Question 9 - A-Level Maths

Edexcel C3 — Question 9

Exam Board	Edexcel
Module	C3 (Core Mathematics 3)
Topic	Exponential Equations & Modelling

9. (i) Find the exact solutions to the equations

$\ln ( 3 x - 7 ) = 5$
$3 ^ { x } \mathrm { e } ^ { 7 x + 2 } = 15$
(ii) The functions f and g are defined by $$\begin{array} { l l } \mathrm { f } ( x ) = \mathrm { e } ^ { 2 x } + 3 , & x \in \mathbb { R }
\mathrm {~g} ( x ) = \ln ( x - 1 ) , & x \in \mathbb { R } , x > 1 \end{array}$$
Find $\mathrm { f } ^ { - 1 }$ and state its domain.
Find fg and state its range.
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1. (a) Show that
$$\frac { \sin 2 \theta } { 1 + \cos 2 \theta } = \tan \theta$$
Hence find, for $- 180 ^ { \circ } \leqslant \theta < 180 ^ { \circ }$, all the solutions of $$\frac { 2 \sin 2 \theta } { 1 + \cos 2 \theta } = 1$$ Give your answers to 1 decimal place.
2. A curve $C$ has equation $$y = \frac { 3 } { ( 5 - 3 x ) ^ { 2 } } , \quad x \neq \frac { 5 } { 3 }$$ The point $P$ on $C$ has $x$-coordinate 2. Find an equation of the normal to $C$ at $P$ in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers.
3. $\mathrm { f } ( x ) = 4 \operatorname { cosec } x - 4 x + 1$, where $x$ is in radians.
Show that there is a root $\alpha$ of $\mathrm { f } ( x ) = 0$ in the interval [1.2,1.3].
Show that the equation $\mathrm { f } ( x ) = 0$ can be written in the form $$x = \frac { 1 } { \sin x } + \frac { 1 } { 4 }$$
Use the iterative formula $$x _ { n + 1 } = \frac { 1 } { \sin x _ { n } } + \frac { 1 } { 4 } , \quad x _ { 0 } = 1.25$$ to calculate the values of $x _ { 1 } , x _ { 2 }$ and $x _ { 3 }$, giving your answers to 4 decimal places.
By considering the change of sign of $\mathrm { f } ( x )$ in a suitable interval, verify that $\alpha = 1.291$ correct to 3 decimal places. 4. The function $f$ is defined by $$f : x \mapsto | 2 x - 5 | , \quad x \in \mathbb { R }$$
Sketch the graph with equation $y = \mathrm { f } ( x )$, showing the coordinates of the points where the graph cuts or meets the axes.
Solve $\mathrm { f } ( x ) = 15 + x$. The function $g$ is defined by $$g : x \mapsto x ^ { 2 } - 4 x + 1 , \quad x \in \mathbb { R } , \quad 0 \leqslant x \leqslant 5$$
Find fg(2).
Find the range of g. 5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9527d80b-a2a2-442f-9e32-d8768fbbd01a-138_701_1125_246_443} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve $C$ with the equation $y = \left( 2 x ^ { 2 } - 5 x + 2 \right) \mathrm { e } ^ { - x }$.
Find the coordinates of the point where $C$ crosses the $y$-axis.
Show that $C$ crosses the $x$-axis at $x = 2$ and find the $x$-coordinate of the other point where $C$ crosses the $x$-axis.
Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.
Hence find the exact coordinates of the turning points of $C$.
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9527d80b-a2a2-442f-9e32-d8768fbbd01a-140_781_858_239_575} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of the curve with the equation $y = \mathrm { f } ( x ) , x \in \mathbb { R }$. The curve has a turning point at $A ( 3 , - 4 )$ and also passes through the point $( 0,5 )$.
Write down the coordinates of the point to which $A$ is transformed on the curve with equation
1. $y = | \mathrm { f } ( x ) |$,
2. $y = 2 f \left( \frac { 1 } { 2 } x \right)$.
Sketch the curve with equation $$y = \mathrm { f } ( | x | )$$ On your sketch show the coordinates of all turning points and the coordinates of the point at which the curve cuts the $y$-axis. The curve with equation $y = \mathrm { f } ( x )$ is a translation of the curve with equation $y = x ^ { 2 }$.
Find $\mathrm { f } ( x )$.
