Questions - Edexcel C3 - A-Level Maths

Find the equation of $l$, writing your answer in the form $y = m x + c$, where $m$ and $c$ are constants. The line $l$ meets $C$ again at the point $B$, as shown in Figure 1 .
Show that the $x$ coordinate of $B$ is a solution of $$x = \sqrt { 1 + \frac { 1 } { 2 } x - \mathrm { e } ^ { - 2 x } }$$ Using the iterative formula $$x _ { n + 1 } = \sqrt { 1 + \frac { 1 } { 2 } x _ { n } - \mathrm { e } ^ { - 2 x _ { n } } }$$ with $x _ { 1 } = 1$
find $x _ { 2 }$ and $x _ { 3 }$ to 3 decimal places.

Edexcel C3 2018 June Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{42aff260-e734-48ff-a92a-674032cb0377-16_561_848_214_699} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows part of the graph with equation $y = \mathrm { f } ( x )$, where $$\mathrm { f } ( x ) = 2 | 5 - x | + 3 , \quad x \geqslant 0$$ Given that the equation $\mathrm { f } ( x ) = k$, where $k$ is a constant, has exactly one root,

state the set of possible values of $k$.
Solve the equation $\mathrm { f } ( x ) = \frac { 1 } { 2 } x + 10$ The graph with equation $y = \mathrm { f } ( x )$ is transformed onto the graph with equation $y = 4 \mathrm { f } ( x - 1 )$. The vertex on the graph with equation $y = 4 \mathrm { f } ( x - 1 )$ has coordinates $( p , q )$.
State the value of $p$ and the value of $q$.

Edexcel C3 2018 June Q6

(i) Using the identity for $\tan ( A \pm B )$, solve, for $- 90 ^ { \circ } < x < 90 ^ { \circ }$,

$$\frac { \tan 2 x + \tan 32 ^ { \circ } } { 1 - \tan 2 x \tan 32 ^ { \circ } } = 5$$ Give your answers, in degrees, to 2 decimal places.
(ii) (a) Using the identity for $\tan ( A \pm B )$, show that $$\tan \left( 3 \theta - 45 ^ { \circ } \right) \equiv \frac { \tan 3 \theta - 1 } { 1 + \tan 3 \theta } , \quad \theta \neq ( 60 n + 45 ) ^ { \circ } , n \in \mathbb { Z }$$ (b) Hence solve, for $0 < \theta < 180 ^ { \circ }$, $$( 1 + \tan 3 \theta ) \tan \left( \theta + 28 ^ { \circ } \right) = \tan 3 \theta - 1$$

Edexcel C3 2018 June Q7

The curve $C$ has equation $y = \frac { \ln \left( x ^ { 2 } + 1 \right) } { x ^ { 2 } + 1 } , \quad x \in \mathbb { R }$
1. Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ as a single fraction, simplifying your answer.
2. Hence find the exact coordinates of the stationary points of $C$.

Edexcel C3 2018 June Q8

(a) By writing $\sec \theta = \frac { 1 } { \cos \theta }$, show that $\frac { \mathrm { d } } { \mathrm { d } \theta } ( \sec \theta ) = \sec \theta \tan \theta$
(b) Given that

$$x = \mathrm { e } ^ { \sec y } \quad x > \mathrm { e } , \quad 0 < y < \frac { \pi } { 2 }$$ show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { x \sqrt { \mathrm {~g} ( x ) } } , \quad x > \mathrm { e }$$ where $\mathrm { g } ( x )$ is a function of $\ln x$.

