Questions - Edexcel - A-Level Maths

Edexcel C1 Q4

6 marks Moderate -0.8

4. Find the set of values of $x$ for which

$6 x - 11 > x + 4$,
$x ^ { 2 } - 6 x - 16 < 0$,
both $6 x - 11 > x + 4$ and $x ^ { 2 } - 6 x - 16 < 0$.

Edexcel C1 Q5

8 marks Moderate -0.3

5. $$f ( x ) = ( 2 - \sqrt { x } ) ^ { 2 } , \quad x > 0$$

Solve the equation $\mathrm { f } ( x ) = 0$.
Find $\mathrm { f } ( 3 )$, giving your answer in the form $a + b \sqrt { 3 }$, where $a$ and $b$ are integers.
Find $$\int \mathrm { f } ( x ) \mathrm { d } x$$

Edexcel C1 Q6

8 marks Moderate -0.3

The straight line $l$ passes through the point $P ( - 3,6 )$ and the point $Q ( 1 , - 4 )$.
1. Find an equation for $l$ in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers.
The straight line $m$ has the equation $2 x + k y + 7 = 0$, where $k$ is a constant.
Given that $l$ and $m$ are perpendicular,
find the value of $k$.

Edexcel C1 Q7

8 marks Moderate -0.8

7. Given that $$\mathrm { f } ^ { \prime } ( x ) = 5 + \frac { 4 } { x ^ { 2 } } , \quad x \neq 0$$

find an expression for $\mathrm { f } ( x )$. Given also that
f(2) = 2f(1),
find $\mathrm { f } ( 4 )$.

Edexcel C1 Q8

10 marks Moderate -0.3

8. $$f ( x ) = x ^ { 3 } - 6 x ^ { 2 } + 5 x + 12$$

Show that $$( x + 1 ) ( x - 3 ) ( x - 4 ) \equiv x ^ { 3 } - 6 x ^ { 2 } + 5 x + 12$$
Sketch the curve $y = \mathrm { f } ( x )$, showing the coordinates of any points of intersection with the coordinate axes.
Showing the coordinates of any points of intersection with the coordinate axes, sketch on separate diagrams the curves
1. $\quad y = \mathrm { f } ( x + 3 )$,
2. $y = \mathrm { f } ( - x )$.

Edexcel C1 Q9

11 marks Standard +0.3

9. The first two terms of an arithmetic series are $( t - 1 )$ and $\left( t ^ { 2 } - 5 \right)$ respectively, where $t$ is a positive constant.

Find and simplify expressions in terms of $t$ for
1. the common difference of the series,
2. the third term of the series. Given also that the third term of the series is 19 ,
find the value of $t$,
show that the 10th term of the series is 75,
find the sum of the first 40 terms of the series.

Edexcel C1 Q10

11 marks Standard +0.3

10. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e0ee09b1-25a2-4244-aa69-63e8f5b3543a-4_595_727_1119_493} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows the curve with equation $y = 2 + 3 x - x ^ { 2 }$ and the straight lines $l$ and $m$. The line $l$ is the tangent to the curve at the point $A$ where the curve crosses the $y$-axis.

Find an equation for $l$. The line $m$ is the normal to the curve at the point $B$.
Given that $l$ and $m$ are parallel,
find the coordinates of $B$.

Edexcel C2 Q1

6 marks Moderate -0.3

Given that $p = \log _ { q } 16$, express in terms of $p$,
1. $\log _ { q } 2$,
2. $\log _ { q } ( 8 q )$.
3. The expansion of $( 2 - p x ) ^ { 6 }$ in ascending powers of $x$, as far as the term in $x ^ { 2 }$, is
$$64 + A x + 135 x ^ { 2 }$$ Given that $p > 0$, find the value of $p$ and the value of $A$.
(7)

Edexcel C2 Q3

7 marks Standard +0.3

3. A circle $C$ has equation $$x ^ { 2 } + y ^ { 2 } - 6 x + 8 y - 75 = 0$$

Write down the coordinates of the centre of $C$, and calculate the radius of $C$. A second circle has centre at the point $( 15,12 )$ and radius 10 .
Sketch both circles on a single diagram and find the coordinates of the point where they touch.
(4)

Edexcel C2 Q4

8 marks Moderate -0.3

4. (a) Sketch, for $0 \leq x \leq 360 ^ { \circ }$, the graph of $y = \sin \left( x + 30 ^ { \circ } \right)$.
(b) Write down the coordinates of the points at which the graph meets the axes.
(c) Solve, for $0 \leq x < 360 ^ { \circ }$, the equation $$\sin \left( x + 30 ^ { \circ } \right) = - \frac { 1 } { 2 }$$

Edexcel C2 Q5

6 marks Standard +0.3

5. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{c85316fe-5c59-4cb3-8cb8-d95a4e97af70-3_549_620_276_689}

\end{figure} The shape of a badge is a sector $A B C$ of a circle with centre $A$ and radius $A B$, as shown in Fig 1. The triangle $A B C$ is equilateral and has a perpendicular height 3 cm .