Explain why the function f does not have an inverse. 7. (a) Express $2 \sin \theta - 1.5 \cos \theta$ in the form $R \sin ( \theta - \alpha )$, where $R > 0$ and $0 < \alpha < \frac { \pi } { 2 }$. Give the value of $\alpha$ to 4 decimal places.
1. Find the maximum value of $2 \sin \theta - 1.5 \cos \theta$.
2. Find the value of $\theta$, for $0 \leqslant \theta < \pi$, at which this maximum occurs. Tom models the height of sea water, $H$ metres, on a particular day by the equation $$H = 6 + 2 \sin \left( \frac { 4 \pi t } { 25 } \right) - 1.5 \cos \left( \frac { 4 \pi t } { 25 } \right) , \quad 0 \leqslant t < 12$$ where $t$ hours is the number of hours after midday.
Calculate the maximum value of $H$ predicted by this model and the value of $t$, to 2 decimal places, when this maximum occurs.
Calculate, to the nearest minute, the times when the height of sea water is predicted, by this model, to be 7 metres. 8. (a) Simplify fully $$\frac { 2 x ^ { 2 } + 9 x - 5 } { x ^ { 2 } + 2 x - 15 }$$ Given that $$\ln \left( 2 x ^ { 2 } + 9 x - 5 \right) = 1 + \ln \left( x ^ { 2 } + 2 x - 15 \right) , \quad x \neq - 5$$
find $x$ in terms of e.
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1. (a) Express $7 \cos x - 24 \sin x$ in the form $R \cos ( x + \alpha )$ where $R > 0$ and $0 < \alpha < \frac { \pi } { 2 }$. Give the value of $\alpha$ to 3 decimal places.
2. Hence write down the minimum value of $7 \cos x - 24 \sin x$.
3. Solve, for $0 \leqslant x < 2 \pi$, the equation
$$7 \cos x - 24 \sin x = 10$$ giving your answers to 2 decimal places.
2. (a) Express $$\frac { 4 x - 1 } { 2 ( x - 1 ) } - \frac { 3 } { 2 ( x - 1 ) ( 2 x - 1 ) }$$ as a single fraction in its simplest form. Given that $$f ( x ) = \frac { 4 x - 1 } { 2 ( x - 1 ) } - \frac { 3 } { 2 ( x - 1 ) ( 2 x - 1 ) } - 2 , \quad x > 1$$
show that $$f ( x ) = \frac { 3 } { 2 x - 1 }$$
Hence differentiate $\mathrm { f } ( x )$ and find $\mathrm { f } ^ { \prime } ( 2 )$.
1. Find all the solutions of
$$2 \cos 2 \theta = 1 - 2 \sin \theta$$ in the interval $0 \leqslant \theta < 360 ^ { \circ }$.
4. Joan brings a cup of hot tea into a room and places the cup on a table. At time $t$ minutes after Joan places the cup on the table, the temperature, $\theta ^ { \circ } \mathrm { C }$, of the tea is modelled by the equation $$\theta = 20 + A \mathrm { e } ^ { - k t } ,$$ where $A$ and $k$ are positive constants. Given that the initial temperature of the tea was $90 ^ { \circ } \mathrm { C }$,
find the value of $A$. The tea takes 5 minutes to decrease in temperature from $90 ^ { \circ } \mathrm { C }$ to $55 ^ { \circ } \mathrm { C }$.
Show that $k = \frac { 1 } { 5 } \ln 2$.
Find the rate at which the temperature of the tea is decreasing at the instant when $t = 10$. Give your answer, in ${ } ^ { \circ } \mathrm { C }$ per minute, to 3 decimal places. 5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9527d80b-a2a2-442f-9e32-d8768fbbd01a-152_624_1054_274_447} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of part of the curve with equation $y = \mathrm { f } ( x )$, where $$\mathrm { f } ( x ) = ( 8 - x ) \ln x , \quad x > 0$$ The curve cuts the $x$-axis at the points $A$ and $B$ and has a maximum turning point at $Q$, as shown in Figure 1.
Write down the coordinates of $A$ and the coordinates of $B$.
Find f'(x).
Show that the $x$-coordinate of $Q$ lies between 3.5 and 3.6
Show that the $x$-coordinate of $Q$ is the solution of $$x = \frac { 8 } { 1 + \ln x }$$ To find an approximation for the $x$-coordinate of $Q$, the iteration formula $$x _ { n + 1 } = \frac { 8 } { 1 + \ln x _ { n } }$$ is used.