Edexcel C3 2018 June Q9

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

Give the exact value of $R$ and the value of $\alpha$, in radians, to 3 decimal places. $$\mathrm { M } ( \theta ) = 40 + ( 3 \sin \theta - 6 \cos \theta ) ^ { 2 }$$ (b) Find

the maximum value of $\mathrm { M } ( \theta )$,
the smallest value of $\theta$, in the range $0 < \theta \leqslant 2 \pi$, at which the maximum value of $\mathrm { M } ( \theta )$ occurs. $$N ( \theta ) = \frac { 30 } { 5 + 2 ( \sin 2 \theta - 2 \cos 2 \theta ) ^ { 2 } }$$ (c) Find
the maximum value of $\mathrm { N } ( \theta )$,
the largest value of $\theta$, in the range $0 < \theta \leqslant 2 \pi$, at which the maximum value of $\mathrm { N } ( \theta )$ occurs.
(Solutions based entirely on graphical or numerical methods are not acceptable.)
END

Edexcel C3 Q1

Express

$$\frac { 3 x ^ { 2 } } { \left( 2 x ^ { 2 } + 7 x + 6 \right) } \times \frac { 7 ( 3 + 2 x ) } { 3 x ^ { 5 } }$$ as a single fraction in its simplest form.

Edexcel C3 Q2

2. The function $f$ is defined by $$\mathrm { f } : x \mapsto 2 x , \quad x \in \mathbb { R }$$

Find $\mathrm { f } ^ { - 1 } ( x )$ and state the domain of $\mathrm { f } ^ { - 1 }$. The function g is defined by $$\mathrm { g } : x \mapsto 3 x ^ { 2 } + 2 , \quad x \in \mathbb { R }$$
Find $\mathrm { gf } ^ { - 1 } ( x )$.
State the range of $\mathrm { gf } ^ { - 1 } ( x )$.

Edexcel C3 Q3

3. Find the exact solutions of

$\mathrm { e } ^ { 2 x + 3 } = 6$,
$\ln ( 3 x + 2 ) = 4$.

Edexcel C3 Q4

4. Differentiate with respect to $x$

$x ^ { 3 } \mathrm { e } ^ { 3 x }$,
$\frac { 2 x } { \cos x }$,
$\tan ^ { 2 } x$. Given that $x = \cos y ^ { 2 }$,
find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $y$.

Edexcel C3 Q5

5. (a) Using the formulae $$\begin{gathered} \sin ( A \pm B ) = \sin A \cos B \pm \cos A \sin B
\cos ( A \pm B ) = \cos A \cos B \mp \sin A \sin B \end{gathered}$$ show that

$\sin ( A + B ) - \sin ( A - B ) = 2 \cos A \sin B$,
$\cos ( A - B ) - \cos ( A + B ) = 2 \sin A \sin B$.
(b) Use the above results to show that $$\frac { \sin ( A + B ) - \sin ( A - B ) } { \cos ( A - B ) - \cos ( A + B ) } = \cot A$$ Using the result of part (b) and the exact values of $\sin 60 ^ { \circ }$ and $\cos 60 ^ { \circ }$,
(c) find an exact value for $\cot 75 ^ { \circ }$ in its simplest form.

5. continued Leave blank

Edexcel C3 Q6

6. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{ddc10fc0-f3f2-4c5f-b152-eba68a21990f-08_871_1495_286_273}

\end{figure} Figure 1 shows a sketch of part of the curve with equation $y = \mathrm { f } ( x ) , x \in \mathbb { R }$. The curve has a minimum point at $( - 0.5 , - 2 )$ and a maximum point at $( 0.4 , - 4 )$. The lines $x = 1$, the $x$-axis and the $y$-axis are asymptotes of the curve, as shown in Fig. 1. On a separate diagram sketch the graphs of

$y = | \mathrm { f } ( x ) |$,
$y = \mathrm { f } ( x - 3 )$,
$y = \mathrm { f } ( | x | )$. In each case show clearly
1. the coordinates of any points at which the curve has a maximum or minimum point,
2. how the curve approaches the asymptotes of the curve.
  6. continued