Find, in surd form, the length $A B$.
Find, in terms of $\pi$, the area of the badge.
Prove that the perimeter of the badge is $\frac { 2 \sqrt { 3 } } { 3 } ( \pi + 6 ) \mathrm { cm }$.

Edexcel C2 Q6

9 marks Moderate -0.3

6. $\mathrm { f } ( x ) = 6 x ^ { 3 } + p x ^ { 2 } + q x + 8$, where $p$ and $q$ are constants. Given that $\mathrm { f } ( x )$ is exactly divisible by ( $2 x - 1$ ), and also that when $\mathrm { f } ( x )$ is divided by ( $x - 1$ ) the remainder is - 7 ,

find the value of $p$ and the value of $q$.
Hence factorise $\mathrm { f } ( x )$ completely.

Edexcel C2 Q7

10 marks Standard +0.3

7. A geometric series has first term 1200. Its sum to infinity is 960 .

Show that the common ratio of the series is $- \frac { 1 } { 4 }$.
Find, to 3 decimal places, the difference between the ninth and tenth terms of the series.
Write down an expression for the sum of the first $n$ terms of the series. Given that $n$ is odd,
prove that the sum of the first $n$ terms of the series is $$960 \left( 1 + 0.25 ^ { n } \right)$$

Edexcel C2 Q8

9 marks Moderate -0.3

A circle $C$ has centre $( 3,4 )$ and radius $3 \sqrt { } 2$. A straight line $l$ has equation $y = x + 3$.
1. Write down an equation of the circle $C$.
2. Calculate the exact coordinates of the two points where the line $l$ intersects $C$, giving your answers in surds.
3. Find the distance between these two points.
\begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{c85316fe-5c59-4cb3-8cb8-d95a4e97af70-5_730_983_278_404}
\end{figure} The curve $C$, shown in Fig. 2, represents the graph of $$y = \frac { x ^ { 2 } } { 25 } , x \geq 0 .$$ The points $A$ and $B$ on the curve $C$ have $x$-coordinates 5 and 10 respectively.
Write down the $y$-coordinates of $A$ and $B$.
Find an equation of the tangent to $C$ at $A$. The finite region $R$ is enclosed by $C$, the $y$-axis and the lines through $A$ and $B$ parallel to the $x$-axis.
For points $( x , y )$ on $C$, express $x$ in terms of $y$.
Use integration to find the area of $R$. END

Edexcel C2 Q1

5 marks Moderate -0.3

1. $$f ( x ) = p x ^ { 3 } + 6 x ^ { 2 } + 12 x + q .$$ Given that the remainder when $\mathrm { f } ( x )$ is divided by ( $x - 1$ ) is equal to the remainder when $\mathrm { f } ( x )$ is divided by ( $2 x + 1$ ),

find the value of $p$. Given also that $q = 3$, and $p$ has the value found in part (a),
find the value of the remainder.

Edexcel C2 Q2

6 marks Moderate -0.8

2. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{f01026a9-e5fe-4c19-b096-2bb4ad22c389-2_615_833_941_598}

\end{figure} The circle $C$, with centre $( a , b )$ and radius 5 , touches the $x$-axis at $( 4,0 )$, as shown in Fig. 1.

Write down the value of $a$ and the value of $b$.
Find a cartesian equation of $C$. A tangent to the circle, drawn from the point $P ( 8,17 )$, touches the circle at $T$.
Find, to 3 significant figures, the length of $P T$.

