Edexcel C3 (Core Mathematics 3)

Question 6

6. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{9527d80b-a2a2-442f-9e32-d8768fbbd01a-009_458_876_285_539}

\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 $( 1 , a ) , a < 0$. One line meets the $x$-axis at $( 3,0 )$. The other line meets the $x$-axis at $( - 1,0 )$ and the $y$-axis at $( 0 , b ) , b < 0$. In separate diagrams, sketch the graph with equation

$y = \mathrm { f } ( x + 1 )$,
$y = \mathrm { f } ( | x | )$. Indicate clearly on each sketch the coordinates of any points of intersection with the axes. Given that $\mathrm { f } ( x ) = | x - 1 | - 2$, find
the value of $a$ and the value of $b$,
the value of $x$ for which $\mathrm { f } ( x ) = 5 x$.

Question 7

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A particular species of orchid is being studied. The population $p$ at time $t$ years after the study started is assumed to be

$$p = \frac { 2800 a \mathrm { e } ^ { 0.2 t } } { 1 + a \mathrm { e } ^ { 0.2 t } } , \text { where } a \text { is a constant. }$$ Given that there were 300 orchids when the study started,

show that $a = 0.12$,
use the equation with $a = 0.12$ to predict the number of years before the population of orchids reaches 1850.
Show that $p = \frac { 336 } { 0.12 + \mathrm { e } ^ { - 0.2 t } }$.
Hence show that the population cannot exceed 2800.
\end{table} Turn over
1. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{9527d80b-a2a2-442f-9e32-d8768fbbd01a-014_689_766_276_594}
\end{figure} Figure 1 shows the graph of $y = \mathrm { f } ( x ) , - 5 \leqslant x \leqslant 5$.
The point $M ( 2,4 )$ is the maximum turning point of the graph.
Sketch, on separate diagrams, the graphs of
$y = \mathrm { f } ( x ) + 3$,
$y = | \mathrm { f } ( x ) |$,
$y = \mathrm { f } ( | x | )$. Show on each graph the coordinates of any maximum turning points.
1. Express
$$\frac { 2 x ^ { 2 } + 3 x } { ( 2 x + 3 ) ( x - 2 ) } - \frac { 6 } { x ^ { 2 } - x - 2 }$$ as a single fraction in its simplest form.
3. The point $P$ lies on the curve with equation $y = \ln \left( \frac { 1 } { 3 } x \right)$. The $x$-coordinate of $P$ is 3 . Find an equation of the normal to the curve at the point $P$ in the form $y = a x + b$, where $a$ and $b$ are constants.
(5)
4. (a) Differentiate with respect to $x$
(i) $x ^ { 2 } \mathrm { e } ^ { 3 x + 2 }$,
(ii) $\frac { \cos \left( 2 x ^ { 3 } \right) } { 3 x }$.
Given that $x = 4 \sin ( 2 y + 6 )$, find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $x$.
5. $$f ( x ) = 2 x ^ { 3 } - x - 4$$
Show that the equation $\mathrm { f } ( x ) = 0$ can be written as $$x = \sqrt { \left( \frac { 2 } { x } + \frac { 1 } { 2 } \right) }$$ The equation $2 x ^ { 3 } - x - 4 = 0$ has a root between 1.35 and 1.4.
Use the iteration formula $$x _ { n + 1 } = \sqrt { } \left( \frac { 2 } { x _ { n } } + \frac { 1 } { 2 } \right)$$ with $x _ { 0 } = 1.35$, to find, to 2 decimal places, the values of $x _ { 1 } , x _ { 2 }$ and $x _ { 3 }$. The only real root of $\mathrm { f } ( x ) = 0$ is $\alpha$.
By choosing a suitable interval, prove that $\alpha = 1.392$, to 3 decimal places.
6. $$f ( x ) = 12 \cos x - 4 \sin x$$ Given that $\mathrm { f } ( x ) = R \cos ( x + \alpha )$, where $R \geqslant 0$ and $0 \leqslant \alpha \leqslant 90 ^ { \circ }$,
find the value of $R$ and the value of $\alpha$.
Hence solve the equation $$12 \cos x - 4 \sin x = 7$$ for $0 \leqslant x < 360 ^ { \circ }$, giving your answers to one decimal place.
1. Write down the minimum value of $12 \cos x - 4 \sin x$.
2. Find, to 2 decimal places, the smallest positive value of $x$ for which this minimum value occurs.
  \includegraphics[max width=\textwidth, alt={}, center]{9527d80b-a2a2-442f-9e32-d8768fbbd01a-021_60_35_2669_1853}
  7. (a) Show that
3. $\frac { \cos 2 x } { \cos x + \sin x } \equiv \cos x - \sin x , \quad x \neq \left( n - \frac { 1 } { 4 } \right) \pi , n \in \mathbb { Z }$,
4. $\frac { 1 } { 2 } ( \cos 2 x - \sin 2 x ) \equiv \cos ^ { 2 } x - \cos x \sin x - \frac { 1 } { 2 }$.
Hence, or otherwise, show that the equation $$\cos \theta \left( \frac { \cos 2 \theta } { \cos \theta + \sin \theta } \right) = \frac { 1 } { 2 }$$ can be written as $$\sin 2 \theta = \cos 2 \theta$$
Solve, for $0 \leqslant \theta < 2 \pi$, $$\sin 2 \theta = \cos 2 \theta$$ giving your answers in terms of $\pi$.