Taking $x _ { 0 } = 3.55$, find the values of $x _ { 1 } , x _ { 2 }$ and $x _ { 3 }$. Give your answers to 3 decimal places.
1. The function $f$ is defined by
$$\mathrm { f } : x \mapsto \frac { 3 - 2 x } { x - 5 } , \quad x \in \mathbb { R } , x \neq 5$$
Find $\mathrm { f } ^ { - 1 } ( x )$.
(3) \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9527d80b-a2a2-442f-9e32-d8768fbbd01a-154_901_1091_593_429} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} The function g has domain $- 1 \leqslant x \leqslant 8$, and is linear from $( - 1 , - 9 )$ to $( 2,0 )$ and from $( 2,0 )$ to $( 8,4 )$. Figure 2 shows a sketch of the graph of $y = \mathrm { g } ( x )$.
Write down the range of g.
Find $\operatorname { gg } ( 2 )$.
Find $\mathrm { fg } ( 8 )$.
On separate diagrams, sketch the graph with equation
1. $y = | \mathrm { g } ( x ) |$,
2. $y = \mathrm { g } ^ { - 1 } ( x )$. Show on each sketch the coordinates of each point at which the graph meets or cuts the axes.
State the domain of the inverse function $\mathrm { g } ^ { - 1 }$.
1. The curve $C$ has equation
$$y = \frac { 3 + \sin 2 x } { 2 + \cos 2 x }$$
Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 6 \sin 2 x + 4 \cos 2 x + 2 } { ( 2 + \cos 2 x ) ^ { 2 } }$$
Find an equation of the tangent to $C$ at the point on $C$ where $x = \frac { \pi } { 2 }$. Write your answer in the form $y = a x + b$, where $a$ and $b$ are exact constants.
8. (a) Given that $$\frac { \mathrm { d } } { \mathrm {~d} x } ( \cos x ) = - \sin x$$ show that $\frac { \mathrm { d } } { \mathrm { d } x } ( \sec x ) = \sec x \tan x$. Given that $$x = \sec 2 y$$
find $\frac { \mathrm { d } x } { \mathrm {~d} y }$ in terms of $y$.
Hence find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $x$. \includegraphics[max width=\textwidth, alt={}, center]{9527d80b-a2a2-442f-9e32-d8768fbbd01a-158_102_93_2473_1804} Turn over
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1. Differentiate with respect to $x$
2. $\quad \ln \left( x ^ { 2 } + 3 x + 5 \right)$
3. $\frac { \cos x } { x ^ { 2 } }$
$$\mathrm { f } ( x ) = 2 \sin \left( x ^ { 2 } \right) + x - 2 , \quad 0 \leqslant x < 2 \pi$$
Show that $\mathrm { f } ( x ) = 0$ has a root $\alpha$ between $x = 0.75$ and $x = 0.85$ The equation $\mathrm { f } ( x ) = 0$ can be written as $x = [ \arcsin ( 1 - 0.5 x ) ] ^ { \frac { 1 } { 2 } }$.
Use the iterative formula $$x _ { n + 1 } = \left[ \arcsin \left( 1 - 0.5 x _ { n } \right) \right] ^ { \frac { 1 } { 2 } } , \quad x _ { 0 } = 0.8$$ to find the values of $x _ { 1 } , x _ { 2 }$ and $x _ { 3 }$, giving your answers to 5 decimal places.
Show that $\alpha = 0.80157$ is correct to 5 decimal places.
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9527d80b-a2a2-442f-9e32-d8768fbbd01a-162_460_725_260_607} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows part of the graph of $y = \mathrm { f } ( x ) , x \in \mathbb { R }$.
The graph consists of two line segments that meet at the point $R ( 4 , - 3 )$, as shown in Figure 1. Sketch, on separate diagrams, the graphs of
$y = 2 \mathrm { f } ( x + 4 )$,
$y = | f ( - x ) |$. On each diagram, show the coordinates of the point corresponding to $R$.
4. The function $f$ is defined by $$\mathrm { f } : x \mapsto 4 - \ln ( x + 2 ) , \quad x \in \mathbb { R } , x \geqslant - 1$$
Find $\mathrm { f } ^ { - 1 } ( x )$.
Find the domain of $\mathrm { f } ^ { - 1 }$. The function $g$ is defined by $$\mathrm { g } : x \mapsto \mathrm { e } ^ { x ^ { 2 } } - 2 , \quad x \in \mathbb { R }$$
Find $\mathrm { fg } ( x )$, giving your answer in its simplest form.
Find the range of fg.
5. The mass, $m$ grams, of a leaf $t$ days after it has been picked from a tree is given by $$m = p \mathrm { e } ^ { - k t }$$ where $k$ and $p$ are positive constants.