Edexcel C2 Q4

7 marks Moderate -0.8

4
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Edexcel C2 Q5

8 marks Moderate -0.8

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Edexcel C2 Q6

9 marks Moderate -0.8

6
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Edexcel C2 Q7

9 marks Moderate -0.3

7
7
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\hline \end{tabular} \end{center} 1. $$f ( x ) = p x ^ { 3 } + 6 x ^ { 2 } + 12 x + q .$$ Given that the remainder when $\mathrm { f } ( x )$ is divided by ( $x - 1$ ) is equal to the remainder when $\mathrm { f } ( x )$ is divided by ( $2 x + 1$ ),

find the value of $p$. Given also that $q = 3$, and $p$ has the value found in part (a),
find the value of the remainder.
2. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{f01026a9-e5fe-4c19-b096-2bb4ad22c389-2_615_833_941_598}
\end{figure} The circle $C$, with centre $( a , b )$ and radius 5 , touches the $x$-axis at $( 4,0 )$, as shown in Fig. 1.
Write down the value of $a$ and the value of $b$.
Find a cartesian equation of $C$. A tangent to the circle, drawn from the point $P ( 8,17 )$, touches the circle at $T$.
Find, to 3 significant figures, the length of $P T$.
3. (a) Expand $( 2 \sqrt { } x + 3 ) ^ { 2 }$.
Hence evaluate $\int _ { 1 } ^ { 2 } ( 2 \sqrt { } x + 3 ) ^ { 2 } \mathrm {~d} x$, giving your answer in the form $a + b \sqrt { } 2$, where $a$ and $b$ are integers.
4. The first three terms in the expansion, in ascending powers of $x$, of $( 1 + p x ) ^ { n }$, are $1 - 18 x + 36 p ^ { 2 } x ^ { 2 }$. Given that $n$ is a positive integer, find the value of $n$ and the value of $p$.
(7 marks)
5. Find all values of $\theta$ in the interval $0 \leq \theta < 360$ for which
$\cos ( \theta + 75 ) ^ { \circ } = 0$.
$\sin 2 \theta ^ { \circ } = 0.7$, giving your answers to one decima1 place.
6. Given that $\log _ { 2 } x = a$, find, in terms of $a$, the simplest form of
$\log _ { 2 } ( 16 x )$,
$\log _ { 2 } \left( \frac { x ^ { 4 } } { 2 } \right)$.
Hence, or otherwise, solve $$\log _ { 2 } ( 16 x ) - \log _ { 2 } \left( \frac { x ^ { 4 } } { 2 } \right) = \frac { 1 } { 2 }$$ giving your answer in its simplest surd form.
7. The curve $C$ has equation $y = \cos \left( x + \frac { \pi } { 4 } \right) , 0 \leq x \leq 2 \pi$.
Sketch $C$.
Write down the exact coordinates of the points at which $C$ meets the coordinate axes.
Solve, for $x$ in the interval $0 \leq x \leq 2 \pi$, $$\cos \left( x + \frac { \pi } { 4 } \right) = 0.5$$ giving your answers in terms of $\pi$.

Edexcel C2 Q8

12 marks Moderate -0.8

8. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{f01026a9-e5fe-4c19-b096-2bb4ad22c389-4_769_1150_269_379}

\end{figure} Figure 2 shows part of the curve with equation $$y = x ^ { 3 } - 6 x ^ { 2 } + 9 x .$$ The curve touches the $x$-axis at $A$ and has a maximum turning point at $B$.

Show that the equation of the curve may be written as $$y = x ( x - 3 ) ^ { 2 } ,$$ and hence write down the coordinates of $A$.
Find the coordinates of $B$. The shaded region $R$ is bounded by the curve and the $x$-axis.
Find the area of $R$.

Edexcel C2 Q9

12 marks Standard +0.3

9 \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{f01026a9-e5fe-4c19-b096-2bb4ad22c389-5_686_1240_178_312} \captionsetup{labelformat=empty} \caption{Fig. 3}

\end{figure} Triangle $A B C$ has $A B = 9 \mathrm {~cm} , B C 10 \mathrm {~cm}$ and $C A = 5 \mathrm {~cm}$. A circle, centre $A$ and radius 3 cm , intersects $A B$ and $A C$ at $P$ and $Q$ respectively, as shown in Fig. 3.

Show that, to 3 decimal places, $\angle B A C = 1.504$ radians. Calculate,
the area, in $\mathrm { cm } ^ { 2 }$, of the sector $A P Q$,
the area, in $\mathrm { cm } ^ { 2 }$, of the shaded region $B P Q C$,
the perimeter, in cm , of the shaded region $B P Q C$.

Edexcel C2 Q1

7 marks Moderate -0.3

$\mathrm { f } ( x ) = x ^ { 3 } + a x ^ { 2 } + b x - 10$, where $a$ and $b$ are constants.