Question 8

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8. The functions $f$ and $g$ are defined by $$\begin{array} { l l } \mathrm { f } : x \rightarrow 2 x + \ln 2 , & x \in \mathbb { R } ,
\mathrm {~g} : x \rightarrow \mathrm { e } ^ { 2 x } , & x \in \mathbb { R } . \end{array}$$

Prove that the composite function gf is $$\operatorname { gf } : x \rightarrow 4 \mathrm { e } ^ { 4 x } , \quad x \in \mathbb { R }$$
In the space provided on page 19, sketch the curve with equation $y = \operatorname { gf } ( x )$, and show the coordinates of the point where the curve cuts the $y$-axis.
Write down the range of gf.
Find the value of $x$ for which $\frac { \mathrm { d } } { \mathrm { d } x } [ \operatorname { gf } ( x ) ] = 3$, giving your answer to 3 significant figures.
Turn over
1. (a) By writing $\sin 3 \theta$ as $\sin ( 2 \theta + \theta )$, show that
$$\sin 3 \theta = 3 \sin \theta - 4 \sin ^ { 3 } \theta$$
Given that $\sin \theta = \frac { \sqrt { } 3 } { 4 }$, find the exact value of $\sin 3 \theta$.
2. $$f ( x ) = 1 - \frac { 3 } { x + 2 } + \frac { 3 } { ( x + 2 ) ^ { 2 } } , x \neq - 2$$
Show that $\mathrm { f } ( x ) = \frac { x ^ { 2 } + x + 1 } { ( x + 2 ) ^ { 2 } } , x \neq - 2$.
Show that $x ^ { 2 } + x + 1 > 0$ for all values of $x$.
Show that $\mathrm { f } ( x ) > 0$ for all values of $x , x \neq - 2$.
3. The curve $C$ has equation $$x = 2 \sin y .$$
Show that the point $P \left( \sqrt { } 2 , \frac { \pi } { 4 } \right)$ lies on $C$.
Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sqrt { 2 } }$ at $P$.
Find an equation of the normal to $C$ at $P$. Give your answer in the form $y = m x + c$, where $m$ and $c$ are exact constants.
4. (i) The curve $C$ has equation $$y = \frac { x } { 9 + x ^ { 2 } }$$ Use calculus to find the coordinates of the turning points of $C$.
(ii) Given that $$y = \left( 1 + \mathrm { e } ^ { 2 x } \right) ^ { \frac { 3 } { 2 } }$$ find the value of $\frac { \mathrm { d } y } { \mathrm {~d} x }$ at $x = \frac { 1 } { 2 } \ln 3$.
5. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{9527d80b-a2a2-442f-9e32-d8768fbbd01a-045_865_926_301_516}
\end{figure} Figure 1 shows an oscilloscope screen. The curve shown on the screen satisfies the equation $$y = \sqrt { 3 } \cos x + \sin x$$
Express the equation of the curve in the form $y = R \sin ( x + \alpha )$, where $R$ and $\alpha$ are constants, $R > 0$ and $0 < \alpha < \frac { \pi } { 2 }$.
Find the values of $x , 0 \leqslant x < 2 \pi$, for which $y = 1$.
1. The function $f$ is defined by
$$\mathrm { f } : x \mapsto \ln ( 4 - 2 x ) , \quad x < 2 \quad \text { and } \quad x \in \mathbb { R } .$$
Show that the inverse function of f is defined by $$\mathrm { f } ^ { - 1 } : x \mapsto 2 - \frac { 1 } { 2 } \mathrm { e } ^ { x }$$ and write down the domain of $\mathrm { f } ^ { - 1 }$.
Write down the range of $\mathrm { f } ^ { - 1 }$.
In the space provided on page 16, sketch the graph of $y = f ^ { - 1 } ( x )$. State the coordinates of the points of intersection with the $x$ and $y$ axes. The graph of $y = x + 2$ crosses the graph of $y = f ^ { - 1 } ( x )$ at $x = k$. The iterative formula $$x _ { n + 1 } = - \frac { 1 } { 2 } e ^ { x _ { n } } , x _ { 0 } = - 0.3$$ is used to find an approximate value for $k$.