When the leaf is picked from the tree, its mass is 7.5 grams and 4 days later its mass is 2.5 grams.
Write down the value of $p$.
Show that $k = \frac { 1 } { 4 } \ln 3$.
Find the value of $t$ when $\frac { \mathrm { d } m } { \mathrm {~d} t } = - 0.6 \ln 3$.
6. (a) Prove that $$\frac { 1 } { \sin 2 \theta } - \frac { \cos 2 \theta } { \sin 2 \theta } = \tan \theta , \quad \theta \neq 90 n ^ { \circ } , n \in \mathbb { Z }$$
Hence, or otherwise,
1. show that $\tan 15 ^ { \circ } = 2 - \sqrt { 3 }$,
2. solve, for $0 < x < 360 ^ { \circ }$, $$\operatorname { cosec } 4 x - \cot 4 x = 1$$ 7. $$f ( x ) = \frac { 4 x - 5 } { ( 2 x + 1 ) ( x - 3 ) } - \frac { 2 x } { x ^ { 2 } - 9 } , \quad x \neq \pm 3 , x \neq - \frac { 1 } { 2 }$$
Show that $$f ( x ) = \frac { 5 } { ( 2 x + 1 ) ( x + 3 ) }$$ The curve $C$ has equation $y = \mathrm { f } ( x )$. The point $P \left( - 1 , - \frac { 5 } { 2 } \right)$ lies on $C$.
Find an equation of the normal to $C$ at $P$.
1. (a) Express $2 \cos 3 x - 3 \sin 3 x$ in the form $R \cos ( 3 x + \alpha )$, where $R$ and $\alpha$ are constants, $R > 0$ and $0 < \alpha < \frac { \pi } { 2 }$. Give your answers to 3 significant figures.
$$\mathrm { f } ( x ) = \mathrm { e } ^ { 2 x } \cos 3 x$$
Show that $\mathrm { f } ^ { \prime } ( x )$ can be written in the form $$\mathrm { f } ^ { \prime } ( x ) = R \mathrm { e } ^ { 2 x } \cos ( 3 x + \alpha )$$ where $R$ and $\alpha$ are the constants found in part (a).
Hence, or otherwise, find the smallest positive value of $x$ for which the curve with equation $y = \mathrm { f } ( x )$ has a turning point. Turn over
1. Differentiate with respect to $x$, giving your answer in its simplest form,
2. $x ^ { 2 } \ln ( 3 x )$
3. $\frac { \sin 4 x } { x ^ { 3 } }$
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9527d80b-a2a2-442f-9e32-d8768fbbd01a-173_716_1122_212_411} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the graph of equation $y = \mathrm { f } ( x )$.
The points $P ( - 3,0 )$ and $Q ( 2 , - 4 )$ are stationary points on the graph.
Sketch, on separate diagrams, the graphs of
$y = 3 \mathrm { f } ( x + 2 )$
$y = | \mathrm { f } ( x ) |$ On each diagram, show the coordinates of any stationary points.
3. The area, $A \mathrm {~mm} ^ { 2 }$, of a bacterial culture growing in milk, $t$ hours after midday, is given by $$A = 20 \mathrm { e } ^ { 1.5 t } , \quad t \geqslant 0$$
Write down the area of the culture at midday.
Find the time at which the area of the culture is twice its area at midday. Give your answer to the nearest minute.
4. The point $P$ is the point on the curve $x = 2 \tan \left( y + \frac { \pi } { 12 } \right)$ with $y$-coordinate $\frac { \pi } { 4 }$. Find an equation of the normal to the curve at $P$.
5. Solve, for $0 \leqslant \theta \leqslant 180 ^ { \circ }$, $$2 \cot ^ { 2 } 3 \theta = 7 \operatorname { cosec } 3 \theta - 5$$ Give your answers in degrees to 1 decimal place.
6. $$f ( x ) = x ^ { 2 } - 3 x + 2 \cos \left( \frac { 1 } { 2 } x \right) , \quad 0 \leqslant x \leqslant \pi$$
Show that the equation $\mathrm { f } ( x ) = 0$ has a solution in the interval $0.8 < x < 0.9$ The curve with equation $y = \mathrm { f } ( x )$ has a minimum point $P$.
Show that the $x$-coordinate of $P$ is the solution of the equation $$x = \frac { 3 + \sin \left( \frac { 1 } { 2 } x \right) } { 2 }$$
Using the iteration formula $$x _ { n + 1 } = \frac { 3 + \sin \left( \frac { 1 } { 2 } x _ { n } \right) } { 2 } , \quad x _ { 0 } = 2$$ find the values of $x _ { 1 } , x _ { 2 }$ and $x _ { 3 }$, giving your answers to 3 decimal places.