When $\mathrm { f } ( x )$ is divided by ( $x - 3$ ), the remainder is 14 .
When $\mathrm { f } ( x )$ is divided by ( $x + 1$ ), the remainder is - 18 .

Find the value of $a$ and the value of $b$.
Show that ( $x - 2$ ) is a factor of $\mathrm { f } ( x )$.

Edexcel C2 Q2

8 marks Moderate -0.3

2. (a) Write down the first four terms of the binomial expansion, in ascending powers of $x$, of $( 1 + a x ) ^ { n }$, where $n > 2$. Given that, in this expansion, the coefficient of $x$ is 8 and the coefficient of $x ^ { 2 }$ is 30 ,
(b) find the value of $n$ and the value of $a$,
(c) find the coefficient of $x ^ { 3 }$.

Edexcel C2 Q7

14 marks Moderate -0.8

7
7
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$\mathrm { f } ( x ) = x ^ { 3 } + a x ^ { 2 } + b x - 10$, where $a$ and $b$ are constants.

When $\mathrm { f } ( x )$ is divided by ( $x - 3$ ), the remainder is 14 .
When $\mathrm { f } ( x )$ is divided by ( $x + 1$ ), the remainder is - 18 .

Find the value of $a$ and the value of $b$.
Show that ( $x - 2$ ) is a factor of $\mathrm { f } ( x )$.
2. (a) Write down the first four terms of the binomial expansion, in ascending powers of $x$, of $( 1 + a x ) ^ { n }$, where $n > 2$. Given that, in this expansion, the coefficient of $x$ is 8 and the coefficient of $x ^ { 2 }$ is 30 ,
find the value of $n$ and the value of $a$,
find the coefficient of $x ^ { 3 }$.
3. A population of deer is introduced into a park. The population $P$ at $t$ years after the deer have been introduced is modelled by $$P = \frac { 2000 a ^ { t } } { 4 + a ^ { t } } ,$$ where $a$ is a constant. Given that there are 800 deer in the park after 6 years,
calculate, to 4 decimal places, the value of $a$,
use the model to predict the number of years needed for the population of deer to increase from 800 to 1800.
With reference to this model, give a reason why the population of deer cannot exceed 2000.
4. Given that $\mathrm { f } ( x ) = \left( 2 x ^ { \frac { 3 } { 2 } } - 3 x ^ { - \frac { 3 } { 2 } } \right) ^ { 2 } + 5 , \quad x > 0$,
find, to 3 significant figures, the value of $x$ for which $\mathrm { f } ( x ) = 5$.
Show that $\mathrm { f } ( x )$ may be written in the form $A x ^ { 3 } + \frac { B } { x ^ { 3 } } + C$, where $A , B$ and $C$ are constants to be found.
Hence evaluate $\int _ { 1 } ^ { 2 } f ( x ) d x$.
5. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{5e4e4cae-d4c6-465d-8cb7-84712e6e55fe-3_736_1266_276_404}
\end{figure} Figure 1 shows the cross-section $A B C D$ of a chocolate bar, where $A B , C D$ and $A D$ are straight lines and $M$ is the mid-point of $A D$. The length $A D$ is 28 mm , and $B C$ is an arc of a circle with centre $M$. Taking $A$ as the origin, $B , C$ and $D$ have coordinates $( 7,24 ) , ( 21,24 )$ and $( 28,0 )$ respectively.
Show that the length of $B M$ is 25 mm .
Show that, to 3 significant figures, $\angle B M C = 0.568$ radians.
Hence calculate, in $\mathrm { mm } ^ { 2 }$, the area of the cross-section of the chocolate bar. Given that this chocolate bar has length 85 mm ,
calculate, to the nearest $\mathrm { cm } ^ { 3 }$, the volume of the bar.
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5e4e4cae-d4c6-465d-8cb7-84712e6e55fe-4_641_1406_196_287} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} Figure 1 shows the curve with equation $y = 5 + 2 x - x ^ { 2 }$ and the line with equation $y = 2$. The curve and the line intersect at the points $A$ and $B$.
Find the x-coordinates of $A$ and $B$. The shaded region $R$ is bounded by the curve and the line.
Find the area of $R$.
7. Find all the values of $\theta$ in the interval $0 \leq \theta < 360 ^ { \circ }$ for which
$\cos \left( \theta - 10 ^ { \circ } \right) = \cos 15 ^ { \circ }$,
$\tan 2 \theta = 0.4$,
$2 \sin \theta \tan \theta = 3$.

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