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Edexcel C3 2011 June Q1

5 marks Moderate -0.8

Differentiate with respect to $x$

$\quad \ln \left( x ^ { 2 } + 3 x + 5 \right)$
$\frac { \cos x } { x ^ { 2 } }$

Edexcel C3 2011 June Q3

6 marks Moderate -0.3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{a0c2a69f-1196-4a07-a368-5dab3efaf316-04_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

$y = 2 \mathrm { f } ( x + 4 )$,
$y = | \mathrm { f } ( - x ) |$. On each diagram, show the coordinates of the point corresponding to $R$.

Edexcel C3 2011 June Q4

8 marks Moderate -0.3

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.

Edexcel C3 2011 June Q5

11 marks Moderate -0.3

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 $\frac { \mathrm { d } m } { \mathrm {~d} t } = - 0.6 \ln 3$.

Edexcel C3 2011 June Q6

12 marks Standard +0.3

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 $0 < x < 360 ^ { \circ }$, $$\operatorname { cosec } 4 x - \cot 4 x = 1$$

Edexcel C3 2011 June Q7

13 marks Standard +0.3

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 $P$.

Edexcel C3 2011 June Q8

12 marks Standard +0.3

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.

Edexcel C3 2012 June Q1

4 marks Moderate -0.8

Express

$$\frac { 2 ( 3 x + 2 ) } { 9 x ^ { 2 } - 4 } - \frac { 2 } { 3 x + 1 }$$ as a single fraction in its simplest form.

Edexcel C3 2012 June Q2

9 marks Moderate -0.3

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.

Edexcel C3 2012 June Q3

9 marks Standard +0.3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{3fbdfb55-5dd5-44ab-b031-d39e64bdfc3b-04_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 $x = 0$

Edexcel C3 2012 June Q4

7 marks Moderate -0.3

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{3fbdfb55-5dd5-44ab-b031-d39e64bdfc3b-06_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.

Edexcel C3 2012 June Q5

9 marks Standard +0.3

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 $\pi$.

Edexcel C3 2012 June Q6

14 marks Moderate -0.3

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.

Edexcel C3 2012 June Q7

11 marks Moderate -0.3

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 $x$.

Edexcel C3 2013 June Q1

4 marks Moderate -0.5

Express

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

Edexcel C3 2013 June Q2

5 marks Moderate -0.3

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{a80a71cb-42e0-4587-8f8e-bacd69b8d07a-03_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.

Edexcel C3 2013 June Q3

10 marks Standard +0.3

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 $0 \leqslant x < 360 ^ { \circ }$

Edexcel C3 2013 June Q4

11 marks Moderate -0.8

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 $$\operatorname { gg } ( x ) + [ \mathrm { g } ( x ) ] ^ { 2 } = 0$$

Edexcel C3 2013 June Q5

10 marks Moderate -0.3

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.

Edexcel C3 2013 June Q6

9 marks Standard +0.3

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 $\lambda$.
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 $\pi$.

Edexcel C3 2013 June Q7

13 marks Standard +0.3

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{a80a71cb-42e0-4587-8f8e-bacd69b8d07a-11_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.

Edexcel C3 2013 June Q8

13 marks Standard +0.3

8. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{a80a71cb-42e0-4587-8f8e-bacd69b8d07a-13_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.

Edexcel C3 2013 June Q1

8 marks Moderate -0.3

1. $$g ( x ) = \frac { 6 x + 12 } { x ^ { 2 } + 3 x + 2 } - 2 , \quad x \geqslant 0$$

Show that $\mathrm { g } ( x ) = \frac { 4 - 2 x } { x + 1 } , x \geqslant 0$
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0f6fd881-4d4b-4f80-96cc-6da41cc33c60-02_494_922_628_511} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve with equation $y = \mathrm { g } ( x ) , x \geqslant 0$ The curve meets the $y$-axis at $( 0,4 )$ and crosses the $x$-axis at $( 2,0 )$. On separate diagrams sketch the graph with equation
1. $y = 2 \mathrm {~g} ( 2 x )$,
2. $y = \mathrm { g } ^ { - 1 } ( x )$. Show on each sketch the coordinates of each point at which the graph meets or crosses the axes.