By choosing a suitable interval, show that the $x$-coordinate of $P$ is 1.9078 correct to 4 decimal places.
1. The function f is defined by
$$\mathrm { f } : x \mapsto \frac { 3 ( x + 1 ) } { 2 x ^ { 2 } + 7 x - 4 } - \frac { 1 } { x + 4 } , \quad x \in \mathbb { R } , x > \frac { 1 } { 2 }$$
Show that $\mathrm { f } ( x ) = \frac { 1 } { 2 x - 1 }$
Find $\mathrm { f } ^ { - 1 } ( x )$
Find the domain of $\mathrm { f } ^ { - 1 }$ $$\mathrm { g } ( x ) = \ln ( x + 1 )$$
Find the solution of $\mathrm { fg } ( x ) = \frac { 1 } { 7 }$, giving your answer in terms of e . 8. (a) Starting from the formulae for $\sin ( A + B )$ and $\cos ( A + B )$, prove that
Deduce that $$\tan ( A + B ) = \frac { \tan A + \tan B } { 1 - \tan A \tan B }$$
Hence, or otherwise, solve, for $0 \leqslant \theta \leqslant \pi$, $$\tan \left( \theta + \frac { \pi } { 6 } \right) = \frac { 1 + \sqrt { } 3 \tan \theta } { \sqrt { } 3 - \tan \theta }$$
Hence, or otherwise, solve, for $0 \leqslant \theta \leqslant \pi$,
$$1 + \sqrt { } 3 \tan \theta = ( \sqrt { } 3 - \tan \theta ) \tan ( \pi - \theta )$$ \section*{的} \begin{table}[h]
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\end{table} Paper Reference(s) \section*{6665/01} \begin{table}[h]
\captionsetup{labelformat=empty} \caption{Examiner's use only}
\end{table} \begin{table}[h]
\captionsetup{labelformat=empty} \caption{ $\frac { \text { Items included with question papers } } { \text { Nil } }$ Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation or symbolic differentiation/integration, or have retrievable mathematical formulae stored in them. In the boxes above, write your centre number, candidate number, your surname, initials and signature.
Check that you have the correct question paper.
Answer ALL the questions.
You must write your answer for each question in the space following the question.
When a calculator is used, the answer should be given to an appropriate degree of accuracy. A booklet 'Mathematical Formulae and Statistical Tables' is provided.
Full marks may be obtained for answers to ALL questions.
The marks for individual questions and the parts of questions are shown in round brackets: e.g. (2). There are 8 questions in this question paper. The total mark for this paper is 75 .
There are 32 pages in this question paper. Any blank pages are indicated. You must ensure that your answers to parts of questions are clearly labelled.
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1. Express
$$\frac { 2 ( 3 x + 2 ) } { 9 x ^ { 2 } - 4 } - \frac { 2 } { 3 x + 1 }$$ as a single fraction in its simplest form.
2. $$f ( x ) = x ^ { 3 } + 3 x ^ { 2 } + 4 x - 12$$
Show that the equation $\mathrm { f } ( x ) = 0$ can be written as $$x = \sqrt { } \left( \frac { 4 ( 3 - x ) } { ( 3 + x ) } \right) , \quad x \neq - 3$$ The equation $x ^ { 3 } + 3 x ^ { 2 } + 4 x - 12 = 0$ has a single root which is between 1 and 2
Use the iteration formula $$x _ { n + 1 } = \sqrt { } \left( \frac { 4 \left( 3 - x _ { n } \right) } { \left( 3 + x _ { n } \right) } \right) , n \geqslant 0$$ with $x _ { 0 } = 1$ to find, to 2 decimal places, the value of $x _ { 1 } , x _ { 2 }$ and $x _ { 3 }$. The root of $\mathrm { f } ( x ) = 0$ is $\alpha$.
By choosing a suitable interval, prove that $\alpha = 1.272$ to 3 decimal places.
3. \begin{figure}[h] \begin{center} \includegraphics[alt={},max width=\textwidth]{9527d80b-a2a2-442f-9e32-d8768fbbd01a-186_538_953_251_532} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of the curve $C$ which has equation $$y = \mathrm { e } ^ { x \sqrt { } 3 } \sin 3 x , \quad - \frac { \pi } { 3 } \leqslant x \leqslant \frac { \pi } { 3 }$$