Edexcel C3 Q7

7. (a) Sketch the curve with equation $y = \ln x$.
(b) Show that the tangent to the curve with equation $y = \ln x$ at the point ( $\mathrm { e } , 1$ ) passes through the origin.
(c) Use your sketch to explain why the line $y = m x$ cuts the curve $y = \ln x$ between $x = 1$ and $x = \mathrm { e }$ if $0 < m < \frac { 1 } { \mathrm { e } }$. Taking $x _ { 0 } = 1.86$ and using the iteration $x _ { n + 1 } = \mathrm { e } ^ { \frac { 1 } { 3 } x _ { n } }$,
(d) calculate $x _ { 1 } , x _ { 2 } , x _ { 3 } , x _ { 4 }$ and $x _ { 5 }$, giving your answer to $x _ { 5 }$ to 3 decimal places. The root of $\ln x - \frac { 1 } { 3 } x = 0$ is $\alpha$.
(e) By considering the change of sign of $\ln x - \frac { 1 } { 3 } x$ over a suitable interval, show that your answer for $x _ { 5 }$ is an accurate estimate of $\alpha$, correct to 3 decimal places.

7. continued

Leave blank

Edexcel C3 Q8

In a particular circuit the current, $I$ amperes, is given by

$$I = 4 \sin \theta - 3 \cos \theta , \quad \theta > 0$$ where $\theta$ is an angle related to the voltage. Given that $I = R \sin ( \theta - \alpha )$, where $R > 0$ and $0 \leqslant \alpha < 360 ^ { \circ }$,

find the value of $R$, and the value of $\alpha$ to 1 decimal place.
Hence solve the equation $4 \sin \theta - 3 \cos \theta = 3$ to find the values of $\theta$ between 0 and $360 ^ { \circ }$.
Write down the greatest value for $I$.
Find the value of $\theta$ between 0 and $360 ^ { \circ }$ at which the greatest value of $I$ occurs.
8. continued

Edexcel C3 Specimen Q1

The function f is defined by

$$\mathrm { f } : x \mapsto | x - 2 | - 3 , x \in \mathbb { R }$$

Solve the equation $\mathrm { f } ( x ) = 1$. The function g is defined by $$\mathrm { g } : x \mapsto x ^ { 2 } - 4 x + 11 , x \geq 0$$
Find the range of g .
Find $g f ( - 1 )$.

Edexcel C3 Specimen Q2

2. $\quad \mathrm { f } ( x ) = x ^ { 3 } - 2 x - 5$.

Show that there is a root $\alpha$ of $\mathrm { f } ( x ) = 0$ for $x$ in the interval $[ 2,3 ]$. The root $\alpha$ is to be estimated using the iterative formula $$x _ { n + 1 } = \sqrt { \left( 2 + \frac { 5 } { x _ { n } } \right) } , \quad x _ { 0 } = 2$$
Calculate the values of $x _ { 1 } , x _ { 2 } , x _ { 3 }$ and $x _ { 4 }$, giving your answers to 4 significant figures.
Prove that, to 5 significant figures, $\alpha$ is 2.0946.

Edexcel C3 Specimen Q3

3. (a) Using the identity for $\cos ( A + B )$, prove that $\cos \theta \equiv 1 - 2 \sin ^ { 2 } \left( \frac { 1 } { 2 } \theta \right)$.
(b) Prove that $1 + \sin \theta - \cos \theta \equiv 2 \sin \left( \frac { 1 } { 2 } \theta \right) \left[ \cos \left( \frac { 1 } { 2 } \theta \right) + \sin \left( \frac { 1 } { 2 } \theta \right) \right]$.
(c) Hence, or otherwise, solve the equation $$1 + \sin \theta - \cos \theta = 0 , \quad 0 \leq \theta < 2 \pi$$

Edexcel C3 Specimen Q4

4. $$\mathrm { f } ( x ) = x + \frac { 3 } { x - 1 } - \frac { 12 } { x ^ { 2 } + 2 x - 3 } , x \in \mathbb { R } , x > 1$$

Show that $\mathrm { f } ( x ) = \frac { x ^ { 2 } + 3 x + 3 } { x + 3 }$.
Solve the equation $\mathrm { f } ^ { \prime } ( x ) = \frac { 22 } { 25 }$.

