Edexcel P3 (Pure Mathematics 3) 2020 October

1. Solve, for \(0 \leqslant x < 360 ^ { \circ }\), the equation

$$2 \cos 2 x = 7 \cos x$$ giving your solutions to one decimal place.
(Solutions based entirely on graphical or numerical methods are not acceptable.)

1. A scientist monitored the growth of bacteria on a dish over a 30 -day period.

The area, \(N \mathrm {~mm} ^ { 2 }\), of the dish covered by bacteria, \(t\) days after monitoring began, is modelled by the equation $$\log _ { 10 } N = 0.0646 t + 1.478 \quad 0 \leqslant t \leqslant 30$$

1. Show that this equation may be written in the form $$N = a b ^ { t }$$ where \(a\) and \(b\) are constants to be found. Give the value of \(a\) to the nearest integer and give the value of \(b\) to 3 significant figures.
2. Use the model to find the area of the dish covered by bacteria 30 days after monitoring began. Give your answer, in \(\mathrm { mm } ^ { 2 }\), to 2 significant figures.

3. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of a curve with equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = \frac { 2 x + 3 } { \sqrt { 4 x - 1 } } \quad x > \frac { 1 } { 4 }$$

1. Find, in simplest form, \(\mathrm { f } ^ { \prime } ( x )\).
2. Hence find the range of f.

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4. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the graph with equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = 21 - 2 | 2 - x | \quad x \geqslant 0$$

1. Find ff(6)
2. Solve the equation \(\mathrm { f } ( x ) = 5 x\) Given that the equation \(\mathrm { f } ( x ) = k\), where \(k\) is a constant, has exactly two roots,
3. state the set of possible values of \(k\). The graph with equation \(y = \mathrm { f } ( x )\) is transformed onto the graph with equation \(y = a \mathrm { f } ( x - b )\) The vertex of the graph with equation \(y = a \mathrm { f } ( x - b )\) is (6, 3). Given that \(a\) and \(b\) are constants,
4. find the value of \(a\) and the value of \(b\). \includegraphics[max width=\textwidth, alt={}, center]{96948fd3-5438-4e95-b41b-2f649ca8dfac-11_2255_50_314_34}

5. (a) Show that $$\sin 3 x \equiv 3 \sin x - 4 \sin ^ { 3 } x$$ (b) Hence find, using algebraic integration, $$\int _ { 0 } ^ { \frac { \pi } { 3 } } \sin ^ { 3 } x d x$$

6. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of curve \(C _ { 1 }\) with equation \(y = 5 \mathrm { e } ^ { x - 1 } + 3\)
and curve \(C _ { 2 }\) with equation \(y = 10 - x ^ { 2 }\)
The point \(P\) lies on \(C _ { 1 }\) and has \(y\) coordinate 18

1. Find the \(x\) coordinate of \(P\), writing your answer in the form \(\ln k\), where \(k\) is a constant to be found. The curve \(C _ { 1 }\) meets the curve \(C _ { 2 }\) at \(x = \alpha\) and at \(x = \beta\), as shown in Figure 3.
2. Using a suitable interval and a suitable function that should be stated, show that to 3 decimal places \(\alpha = 1.134\) The iterative equation $$x _ { n + 1 } = - \sqrt { 7 - 5 \mathrm { e } ^ { x _ { n } - 1 } }$$ is used to find an approximation to \(\beta\). Using this iterative formula with \(x _ { 1 } = - 3\)
3. find the value of \(x _ { 2 }\) and the value of \(\beta\), giving each answer to 6 decimal places.

7. (a) Express \(\cos x + 4 \sin x\) in the form \(R \cos ( x - \alpha )\) where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\) Give the exact value of \(R\) and give the value of \(\alpha\), in radians, to 3 decimal places. A scientist is studying the behaviour of seabirds in a colony. She models the height above sea level, \(H\) metres, of one of the birds in the colony by the equation $$H = \frac { 24 } { 3 + \cos \left( \frac { 1 } { 2 } t \right) + 4 \sin \left( \frac { 1 } { 2 } t \right) } \quad 0 \leqslant t \leqslant 6.5$$ where \(t\) seconds is the time after it leaves the nest. Find, according to the model,
(b) the minimum height of the seabird above sea level, giving your answer to the nearest cm,
(c) the value of \(t\), to 2 decimal places, when \(H = 10\) \includegraphics[max width=\textwidth, alt={}, center]{96948fd3-5438-4e95-b41b-2f649ca8dfac-21_2255_50_314_34}

1. 1. The curve \(C\) has equation \(y = \mathrm { g } ( x )\) where

$$g ( x ) = e ^ { 3 x } \sec 2 x \quad - \frac { \pi } { 4 } < x < \frac { \pi } { 4 }$$

1. Find \(\mathrm { g } ^ { \prime } ( x )\)
2. Hence find the \(x\) coordinate of the stationary point of \(C\).
   (ii) A different curve has equation $$x = \ln ( \sin y ) \quad 0 < y < \frac { \pi } { 2 }$$ Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { \mathrm { e } ^ { x } } { \mathrm { f } ( x ) }$$ where \(\mathrm { f } ( x )\) is a function of \(\mathrm { e } ^ { x }\) that should be found.
   

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9. (a) Given that $$\frac { x ^ { 4 } - x ^ { 3 } - 10 x ^ { 2 } + 3 x - 9 } { x ^ { 2 } - x - 12 } \equiv x ^ { 2 } + P + \frac { Q } { x - 4 } \quad x > - 3$$ find the value of the constant \(P\) and show that \(Q = 5\) The curve \(C\) has equation \(y = \mathrm { g } ( x )\), where $$g ( x ) = \frac { x ^ { 4 } - x ^ { 3 } - 10 x ^ { 2 } + 3 x - 9 } { x ^ { 2 } - x - 12 } \quad - 3 < x < 3.5 \quad x \in \mathbb { R }$$ (b) Find the equation of the tangent to \(C\) at the point where \(x = 2\) Give your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be found. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of the curve \(C\).
The region \(R\), shown shaded in Figure 4, is bounded by \(C\), the \(y\)-axis, the \(x\)-axis and the line with equation \(x = 2\)
(c) Find the exact area of \(R\), writing your answer in the form \(a + b \ln 2\), where \(a\) and \(b\) are constants to be found. \includegraphics[max width=\textwidth, alt={}, center]{96948fd3-5438-4e95-b41b-2f649ca8dfac-31_2255_50_314_34}
\includegraphics[max width=\textwidth, alt={}, center]{96948fd3-5438-4e95-b41b-2f649ca8dfac-32_106_113_2524_1832}