Find the $x$ coordinate of the turning point $P$ on $C$, for which $x > 0$ Give your answer as a multiple of $\pi$.

Find an equation of the normal to $C$ at the point where $x = 0$
4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{9527d80b-a2a2-442f-9e32-d8768fbbd01a-188_560_1145_210_386} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows part of the curve with equation $y = \mathrm { f } ( x )$
The curve passes through the points $P ( - 1.5,0 )$ and $Q ( 0,5 )$ as shown.
On separate diagrams, sketch the curve with equation

$y = | f ( x ) |$

$y = \mathrm { f } ( | x | )$

$y = 2 f ( 3 x )$ Indicate clearly on each sketch the coordinates of the points at which the curve crosses or meets the axes.

(a) Express $4 \operatorname { cosec } ^ { 2 } 2 \theta - \operatorname { cosec } ^ { 2 } \theta$ in terms of $\sin \theta$ and $\cos \theta$.
Hence show that

$$4 \operatorname { cosec } ^ { 2 } 2 \theta - \operatorname { cosec } ^ { 2 } \theta = \sec ^ { 2 } \theta$$

Hence or otherwise solve, for $0 < \theta < \pi$, $$4 \operatorname { cosec } ^ { 2 } 2 \theta - \operatorname { cosec } ^ { 2 } \theta = 4$$ giving your answers in terms of $\pi$.
6. The functions $f$ and $g$ are defined by $$\begin{aligned} & \mathrm { f } : x \mapsto \mathrm { e } ^ { x } + 2 , \quad x \in \mathbb { R }
& \mathrm {~g} : x \mapsto \ln x , \quad x > 0 \end{aligned}$$

State the range of f.

Find $\mathrm { fg } ( x )$, giving your answer in its simplest form.

Find the exact value of $x$ for which $\mathrm { f } ( 2 x + 3 ) = 6$

Find $\mathrm { f } ^ { - 1 }$, the inverse function of f , stating its domain.

On the same axes sketch the curves with equation $y = \mathrm { f } ( x )$ and $y = \mathrm { f } ^ { - 1 } ( x )$, giving the coordinates of all the points where the curves cross the axes.

(a) Differentiate with respect to $x$,
1. $x ^ { \frac { 1 } { 2 } } \ln ( 3 x )$
2. $\frac { 1 - 10 x } { ( 2 x - 1 ) ^ { 5 } }$, giving your answer in its simplest form.
Given that $x = 3 \tan 2 y$ find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $x$.

$$f ( x ) = 7 \cos 2 x - 24 \sin 2 x$$ Given that $\mathrm { f } ( x ) = R \cos ( 2 x + \alpha )$, where $R > 0$ and $0 < \alpha < 90 ^ { \circ }$,

find the value of $R$ and the value of $\alpha$.

Hence solve the equation $$7 \cos 2 x - 24 \sin 2 x = 12.5$$ for $0 \leqslant x < 180 ^ { \circ }$, giving your answers to 1 decimal place.

Express $14 \cos ^ { 2 } x - 48 \sin x \cos x$ in the form $a \cos 2 x + b \sin 2 x + c$, where $a , b$, and $c$ are constants to be found.

Hence, using your answers to parts (a) and (c), deduce the maximum value of $$14 \cos ^ { 2 } x - 48 \sin x \cos x$$ \begin{table}[h]

\captionsetup{labelformat=empty} \caption{physicsandmathstutor.com}

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The curve $C$ has equation

$$y = ( 2 x - 3 ) ^ { 5 }$$ The point $P$ lies on $C$ and has coordinates $( w , - 32 )$.
Find

the value of $w$,

the equation of the tangent to $C$ at the point $P$ in the form $y = m x + c$, where $m$ and $c$ are constants.
2. $$\mathrm { g } ( x ) = \mathrm { e } ^ { x - 1 } + x - 6$$

Show that the equation $\mathrm { g } ( x ) = 0$ can be written as $$x = \ln ( 6 - x ) + 1 , \quad x < 6$$ The root of $\mathrm { g } ( x ) = 0$ is $\alpha$.
The iterative formula $$x _ { n + 1 } = \ln \left( 6 - x _ { n } \right) + 1 , \quad x _ { 0 } = 2$$ is used to find an approximate value for $\alpha$.