Calculate the values of $x _ { 1 }$ and $x _ { 2 }$, giving your answers to 4 decimal places.
Find the value of $k$ to 3 decimal places. 7. $$f ( x ) = x ^ { 4 } - 4 x - 8$$
Show that there is a root of $\mathrm { f } ( x ) = 0$ in the interval $[ - 2 , - 1 ]$.
Find the coordinates of the turning point on the graph of $y = \mathrm { f } ( x )$.
Given that $\mathrm { f } ( x ) = ( x - 2 ) \left( x ^ { 3 } + a x ^ { 2 } + b x + c \right)$, find the values of the constants, $a , b$ and $c$.
In the space provided on page 21, sketch the graph of $y = \mathrm { f } ( x )$.
Hence sketch the graph of $y = | \mathrm { f } ( x ) |$.
1. (i) Prove that
$$\sec ^ { 2 } x - \operatorname { cosec } ^ { 2 } x \equiv \tan ^ { 2 } x - \cot ^ { 2 } x$$ (ii) Given that $$y = \arccos x , \quad - 1 \leqslant x \leqslant 1 \text { and } 0 \leqslant y \leqslant \pi ,$$
express arcsin $x$ in terms of $y$.
Hence evaluate $\arccos x + \arcsin x$. Give your answer in terms of $\pi$.
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1. Find the exact solutions to the equations
2. $\ln x + \ln 3 = \ln 6$,
3. $\mathrm { e } ^ { x } + 3 \mathrm { e } ^ { - x } = 4$.
$$f ( x ) = \frac { 2 x + 3 } { x + 2 } - \frac { 9 + 2 x } { 2 x ^ { 2 } + 3 x - 2 } , \quad x > \frac { 1 } { 2 }$$
Show that $\mathrm { f } ( x ) = \frac { 4 x - 6 } { 2 x - 1 }$.
Hence, or otherwise, find $\mathrm { f } ^ { \prime } ( x )$ in its simplest form.
3. A curve $C$ has equation $$y = x ^ { 2 } \mathrm { e } ^ { x }$$
Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$, using the product rule for differentiation.
Hence find the coordinates of the turning points of $C$.
Find $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$.
Determine the nature of each turning point of the curve $C$.
4. $$f ( x ) = - x ^ { 3 } + 3 x ^ { 2 } - 1$$
Show that the equation $\mathrm { f } ( x ) = 0$ can be rewritten as $$x = \sqrt { } \left( \frac { 1 } { 3 - x } \right)$$
Starting with $x _ { 1 } = 0.6$, use the iteration $$\left. x _ { n + 1 } = \sqrt { ( } \frac { 1 } { 3 - x _ { n } } \right)$$ to calculate the values of $x _ { 2 } , x _ { 3 }$ and $x _ { 4 }$, giving all your answers to 4 decimal places.
Show that $x = 0.653$ is a root of $\mathrm { f } ( x ) = 0$ correct to 3 decimal places.
5. The functions $f$ and $g$ are defined by $$\begin{array} { l l } \mathrm { f } : x \mapsto \ln ( 2 x - 1 ) , & x \in \mathbb { R } , x > \frac { 1 } { 2 }
\mathrm {~g} : x \mapsto \frac { 2 } { x - 3 } , & x \in \mathbb { R } , x \neq 3 \end{array}$$
Find the exact value of fg(4).
Find the inverse function $\mathrm { f } ^ { - 1 } ( x )$, stating its domain.
Sketch the graph of $y = | \mathrm { g } ( x ) |$. Indicate clearly the equation of the vertical asymptote and the coordinates of the point at which the graph crosses the $y$-axis.
Find the exact values of $x$ for which $\left| \frac { 2 } { x - 3 } \right| = 3$.
1. (a) Express $3 \sin x + 2 \cos x$ in the form $R \sin ( x + \alpha )$ where $R > 0$ and $0 < \alpha < \frac { \pi } { 2 }$.
2. Hence find the greatest value of $( 3 \sin x + 2 \cos x ) ^ { 4 }$.
3. Solve, for $0 < x < 2 \pi$, the equation
$$3 \sin x + 2 \cos x = 1$$ giving your answers to 3 decimal places.
