Questions - Edexcel - A-Level Maths

(i) (a) Find, in ascending powers of $x$, the 2nd, 3rd and 5th terms of the binomial expansion of

$$( 3 + 2 x ) ^ { 6 }$$ For a particular value of $x$, these three terms form consecutive terms in a geometric series.
(b) Find this value of $x$.
(ii) In a different geometric series,

the first term is $\sin ^ { 2 } \theta$
the common ratio is $2 \cos \theta$
the sum to infinity is $\frac { 8 } { 5 }$
(a) Show that

$$5 \cos ^ { 2 } \theta - 16 \cos \theta + 3 = 0$$ (b) Hence find the exact value of the 2nd term in the series.

Edexcel P2 2018 Specimen Q1

7 marks Moderate -0.8

1. $$\mathrm { f } ( x ) = x ^ { 4 } + x ^ { 3 } + 2 x ^ { 2 } + a x + b ,$$ where $a$ and $b$ are constants.
When $\mathrm { f } ( x )$ is divided by ( $x - 1$ ), the remainder is 7

Show that $a + b = 3$ When $\mathrm { f } ( x )$ is divided by ( $x + 2$ ), the remainder is - 8
Find the value of $a$ and the value of $b$

VIIIV SIHI NI JIIIM ION OC VIIV SIHI NI JINAM ION OC VEYV SIHI NI JULIM ION OO

Edexcel P2 2018 Specimen Q2

8 marks Moderate -0.3

2. The first term of a geometric series is 20 and the common ratio is $\frac { 7 } { 8 }$. The sum to infinity of the series is $S _ { \infty }$

Find the value of $S _ { \infty }$ The sum to $N$ terms of the series is $S _ { N }$
Find, to 1 decimal place, the value of $S _ { 12 }$
Find the smallest value of $N$, for which $S _ { \infty } - S _ { N } < 0.5$
2. The first term of a geometric series is 20 and the common ratio is $\frac { 7 } { 8 }$. The sum to infinity
of the series is $S _ { \infty }$

Edexcel P2 2018 Specimen Q3

7 marks Moderate -0.8

3. $$y = \sqrt { \left( 3 ^ { x } + x \right) }$$

Complete the table below, giving the values of $y$ to 3 decimal places.
$x$ 0 0.25 0.5 0.75 1
$y$ 1 1.251 2
Use the trapezium rule with all the values of $y$ from your table to find an approximation for the value of $$\int _ { 0 } ^ { 1 } \sqrt { \left( 3 ^ { x } + x \right) } \mathrm { d } x$$ You must show clearly how you obtained your answer.
Explain how the trapezium rule could be used to obtain a more accurate estimate for the value of $$\int _ { 0 } ^ { 1 } \sqrt { \left( 3 ^ { x } + x \right) } d x$$

\includegraphics[max width=\textwidth, alt={}]{0aafa21b-25f4-4f36-b914-bbaf6cae7a66-10_2673_1948_107_118}

Edexcel P2 2018 Specimen Q4

4 marks Moderate -0.8

Given $n \in \mathbb { N }$, prove, by exhaustion, that $n ^ { 2 } + 2$ is not divisible by 4 .
\includegraphics[max width=\textwidth, alt={}, center]{0aafa21b-25f4-4f36-b914-bbaf6cae7a66-12_2658_1943_111_118}

Edexcel P2 2018 Specimen Q6

7 marks Moderate -0.8

6. (i) Find the exact value of $x$ for which $$\log _ { 2 } ( 2 x ) = \log _ { 2 } ( 5 x + 4 ) - 3$$ (ii) Given that $$\log _ { a } y + 3 \log _ { a } 2 = 5$$ express $y$ in terms of $a$. Give your answer in its simplest form.
\includegraphics[max width=\textwidth, alt={}, center]{0aafa21b-25f4-4f36-b914-bbaf6cae7a66-18_2674_1948_107_118}

Edexcel P2 2018 Specimen Q7

10 marks Moderate -0.5

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{0aafa21b-25f4-4f36-b914-bbaf6cae7a66-19_739_871_260_532} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} The circle with equation $$x ^ { 2 } + y ^ { 2 } - 20 x - 16 y + 139 = 0$$ had centre $C$ and radius $r$.

Find the coordinates of $C$.
Show that $r = 5$ The line with equation $x = 13$ crosses the circle at the points $P$ and $Q$ as shown in Figure 1 .
Find the $y$ coordinate of $P$ and the $y$ coordinate of $Q$. A tangent to the circle from $O$ touches the circle at point $X$.
Find, in surd form, the length $O X$.
\includegraphics[max width=\textwidth, alt={}, center]{0aafa21b-25f4-4f36-b914-bbaf6cae7a66-22_2673_1948_107_118}

Edexcel P2 2018 Specimen Q8

12 marks Moderate -0.3

8. Figure 2 Figure 2 shows a sketch of part of the curves $C _ { 1 }$ and $C _ { 2 }$ with equations $$\begin{array} { l l } C _ { 1 } : y = 10 x - x ^ { 2 } - 8 & x > 0 \\ C _ { 2 } : y = x ^ { 3 } & x > 0 \end{array}$$ The curves $C _ { 1 }$ and $C _ { 2 }$ intersect at the points $A$ and $B$.