Calculate the values of $x _ { 1 } , x _ { 2 }$ and $x _ { 3 }$ to 4 decimal places.

By choosing a suitable interval, show that $\alpha = 2.307$ correct to 3 decimal places.
3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{9527d80b-a2a2-442f-9e32-d8768fbbd01a-199_620_1095_223_420} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows part of the curve with equation $y = \mathrm { f } ( x ) , x \in \mathbb { R }$.
The curve passes through the points $Q ( 0,2 )$ and $P ( - 3,0 )$ as shown.

Find the value of ff(-3). On separate diagrams, sketch the curve with equation

$y = \mathrm { f } ^ { - 1 } ( x )$,

$y = \mathrm { f } ( | x | ) - 2$,

$y = 2 \mathrm { f } \left( \frac { 1 } { 2 } x \right)$. Indicate clearly on each sketch the coordinates of the points at which the curve crosses or meets the axes.

(a) Express $6 \cos \theta + 8 \sin \theta$ in the form $R \cos ( \theta - \alpha )$, where $R > 0$ and $0 < \alpha < \frac { \pi } { 2 }$.

Give the value of $\alpha$ to 3 decimal places.

$$\mathrm { p } ( \theta ) = \frac { 4 } { 12 + 6 \cos \theta + 8 \sin \theta } , \quad 0 \leqslant \theta \leqslant 2 \pi$$ Calculate

the maximum value of $\mathrm { p } ( \theta )$,
the value of $\theta$ at which the maximum occurs.
5. (i) Differentiate with respect to $x$

$y = x ^ { 3 } \ln 2 x$

$y = ( x + \sin 2 x ) ^ { 3 }$ Given that $x = \cot y$,
(ii) show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { - 1 } { 1 + x ^ { 2 } }$
6. (i) Without using a calculator, find the exact value of $$\left( \sin 22.5 ^ { \circ } + \cos 22.5 ^ { \circ } \right) ^ { 2 }$$ You must show each stage of your working.
(ii) (a) Show that $\cos 2 \theta + \sin \theta = 1$ may be written in the form $$k \sin ^ { 2 } \theta - \sin \theta = 0 , \text { stating the value of } k$$

Hence solve, for $0 \leqslant \theta < 360 ^ { \circ }$, the equation $$\cos 2 \theta + \sin \theta = 1$$ 7. $$\mathrm { h } ( x ) = \frac { 2 } { x + 2 } + \frac { 4 } { x ^ { 2 } + 5 } - \frac { 18 } { \left( x ^ { 2 } + 5 \right) ( x + 2 ) } , \quad x \geqslant 0$$

Show that $\mathrm { h } ( x ) = \frac { 2 x } { x ^ { 2 } + 5 }$

Hence, or otherwise, find $\mathrm { h } ^ { \prime } ( x )$ in its simplest form. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{9527d80b-a2a2-442f-9e32-d8768fbbd01a-205_729_1235_644_351} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows a graph of the curve with equation $y = \mathrm { h } ( x )$.

Calculate the range of $\mathrm { h } ( x )$.

The value of Bob's car can be calculated from the formula

$$V = 17000 \mathrm { e } ^ { - 0.25 t } + 2000 \mathrm { e } ^ { - 0.5 t } + 500$$ where $V$ is the value of the car in pounds $( \pounds )$ and $t$ is the age in years.

Find the value of the car when $t = 0$

Calculate the exact value of $t$ when $V = 9500$

Find the rate at which the value of the car is decreasing at the instant when $t = 8$. Give your answer in pounds per year to the nearest pound. Turn over

Express

$$\frac { 3 x + 5 } { x ^ { 2 } + x - 12 } - \frac { 2 } { x - 3 }$$ as a single fraction in its simplest form.
2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{9527d80b-a2a2-442f-9e32-d8768fbbd01a-211_499_1099_210_443} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of the curve with equation $y = \mathrm { f } ( x ) , x > 0$, where f is an increasing function of $x$. The curve crosses the $x$-axis at the point $( 1,0 )$ and the line $x = 0$ is an asymptote to the curve. On separate diagrams, sketch the curve with equation

$y = \mathrm { f } ( 2 x ) , x > 0$

$y = | \mathrm { f } ( x ) | , x > 0$ Indicate clearly on each sketch the coordinates of the point at which the curve crosses or meets the $x$-axis.
3. $$f ( x ) = 7 \cos x + \sin x$$ Given that $\mathrm { f } ( x ) = R \cos ( x - \alpha )$, where $R > 0$ and $0 < \alpha < 90 ^ { \circ }$,

find the exact value of $R$ and the value of $\alpha$ to one decimal place.

