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- \(\mathrm { f } ( x ) = x ^ { 3 } + a x ^ { 2 } + b x - 10\), where \(a\) and \(b\) are constants.
When \(\mathrm { f } ( x )\) is divided by ( \(x - 3\) ), the remainder is 14 .
When \(\mathrm { f } ( x )\) is divided by ( \(x + 1\) ), the remainder is - 18 .
- Find the value of \(a\) and the value of \(b\).
- Show that ( \(x - 2\) ) is a factor of \(\mathrm { f } ( x )\).
2. (a) Write down the first four terms of the binomial expansion, in ascending powers of \(x\), of \(( 1 + a x ) ^ { n }\), where \(n > 2\).
Given that, in this expansion, the coefficient of \(x\) is 8 and the coefficient of \(x ^ { 2 }\) is 30 , - find the value of \(n\) and the value of \(a\),
- find the coefficient of \(x ^ { 3 }\).
3. A population of deer is introduced into a park. The population \(P\) at \(t\) years after the deer have been introduced is modelled by
$$P = \frac { 2000 a ^ { t } } { 4 + a ^ { t } } ,$$
where \(a\) is a constant. Given that there are 800 deer in the park after 6 years, - calculate, to 4 decimal places, the value of \(a\),
- use the model to predict the number of years needed for the population of deer to increase from 800 to 1800.
- With reference to this model, give a reason why the population of deer cannot exceed 2000.
4. Given that \(\mathrm { f } ( x ) = \left( 2 x ^ { \frac { 3 } { 2 } } - 3 x ^ { - \frac { 3 } { 2 } } \right) ^ { 2 } + 5 , \quad x > 0\), - find, to 3 significant figures, the value of \(x\) for which \(\mathrm { f } ( x ) = 5\).
- Show that \(\mathrm { f } ( x )\) may be written in the form \(A x ^ { 3 } + \frac { B } { x ^ { 3 } } + C\), where \(A , B\) and \(C\) are constants to be found.
- Hence evaluate \(\int _ { 1 } ^ { 2 } f ( x ) d x\).
5.
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 1}
\includegraphics[alt={},max width=\textwidth]{5e4e4cae-d4c6-465d-8cb7-84712e6e55fe-3_736_1266_276_404}
\end{figure}
Figure 1 shows the cross-section \(A B C D\) of a chocolate bar, where \(A B , C D\) and \(A D\) are straight lines and \(M\) is the mid-point of \(A D\). The length \(A D\) is 28 mm , and \(B C\) is an arc of a circle with centre \(M\).
Taking \(A\) as the origin, \(B , C\) and \(D\) have coordinates \(( 7,24 ) , ( 21,24 )\) and \(( 28,0 )\) respectively. - Show that the length of \(B M\) is 25 mm .
- Show that, to 3 significant figures, \(\angle B M C = 0.568\) radians.
- Hence calculate, in \(\mathrm { mm } ^ { 2 }\), the area of the cross-section of the chocolate bar.
Given that this chocolate bar has length 85 mm ,
- calculate, to the nearest \(\mathrm { cm } ^ { 3 }\), the volume of the bar.
6.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5e4e4cae-d4c6-465d-8cb7-84712e6e55fe-4_641_1406_196_287}
\captionsetup{labelformat=empty}
\caption{Fig. 1}
\end{figure}
Figure 1 shows the curve with equation \(y = 5 + 2 x - x ^ { 2 }\) and the line with equation \(y = 2\). The curve and the line intersect at the points \(A\) and \(B\). - Find the x-coordinates of \(A\) and \(B\).
The shaded region \(R\) is bounded by the curve and the line.
- Find the area of \(R\).
7. Find all the values of \(\theta\) in the interval \(0 \leq \theta < 360 ^ { \circ }\) for which - \(\cos \left( \theta - 10 ^ { \circ } \right) = \cos 15 ^ { \circ }\),
- \(\tan 2 \theta = 0.4\),
- \(2 \sin \theta \tan \theta = 3\).