1. (a) Prove that
$$\frac { \sin \theta } { \cos \theta } + \frac { \cos \theta } { \sin \theta } = 2 \operatorname { cosec } 2 \theta , \quad \theta \neq 90 n ^ { \circ }$$
On the axes on page 20, sketch the graph of $y = 2 \operatorname { cosec } 2 \theta$ for $0 ^ { \circ } < \theta < 360 ^ { \circ }$.
Solve, for $0 ^ { \circ } < \theta < 360 ^ { \circ }$, the equation $$\frac { \sin \theta } { \cos \theta } + \frac { \cos \theta } { \sin \theta } = 3 ,$$ giving your answers to 1 decimal place.
\includegraphics[max width=\textwidth, alt={}, center]{9527d80b-a2a2-442f-9e32-d8768fbbd01a-063_899_1253_315_347}
8. The amount of a certain type of drug in the bloodstream $t$ hours after it has been taken is given by the formula $$x = D \mathrm { e } ^ { - \frac { 1 } { 8 } t } ,$$ where $x$ is the amount of the drug in the bloodstream in milligrams and $D$ is the dose given in milligrams. A dose of 10 mg of the drug is given.
Find the amount of the drug in the bloodstream 5 hours after the dose is given. Give your answer in mg to 3 decimal places. A second dose of 10 mg is given after 5 hours.
Show that the amount of the drug in the bloodstream 1 hour after the second dose is 13.549 mg to 3 decimal places. No more doses of the drug are given. At time $T$ hours after the second dose is given, the amount of the drug in the bloodstream is 3 mg .
Find the value of $T$. \end{table} Paper Reference(s) \section*{6665/01} \section*{Edexcel GCE } \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Examiner's use only} \includegraphics[alt={},max width=\textwidth]{9527d80b-a2a2-442f-9e32-d8768fbbd01a-079_97_309_495_1636}
\end{figure} $$y = 4 \mathrm { e } ^ { 2 x + 1 }$$ The $y$-coordinate of $P$ is 8 .
Find, in terms of $\ln 2$, the $x$-coordinate of $P$.
Find the equation of the tangent to the curve at the point $P$ in the form $y = a x + b$, where $a$ and $b$ are exact constants to be found.
2. $$f ( x ) = 5 \cos x + 12 \sin x$$ Given that $\mathrm { f } ( x ) = R \cos ( x - \alpha )$, where $R > 0$ and $0 < \alpha < \frac { \pi } { 2 }$,
find the value of $R$ and the value of $\alpha$ to 3 decimal places.
Hence solve the equation $$5 \cos x + 12 \sin x = 6$$ for $0 \leqslant x < 2 \pi$.
1. Write down the maximum value of $5 \cos x + 12 \sin x$.
2. Find the smallest positive value of $x$ for which this maximum value occurs.
  3. \begin{figure}[h]
  \includegraphics[alt={},max width=\textwidth]{9527d80b-a2a2-442f-9e32-d8768fbbd01a-083_623_977_207_479} \captionsetup{labelformat=empty} \caption{Figure 1}
  \end{figure} Figure 1 shows the graph of $y = f ( x ) , x \in \mathbb { R }$.
  The graph consists of two line segments that meet at the point $P$.
  The graph cuts the $y$-axis at the point $Q$ and the $x$-axis at the points $( - 3,0 )$ and $R$. Sketch, on separate diagrams, the graphs of
$y = | f ( x ) |$,
$y = \mathrm { f } ( - x )$. Given that $\mathrm { f } ( x ) = 2 - | x + 1 |$,
find the coordinates of the points $P , Q$ and $R$,
solve $\mathrm { f } ( x ) = \frac { 1 } { 2 } x$.
4. The function $f$ is defined by $$f : x \mapsto \frac { 2 ( x - 1 ) } { x ^ { 2 } - 2 x - 3 } - \frac { 1 } { x - 3 } , \quad x > 3$$
Show that $\mathrm { f } ( x ) = \frac { 1 } { x + 1 } , \quad x > 3$.
