7
7
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\end{center}
- (a) Using the factor theorem, show that \(( x + 3 )\) is a factor of
$$x ^ { 3 } - 3 x ^ { 2 } - 10 x + 24$$
(b) Factorise \(x ^ { 3 } - 3 x ^ { 2 } - 10 x + 24\) completely.
2.
$$f ( n ) = n ^ { 3 } + p n ^ { 2 } + 11 n + 9 , \text { where } p \text { is a constant. }$$
(a) Given that \(\mathrm { f } ( n )\) has a remainder of 3 when it is divided by ( \(n + 2\) ), prove that \(p = 6\).
(b) Show that \(\mathrm { f } ( n )\) can be written in the form \(( n + 2 ) ( n + q ) ( n + r ) + 3\), where \(q\) and \(r\) are integers to be found.
(c) Hence show that \(\mathrm { f } ( n )\) is divisible by 3 for all positive integer values of \(n\).
3. Find the values of \(\theta\), to 1 decimal place, in the interval \(- 180 \leq \theta < 180\) for which
$$2 \sin ^ { 2 } \theta ^ { \circ } - 2 \sin \theta ^ { \circ } = \cos ^ { 2 } \theta ^ { \circ } .$$
- Every \(\pounds 1\) of money invested in a savings scheme continuously gains interest at a rate of \(4 \%\) per year. Hence, after \(x\) years, the total value of an initial \(\pounds 1\) investment is \(\pounds y\), where
$$y = 1.04 ^ { x }$$
(a) Sketch the graph of \(y = 1.04 ^ { x } , x \geq 0\).
(b) Calculate, to the nearest \(\pounds\), the total value of an initial \(\pounds 800\) investment after 10 years.
(c) Use logarithms to find the number of years it takes to double the total value of any initial investment.
5. The curve \(C\) with equation \(y = p + q e ^ { x }\), where \(p\) and \(q\) are constants, passes through the point \(( 0,2 )\). At the point \(P ( \ln 2 , p + 2 q\) on \(C\), the gradient is 5 .
(a) Find the value of p and the value of \(q\).
The normal to \(C\) at \(P\) crosses the \(x\)-axis at \(L\) and the \(y\)-axis at \(M\).
(b) Show that the area of \(\triangle O L M\), where \(O\) is the origin, is approximately 53.8
6.
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 3}
\includegraphics[alt={},max width=\textwidth]{04f9d12c-3423-441a-b1ff-fe39ef6e51ed-3_517_1300_760_370}
\end{figure}
Figure 3 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 \(A ( p , 0 )\).
(a) Show that \(p = 6\).
(b) Find an equation of the tangent to \(C\) at \(A\).
The curve \(C\) has a maximum at the point \(P\).
(c) Find the \(x\)-coordinate of \(P\).
The shaded region \(R\), in Fig. 3, is bounded by \(C\) and the \(x\)-axis.
(d) Find the area of \(R\).
7.
Figure 1
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{04f9d12c-3423-441a-b1ff-fe39ef6e51ed-4_499_492_319_420}
\captionsetup{labelformat=empty}
\caption{Shape X}
\end{figure}
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{04f9d12c-3423-441a-b1ff-fe39ef6e51ed-4_597_474_274_1096}
\captionsetup{labelformat=empty}
\caption{Shape \(Y\)}
\end{figure}
Figure 1 shows the cross-sections of two drawer handles.
Shape \(X\) is a rectangle \(A B C D\) joined to a semicircle with \(B C\) as diameter. The length \(A B = d \mathrm {~cm}\) and \(B C = 2 d \mathrm {~cm}\).
Shape \(Y\) is a sector \(O P Q\) of a circle with centre \(O\) and radius \(2 d \mathrm {~cm}\).
Angle \(P O Q\) is \(\theta\) radians.
Given that the areas of the shapes \(X\) and \(Y\) are equal,
(a) prove that \(\theta = 1 + \frac { 1 } { 4 } \pi\).
Using this value of \(\theta\), and given that \(d = 3\), find in terms of \(\pi\),
(b) the perimeter of shape \(X\),
(c) the perimeter of shape \(Y\).
(d) Hence find the difference, in mm, between the perimeters of shapes \(X\) and \(Y\).