Hence solve the equation $$7 \cos x + \sin x = 5$$ for $0 \leqslant x < 360 ^ { \circ }$, giving your answers to one decimal place.

State the values of $k$ for which the equation $$7 \cos x + \sin x = k$$ has only one solution in the interval $0 \leqslant x < 360 ^ { \circ }$

The functions $f$ and $g$ are defined by

$$\begin{array} { l l } \mathrm { f } : x \mapsto 2 | x | + 3 , & x \in \mathbb { R } ,
\mathrm {~g} : x \mapsto 3 - 4 x , & x \in \mathbb { R } \end{array}$$

State the range of f.

Find $\mathrm { fg } ( 1 )$.

Find $\mathrm { g } ^ { - 1 }$, the inverse function of g .

Solve the equation $$\operatorname { gg } ( x ) + [ \mathrm { g } ( x ) ] ^ { 2 } = 0$$ 5. (a) Differentiate $$\frac { \cos 2 x } { \sqrt { x } }$$ with respect to $x$.

Show that $\frac { \mathrm { d } } { \mathrm { d } x } \left( \sec ^ { 2 } 3 x \right)$ can be written in the form $$\mu \left( \tan 3 x + \tan ^ { 3 } 3 x \right)$$ where $\mu$ is a constant.

Given $x = 2 \sin \left( \frac { y } { 3 } \right)$, find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $x$, simplifying your answer.

(i) Use an appropriate double angle formula to show that

$$\operatorname { cosec } 2 x = \lambda \operatorname { cosec } x \sec x$$ and state the value of the constant $\lambda$.
(ii) Solve, for $0 \leqslant \theta < 2 \pi$, the equation $$3 \sec ^ { 2 } \theta + 3 \sec \theta = 2 \tan ^ { 2 } \theta$$ You must show all your working. Give your answers in terms of $\pi$.
7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{9527d80b-a2a2-442f-9e32-d8768fbbd01a-219_481_858_228_552} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows a sketch of part of the curve with equation $y = \mathrm { f } ( x )$ where $$\mathrm { f } ( x ) = \left( x ^ { 2 } + 3 x + 1 \right) \mathrm { e } ^ { x ^ { 2 } }$$ The curve cuts the $x$-axis at points $A$ and $B$ as shown in Figure 2 .

Calculate the $x$ coordinate of $A$ and the $x$ coordinate of $B$, giving your answers to 3 decimal places.

Find $\mathrm { f } ^ { \prime } ( x )$. The curve has a minimum turning point at the point $P$ as shown in Figure 2.

Show that the $x$ coordinate of $P$ is the solution of $$x = - \frac { 3 \left( 2 x ^ { 2 } + 1 \right) } { 2 \left( x ^ { 2 } + 2 \right) }$$

Use the iteration formula $$x _ { n + 1 } = - \frac { 3 \left( 2 x _ { n } ^ { 2 } + 1 \right) } { 2 \left( x _ { n } ^ { 2 } + 2 \right) } , \quad \text { with } x _ { 0 } = - 2.4$$ to calculate the values of $x _ { 1 } , x _ { 2 }$ and $x _ { 3 }$, giving your answers to 3 decimal places. The $x$ coordinate of $P$ is $\alpha$.

By choosing a suitable interval, prove that $\alpha = - 2.43$ to 2 decimal places. 8. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{9527d80b-a2a2-442f-9e32-d8768fbbd01a-221_721_1227_116_322} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} The population of a town is being studied. The population $P$, at time $t$ years from the start of the study, is assumed to be $$P = \frac { 8000 } { 1 + 7 \mathrm { e } ^ { - k t } } , \quad t \geqslant 0$$ where $k$ is a positive constant.
The graph of $P$ against $t$ is shown in Figure 3. Use the given equation to

find the population at the start of the study,

find a value for the expected upper limit of the population. Given also that the population reaches 2500 at 3 years from the start of the study,

calculate the value of $k$ to 3 decimal places. Using this value for $k$,

find the population at 10 years from the start of the study, giving your answer to 3 significant figures.

Find, using $\frac { \mathrm { d } P } { \mathrm {~d} t }$, the rate at which the population is growing at 10 years from the start of the study. \begin{table}[h]

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\end{table} Paper Reference(s) \section*{6665/01} \begin{table}[h]

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Edexcel C3 — Question 9

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