Find the range of f.
Find $\mathrm { f } ^ { - 1 } ( x )$. State the domain of this inverse function. The function $g$ is defined by $$\mathrm { g } : x \mapsto 2 x ^ { 2 } - 3 , \quad x \in \mathbb { R } .$$
Solve $\mathrm { fg } ( x ) = \frac { 1 } { 8 }$.
5. (a) Given that $\sin ^ { 2 } \theta + \cos ^ { 2 } \theta \equiv 1$, show that $1 + \cot ^ { 2 } \theta \equiv \operatorname { cosec } ^ { 2 } \theta$.
Solve, for $0 \leqslant \theta < 180 ^ { \circ }$, the equation $$2 \cot ^ { 2 } \theta - 9 \operatorname { cosec } \theta = 3$$ giving your answers to 1 decimal place.
6. (a) Differentiate with respect to $x$,
1. $\mathrm { e } ^ { 3 x } ( \sin x + 2 \cos x )$,
2. $x ^ { 3 } \ln ( 5 x + 2 )$. Given that $y = \frac { 3 x ^ { 2 } + 6 x - 7 } { ( x + 1 ) ^ { 2 } } , \quad x \neq - 1$,
show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 20 } { ( x + 1 ) ^ { 3 } }$.
Hence find $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ and the real values of $x$ for which $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = - \frac { 15 } { 4 }$.
7. $$f ( x ) = 3 x ^ { 3 } - 2 x - 6$$
Show that $\mathrm { f } ( x ) = 0$ has a root, $\alpha$, between $x = 1.4$ and $x = 1.45$
Show that the equation $\mathrm { f } ( x ) = 0$ can be written as $$x = \sqrt { } \left( \frac { 2 } { x } + \frac { 2 } { 3 } \right) , \quad x \neq 0$$
Starting with $x _ { 0 } = 1.43$, use the iteration $$x _ { \mathrm { n } + 1 } = \sqrt { } \left( \frac { 2 } { x _ { \mathrm { n } } } + \frac { 2 } { 3 } \right)$$ to calculate the values of $x _ { 1 } , x _ { 2 }$ and $x _ { 3 }$, giving your answers to 4 decimal places.
By choosing a suitable interval, show that $\alpha = 1.435$ is correct to 3 decimal places. \end{table} Turn over
advancing learning, changing lives
1. (a) Find the value of $\frac { \mathrm { d } y } { \mathrm {~d} x }$ at the point where $x = 2$ on the curve with equation
$$y = x ^ { 2 } \sqrt { } ( 5 x - 1 )$$
Differentiate $\frac { \sin 2 x } { x ^ { 2 } }$ with respect to $x$.
2. $$f ( x ) = \frac { 2 x + 2 } { x ^ { 2 } - 2 x - 3 } - \frac { x + 1 } { x - 3 }$$
Express $\mathrm { f } ( x )$ as a single fraction in its simplest form.
Hence show that $\mathrm { f } ^ { \prime } ( x ) = \frac { 2 } { ( x - 3 ) ^ { 2 } }$
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9527d80b-a2a2-442f-9e32-d8768fbbd01a-095_767_913_246_511} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the graph of $y = \mathrm { f } ( x ) , \quad 1 < x < 9$.
The points $T ( 3,5 )$ and $S ( 7,2 )$ are turning points on the graph.
Sketch, on separate diagrams, the graphs of
$y = 2 \mathrm { f } ( x ) - 4$,
$y = | \mathrm { f } ( x ) |$. Indicate on each diagram the coordinates of any turning points on your sketch.
4. Find the equation of the tangent to the curve $x = \cos ( 2 y + \pi )$ at $\left( 0 , \frac { \pi } { 4 } \right)$. Give your answer in the form $y = a x + b$, where $a$ and $b$ are constants to be found.
5. The functions $f$ and $g$ are defined by $$\begin{aligned} & \mathrm { f } : x \mapsto 3 x + \ln x , \quad x > 0 , \quad x \in \mathbb { R }
& \mathrm {~g} : x \mapsto \mathrm { e } ^ { x ^ { 2 } } , \quad x \in \mathbb { R } \end{aligned}$$
