm , to 3 significant figures.
The segment \(R\), shaded in Fig. 1, is enclosed by the arc \(A B\) and the straight line \(A B\).
- Calculate, in cm , the perimeter of \(R\).
\section*{9.}
\section*{Figure 2}
\includegraphics[max width=\textwidth, alt={}]{8d60bedd-6496-4fc9-abc8-16fc5ac52e01-5_529_1205_324_269}
Figure 2 shows the line with equation \(y = x + 1\) and the curve with equation \(y = 6 x - x ^ { 2 } - 3\).
The line and the curve intersect at the points \(A\) and \(B\), and \(O\) is the origin. - Calculate the coordinates of \(A\) and the coordinates of \(B\).
The shaded region \(R\) is bounded by the line and the curve.
- Calculate the area of \(R\).