Questions - Edexcel C3 - A-Level Maths

$y = \mathrm { f } ( x ) + 3$,
$y = | \mathrm { f } ( x ) |$,
$y = \mathrm { f } ( | x | )$. Show on each graph the coordinates of any maximum turning points.

Edexcel C3 2006 January Q2

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.

Edexcel C3 2006 January Q3

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)

Edexcel C3 2006 January Q4

4. (a) Differentiate with respect to $x$

$x ^ { 2 } \mathrm { e } ^ { 3 x + 2 }$,
$\frac { \cos \left( 2 x ^ { 3 } \right) } { 3 x }$.
(b) Given that $x = 4 \sin ( 2 y + 6 )$, find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $x$.

Edexcel C3 2006 January Q5

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.

Edexcel C3 2006 January Q6

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.
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Edexcel C3 2006 January Q7

7. (a) Show that

$\frac { \cos 2 x } { \cos x + \sin x } \equiv \cos x - \sin x , \quad x \neq \left( n - \frac { 1 } { 4 } \right) \pi , n \in \mathbb { Z }$,
$\frac { 1 } { 2 } ( \cos 2 x - \sin 2 x ) \equiv \cos ^ { 2 } x - \cos x \sin x - \frac { 1 } { 2 }$.
(b) 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 $$\sin 2 \theta = \cos 2 \theta$$ (c) Solve, for $0 \leqslant \theta < 2 \pi$, $$\sin 2 \theta = \cos 2 \theta$$ giving your answers in terms of $\pi$.

Edexcel C3 2006 January Q8

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.

Edexcel C3 2007 January Q1

(a) By writing $\sin 3 \theta$ as $\sin ( 2 \theta + \theta )$, show that

$$\sin 3 \theta = 3 \sin \theta - 4 \sin ^ { 3 } \theta$$ (b) Given that $\sin \theta = \frac { \sqrt { } 3 } { 4 }$, find the exact value of $\sin 3 \theta$.

Edexcel C3 2007 January Q2

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 $x , x \neq - 2$.

Edexcel C3 2007 January Q3

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.

Edexcel C3 2007 January Q4

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 $x = \frac { 1 } { 2 } \ln 3$.

Edexcel C3 2007 January Q5

5. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{a4ad749b-181b-4680-8771-94d9b581125a-07_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 $y = 1$.

Edexcel C3 2007 January Q6

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.

Edexcel C3 2007 January Q7

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

Edexcel C3 2007 January Q8

1. 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 $\pi$.

Edexcel C3 2008 January Q1

Given that

$$\frac { 2 x ^ { 4 } - 3 x ^ { 2 } + x + 1 } { \left( x ^ { 2 } - 1 \right) } \equiv \left( a x ^ { 2 } + b x + c \right) + \frac { d x + e } { \left( x ^ { 2 } - 1 \right) }$$ find the values of the constants $a , b , c , d$ and $e$.
(4)

Edexcel C3 2008 January Q2

2. A curve $C$ has equation $$y = \mathrm { e } ^ { 2 x } \tan x , \quad x \neq ( 2 n + 1 ) \frac { \pi } { 2 }$$

Show that the turning points on $C$ occur where $\tan x = - 1$.
Find an equation of the tangent to $C$ at the point where $x = 0$.

Edexcel C3 2008 January Q3

3. $$\mathrm { f } ( x ) = \ln ( x + 2 ) - x + 1 , \quad x > - 2 , x \in \mathbb { R } .$$

Show that there is a root of $\mathrm { f } ( x ) = 0$ in the interval $2 < x < 3$.
Use the iterative formula $$x _ { n + 1 } = \ln \left( x _ { n } + 2 \right) + 1 , x _ { 0 } = 2.5$$ to calculate the values of $x _ { 1 } , x _ { 2 }$ and $x _ { 3 }$ giving your answers to 5 decimal places.
Show that $x = 2.505$ is a root of $\mathrm { f } ( x ) = 0$ correct to 3 decimal places.

Edexcel C3 2008 January Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{a15db39c-d54b-4cf4-8da7-01f3db223415-05_735_1171_223_390} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of the curve with equation $y = \mathrm { f } ( x )$.
The curve passes through the origin $O$ and the points $A ( 5,4 )$ and $B ( - 5 , - 4 )$.
In separate diagrams, sketch the graph with equation

$y = | f ( x ) |$,
$y = \mathrm { f } ( | x | )$,
$y = 2 f ( x + 1 )$. On each sketch, show the coordinates of the points corresponding to $A$ and $B$.

Edexcel C3 2008 January Q5

5. The radioactive decay of a substance is given by $$R = 1000 \mathrm { e } ^ { - c t } , \quad t \geqslant 0 .$$ where $R$ is the number of atoms at time $t$ years and $c$ is a positive constant.

Find the number of atoms when the substance started to decay. It takes 5730 years for half of the substance to decay.
Find the value of $c$ to 3 significant figures.
Calculate the number of atoms that will be left when $t = 22920$.
In the space provided on page 13, sketch the graph of $R$ against $t$.

Edexcel C3 2008 January Q6

6. (a) Use the double angle formulae and the identity $$\cos ( A + B ) \equiv \cos A \cos B - \sin A \sin B$$ to obtain an expression for $\cos 3 x$ in terms of powers of $\cos x$ only.
(b) (i) Prove that $$\frac { \cos x } { 1 + \sin x } + \frac { 1 + \sin x } { \cos x } \equiv 2 \sec x , \quad x \neq ( 2 n + 1 ) \frac { \pi } { 2 }$$ (ii) Hence find, for $0 < x < 2 \pi$, all the solutions of $$\frac { \cos x } { 1 + \sin x } + \frac { 1 + \sin x } { \cos x } = 4$$

Edexcel C3 2008 January Q7

A curve $C$ has equation

$$y = 3 \sin 2 x + 4 \cos 2 x , - \pi \leqslant x \leqslant \pi$$ The point $A ( 0,4 )$ lies on $C$.

Find an equation of the normal to the curve $C$ at $A$.
Express $y$ in the form $R \sin ( 2 x + \alpha )$, where $R > 0$ and $0 < \alpha < \frac { \pi } { 2 }$. Give the value of $\alpha$ to 3 significant figures.
Find the coordinates of the points of intersection of the curve $C$ with the $x$-axis. Give your answers to 2 decimal places.

Edexcel C3 2008 January Q8

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

$$\begin{aligned} & \mathrm { f } : x \mapsto 1 - 2 x ^ { 3 } , x \in \mathbb { R }
& \mathrm {~g} : x \mapsto \frac { 3 } { x } - 4 , x > 0 , x \in \mathbb { R } \end{aligned}$$

Find the inverse function $\mathrm { f } ^ { - 1 }$.
Show that the composite function gf is $$\text { gf } : x \mapsto \frac { 8 x ^ { 3 } - 1 } { 1 - 2 x ^ { 3 } }$$
Solve $\operatorname { gf } ( x ) = 0$.
Use calculus to find the coordinates of the stationary point on the graph of $y = \operatorname { gf } ( x )$.

Edexcel C3 2009 January Q1

(a) Find the value of $\frac { \mathrm { d } y } { \mathrm {~d} x }$ at the point where $x = 2$ on the curve with equation

$$y = x ^ { 2 } \sqrt { } ( 5 x - 1 )$$ (b) Differentiate $\frac { \sin 2 x } { x ^ { 2 } }$ with respect to $x$.

Questions — Edexcel C3 (377 questions)

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