Write down the range of g.
Show that the composite function fg is defined by $$\mathrm { fg } : x \mapsto x ^ { 2 } + 3 \mathrm { e } ^ { x ^ { 2 } } , \quad x \in \mathbb { R } .$$
Write down the range of fg.
Solve the equation $\frac { \mathrm { d } } { \mathrm { d } x } [ \mathrm { fg } ( x ) ] = x \left( x \mathrm { e } ^ { x ^ { 2 } } + 2 \right)$.
6. (a) (i) By writing $3 \theta = ( 2 \theta + \theta )$, show that $$\sin 3 \theta = 3 \sin \theta - 4 \sin ^ { 3 } \theta$$ (ii) Hence, or otherwise, for $0 < \theta < \frac { \pi } { 3 }$, solve $$8 \sin ^ { 3 } \theta - 6 \sin \theta + 1 = 0 .$$ Give your answers in terms of $\pi$.
Using $\sin ( \theta - \alpha ) = \sin \theta \cos \alpha - \cos \theta \sin \alpha$, or otherwise, show that $$\sin 15 ^ { \circ } = \frac { 1 } { 4 } ( \sqrt { } 6 - \sqrt { } 2 )$$ 7. $$f ( x ) = 3 x e ^ { x } - 1$$ The curve with equation $y = \mathrm { f } ( x )$ has a turning point $P$.
Find the exact coordinates of $P$. The equation $\mathrm { f } ( x ) = 0$ has a root between $x = 0.25$ and $x = 0.3$
Use the iterative formula $$x _ { n + 1 } = \frac { 1 } { 3 } \mathrm { e } ^ { - x _ { n } }$$ with $x _ { 0 } = 0.25$ to find, to 4 decimal places, the values of $x _ { 1 } , x _ { 2 }$ and $x _ { 3 }$.
By choosing a suitable interval, show that a root of $\mathrm { f } ( x ) = 0$ is $x = 0.2576$ correct to 4 decimal places.
8. (a) Express $3 \cos \theta + 4 \sin \theta$ in the form $R \cos ( \theta - \alpha )$, where $R$ and $\alpha$ are constants, $R > 0$ and $0 < \alpha < 90 ^ { \circ }$.
Hence find the maximum value of $3 \cos \theta + 4 \sin \theta$ and the smallest positive value of $\theta$ for which this maximum occurs. The temperature, $\mathrm { f } ( t )$, of a warehouse is modelled using the equation $$f ( t ) = 10 + 3 \cos ( 15 t ) ^ { \circ } + 4 \sin ( 15 t ) ^ { \circ }$$ where $t$ is the time in hours from midday and $0 \leqslant t < 24$.
Calculate the minimum temperature of the warehouse as given by this model.
Find the value of $t$ when this minimum temperature occurs.
\begin{table}[h]
\captionsetup{labelformat=empty} \caption{physicsandmathstutor.com}
\end{table} Paper Reference(s) \section*{6665/01} \section*{Edexcel GCE } \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Examiner's use only} \includegraphics[alt={},max width=\textwidth]{9527d80b-a2a2-442f-9e32-d8768fbbd01a-105_99_311_495_1635}
\end{figure} $$x _ { n + 1 } = \frac { 2 } { \left( x _ { n } \right) ^ { 2 } } + 2$$ is used.
Taking $x _ { 0 } = 2.5$, find the values of $x _ { 1 } , x _ { 2 } , x _ { 3 }$ and $x _ { 4 }$. Give your answers to 3 decimal places where appropriate.
Show that $\alpha = 2.359$ correct to 3 decimal places.
2. (a) Use the identity $\cos ^ { 2 } \theta + \sin ^ { 2 } \theta = 1$ to prove that $\tan ^ { 2 } \theta = \sec ^ { 2 } \theta - 1$.
Solve, for $0 \leqslant \theta < 360 ^ { \circ }$, the equation $$2 \tan ^ { 2 } \theta + 4 \sec \theta + \sec ^ { 2 } \theta = 2$$
1. Rabbits were introduced onto an island. The number of rabbits, $P , t$ years after they were introduced is modelled by the equation
$$P = 80 \mathrm { e } ^ { \frac { 1 } { 5 } t } , \quad t \in \mathbb { R } , t \geqslant 0$$