Edexcel C3 Specimen Q5

5. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{937edb48-ef4c-4974-a571-60b0fded841b-3_394_680_982_680}

\end{figure} Figure 1 shows part of the curve with equation $y = \mathrm { f } ( x ) , x \in \mathbb { R }$. The curve meets the $x$-axis at $P ( p , 0 )$ and meets the $y$-axis at $Q ( 0 , q )$.

On separate diagrams, sketch the curve with equation
1. $y = | \mathrm { f } ( x ) |$,
2. $y = 3 \mathrm { f } \left( \frac { 1 } { 2 } x \right)$. In each case show, in terms of $p$ or $q$, the coordinates of points at which the curve meets the axes. Given that $\mathrm { f } ( x ) = 3 \ln ( 2 x + 3 )$,
state the exact value of $q$,
find the value of $p$,
find an equation for the tangent to the curve at $P$.

Edexcel C3 Specimen Q6

6. As a substance cools its temperature, $T ^ { \circ } \mathrm { C }$, is related to the time ( $t$ minutes) for which it has been cooling. The relationship is given by the equation $$T = 20 + 60 \mathrm { e } ^ { - 0.1 t } , t \geq 0$$

Find the value of $T$ when the substance started to cool.
Explain why the temperature of the substance is always above $20 ^ { \circ } \mathrm { C }$.
Sketch the graph of $T$ against $t$.
Find the value, to 2 significant figures, of $t$ at the instant $T = 60$.
Find $\frac { \mathrm { d } T } { \mathrm {~d} t }$.
Hence find the value of $T$ at which the temperature is decreasing at a rate of $1.8 ^ { \circ } \mathrm { C }$ per minute.

Edexcel C3 Specimen Q7

7. (i) Given that $y = \tan x + 2 \cos x$, find the exact value of $\frac { \mathrm { d } y } { \mathrm {~d} x }$ at $x = \frac { \pi } { 4 }$.
(ii) Given that $x = \tan \frac { 1 } { 2 } y$, prove that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 } { 1 + x ^ { 2 } }$.
(iii) Given that $y = \mathrm { e } ^ { - x } \sin 2 x$, show that $\frac { \mathrm { d } y } { \mathrm {~d} x }$ can be expressed in the form $R \mathrm { e } ^ { - x } \cos ( 2 x + \alpha )$. Find, to 3 significant figures, the values of $R$ and $\alpha$, where $0 < \alpha < \frac { \pi } { 2 }$.

Edexcel C3 2012 January Q2

\begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{363c5b5d-7ea9-41e5-ae5a-c3efb5626af5-03_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.

Edexcel C3 2006 June Q2

Differentiate, with respect to $x$,

$\mathrm { e } ^ { 3 x } + \ln 2 x$,
$\left( 5 + x ^ { 2 } \right) ^ { \frac { 3 } { 2 } }$.

Edexcel C3 2006 June Q3

\begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{f0f328ed-3550-4b8d-8b80-016df8773b21-04_568_881_312_504}

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

$y = | f ( x ) |$,
$y = \mathrm { f } ^ { - 1 } ( x )$,
$y = \frac { 1 } { 2 } \mathrm { f } ( 3 x )$. Indicate clearly on each sketch the coordinates of the points at which the curve crosses or meets the axes.

Edexcel C3 2007 June Q2

$$f ( x ) = \frac { 2 x + 3 } { x + 2 } - \frac { 9 + 2 x } { 2 x ^ { 2 } + 3 x - 2 } , \quad x > \frac { 1 } { 2 }$$

Show that $\mathrm { f } ( x ) = \frac { 4 x - 6 } { 2 x - 1 }$.
Hence, or otherwise, find $\mathrm { f } ^ { \prime } ( x )$ in its simplest form.

Questions — Edexcel C3 (377 questions)

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