Write down the number of rabbits that were introduced to the island.
Find the number of years it would take for the number of rabbits to first exceed 1000.
Find $\frac { \mathrm { d } P } { \mathrm {~d} t }$.
Find $P$ when $\frac { \mathrm { d } P } { \mathrm {~d} t } = 50$.
4. (i) Differentiate with respect to $x$
$x ^ { 2 } \cos 3 x$
$\frac { \ln \left( x ^ { 2 } + 1 \right) } { x ^ { 2 } + 1 }$
(ii) A curve $C$ has the equation $$y = \sqrt { } ( 4 x + 1 ) , \quad x > - \frac { 1 } { 4 } , \quad y > 0$$ The point $P$ on the curve has $x$-coordinate 2 . Find an equation of the tangent to $C$ at $P$ in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers. 5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9527d80b-a2a2-442f-9e32-d8768fbbd01a-111_721_1217_237_397} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of part of the curve with equation $y = \mathrm { f } ( x ) , x \in \mathbb { R }$.
The curve meets the coordinate axes at the points $A ( 0,1 - k )$ and $B \left( \frac { 1 } { 2 } \ln k , 0 \right)$, where $k$ is a constant and $k > 1$, as shown in Figure 2. On separate diagrams, sketch the curve with equation
$y = | f ( x ) |$,
$y = \mathrm { f } ^ { - 1 } ( x )$. Show on each sketch the coordinates, in terms of $k$, of each point at which the curve meets or cuts the axes. Given that $\mathrm { f } ( x ) = \mathrm { e } ^ { 2 x } - k$,
state the range of f ,
find $\mathrm { f } ^ { - 1 } ( x )$,
write down the domain of $\mathrm { f } ^ { - 1 }$.
1. (a) Use the identity $\cos ( A + B ) = \cos A \cos B - \sin A \sin B$, to show that
$$\cos 2 A = 1 - 2 \sin ^ { 2 } A$$ The curves $C _ { 1 }$ and $C _ { 2 }$ have equations $$\begin{aligned} & C _ { 1 } : \quad y = 3 \sin 2 x
& C _ { 2 } : \quad y = 4 \sin ^ { 2 } x - 2 \cos 2 x \end{aligned}$$
Show that the $x$-coordinates of the points where $C _ { 1 }$ and $C _ { 2 }$ intersect satisfy the equation $$4 \cos 2 x + 3 \sin 2 x = 2$$
Express $4 \cos 2 x + 3 \sin 2 x$ in the form $R \cos ( 2 x - \alpha )$, where $R > 0$ and $0 < \alpha < 90 ^ { \circ }$, giving the value of $\alpha$ to 2 decimal places.
Hence find, for $0 \leqslant x < 180 ^ { \circ }$, all the solutions of $$4 \cos 2 x + 3 \sin 2 x = 2$$ giving your answers to 1 decimal place.
7. The function f is defined by $$\mathrm { f } ( x ) = 1 - \frac { 2 } { ( x + 4 ) } + \frac { x - 8 } { ( x - 2 ) ( x + 4 ) } , \quad x \in \mathbb { R } , x \neq - 4 , x \neq 2$$
Show that $\mathrm { f } ( x ) = \frac { x - 3 } { x - 2 }$ The function g is defined by $$\mathrm { g } ( x ) = \frac { \mathrm { e } ^ { x } - 3 } { \mathrm { e } ^ { x } - 2 } , \quad x \in \mathbb { R } , x \neq \ln 2$$
Differentiate $\mathrm { g } ( x )$ to show that $\mathrm { g } ^ { \prime } ( x ) = \frac { \mathrm { e } ^ { x } } { \left( \mathrm { e } ^ { x } - 2 \right) ^ { 2 } }$
Find the exact values of $x$ for which $\mathrm { g } ^ { \prime } ( x ) = 1$
8. (a) Write down $\sin 2 x$ in terms of $\sin x$ and $\cos x$.
Find, for $0 < x < \pi$, all the solutions of the equation $$\operatorname { cosec } x - 8 \cos x = 0$$ giving your answers to 2 decimal places.
\end{table} Turn over
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1. Express
$$\frac { x + 1 } { 3 x ^ { 2 } - 3 } - \frac { 1 } { 3 x + 1 }$$ as a single fraction in its simplest form.
2. $$f ( x ) = x ^ { 3 } + 2 x ^ { 2 } - 3 x - 11$$
Show that $\mathrm { f } ( x ) = 0$ can be rearranged as $$x = \sqrt { } \left( \frac { 3 x + 11 } { x + 2 } \right) , \quad x \neq - 2 .$$ The equation $\mathrm { f } ( x ) = 0$ has one positive root $\alpha$. The iterative formula $x _ { n + 1 } = \sqrt { } \left( \frac { 3 x _ { n } + 11 } { x _ { n } + 2 } \right)$ is used to find an approximation to $\alpha$.
Taking $x _ { 1 } = 0$, find, to 3 decimal places, the values of $x _ { 2 } , x _ { 3 }$ and $x _ { 4 }$.
Show that $\alpha = 2.057$ correct to 3 decimal places.
3. (a) Express $5 \cos x - 3 \sin x$ in the form $R \cos ( x + \alpha )$, where $R > 0$ and $0 < \alpha < \frac { 1 } { 2 } \pi$.
Hence, or otherwise, solve the equation $$5 \cos x - 3 \sin x = 4$$ for $0 \leqslant x < 2 \pi$, giving your answers to 2 decimal places.
4. (i) Given that $y = \frac { \ln \left( x ^ { 2 } + 1 \right) } { x }$, find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.
(ii) Given that $x = \tan y$, show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 1 + x ^ { 2 } }$.
5. Sketch the graph of $y = \ln | x |$, stating the coordinates of any points of intersection with the axes.
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9527d80b-a2a2-442f-9e32-d8768fbbd01a-124_380_574_269_722} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the graph of $y = \mathrm { f } ( x )$.
The graph intersects the $y$-axis at the point $( 0,1 )$ and the point $A ( 2,3 )$ is the maximum turning point. Sketch, on separate axes, the graphs of
(i) $y = \mathrm { f } ( - x ) + 1$,
(ii) $y = \mathrm { f } ( x + 2 ) + 3$,
(iii) $y = 2 \mathrm { f } ( 2 x )$. On each sketch, show the coordinates of the point at which your graph intersects the $y$-axis and the coordinates of the point to which $A$ is transformed.
1. (a) By writing $\sec x$ as $\frac { 1 } { \cos x }$, show that $\frac { \mathrm { d } ( \sec x ) } { \mathrm { d } x } = \sec x \tan x$.
Given that $y = \mathrm { e } ^ { 2 x } \sec 3 x$,
find $\frac { \mathrm { d } y } { \mathrm {~d} x }$. The curve with equation $y = \mathrm { e } ^ { 2 x } \sec 3 x , - \frac { \pi } { 6 } < x < \frac { \pi } { 6 }$, has a minimum turning point at $( a , b )$.
Find the values of the constants $a$ and $b$, giving your answers to 3 significant figures. 8. Solve $$\operatorname { cosec } ^ { 2 } 2 x - \cot 2 x = 1$$ for $0 \leqslant x \leqslant 180 ^ { \circ }$.

Question 9

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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]
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\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.
You should show sufficient working to make your methods clear to the Examiner.
Answers without working may not gain full credit. Turn over
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]

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\end{table} Paper Reference(s) \section*{6665/01} Examiner's use only Turn over

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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\end{table} \begin{table}[h] \begin{center} \captionsetup{labelformat=empty} \caption{