Verify that the point $A$ has coordinates (1, 1)
Use algebra to find the coordinates of the point $B$ The finite region $R$ is bounded by $C _ { 1 }$ and $C _ { 2 }$
Use calculus to find the exact area of $R$
\includegraphics[max width=\textwidth, alt={}, center]{0aafa21b-25f4-4f36-b914-bbaf6cae7a66-23_936_759_118_582} \includegraphics[max width=\textwidth, alt={}, center]{0aafa21b-25f4-4f36-b914-bbaf6cae7a66-26_2674_1948_107_118}

Edexcel P2 2018 Specimen Q9

9 marks Moderate -0.3

9. (i) Solve, for $0 \leqslant \theta < \pi$, the equation $$\sin 3 \theta - \sqrt { 3 } \cos 3 \theta = 0$$ giving your answers in terms of $\pi$
(ii) Given that $$4 \sin ^ { 2 } x + \cos x = 4 - k , \quad 0 \leqslant k \leqslant 3$$

find $\cos x$ in terms of $k$
When $k = 3$, find the values of $x$ in the range $0 \leqslant x < 360 ^ { \circ }$

\includegraphics[max width=\textwidth, alt={}]{0aafa21b-25f4-4f36-b914-bbaf6cae7a66-30_2671_1942_107_121}

Edexcel C2 Q3

Moderate -0.8

3. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{c1a3d21d-38fe-4619-9e99-5c4788cdb891-019_675_792_287_568}

\end{figure} In Figure $1 , A ( 4,0 )$ and $B ( 3,5 )$ are the end points of a diameter of the circle $C$. Find

the exact length of $A B$,
the coordinates of the midpoint $P$ of $A B$,
an equation for the circle $C$.

Edexcel C2 2005 January Q1

4 marks Easy -1.2

Find the first three terms, in ascending powers of $x$, of the binomial expansion of $( 3 + 2 x ) ^ { 5 }$, giving each term in its simplest form.
(4)

Edexcel C2 2005 January Q3

7 marks Moderate -0.8

3. Find, giving your answer to 3 significant figures where appropriate, the value of $x$ for which

$3 ^ { x } = 5$,
$\log _ { 2 } ( 2 x + 1 ) - \log _ { 2 } x = 2$.

Edexcel C2 2005 January Q4

7 marks Moderate -0.3

4. (a) Show that the equation $$5 \cos ^ { 2 } x = 3 ( 1 + \sin x )$$ can be written as $$5 \sin ^ { 2 } x + 3 \sin x - 2 = 0 .$$ (b) Hence solve, for $0 \leqslant x < 360 ^ { \circ }$, the equation $$5 \cos ^ { 2 } x = 3 ( 1 + \sin x )$$ giving your answers to 1 decimal place where appropriate.

Edexcel C2 2005 January Q5

8 marks Moderate -0.8

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

When $\mathrm { f } ( x )$ is divided by ( $x - 2$ ), the remainder is 1 .
When $\mathrm { f } ( x )$ is divided by $( x + 1 )$, the remainder is 28 .

Find the value of $a$ and the value of $b$.
Show that ( $x - 3$ ) is a factor of $\mathrm { f } ( x )$.

Edexcel C2 2005 January Q6

8 marks Moderate -0.3

The second and fourth terms of a geometric series are 7.2 and 5.832 respectively.

The common ratio of the series is positive.
For this series, find

the common ratio,
the first term,
the sum of the first 50 terms, giving your answer to 3 decimal places,
the difference between the sum to infinity and the sum of the first 50 terms, giving your answer to 3 decimal places.

Edexcel C2 2005 January Q7

11 marks Standard +0.3

7. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{13bca882-27da-40f2-99d8-4fdeb6629c4e-12_707_1072_301_434}

\end{figure} Figure 1 shows the triangle $A B C$, with $A B = 8 \mathrm {~cm} , A C = 11 \mathrm {~cm}$ and $\angle B A C = 0.7$ radians. The $\operatorname { arc } B D$, where $D$ lies on $A C$, is an arc of a circle with centre $A$ and radius 8 cm . The region $R$, shown shaded in Figure 1, is bounded by the straight lines $B C$ and $C D$ and the $\operatorname { arc } B D$. Find

the length of the $\operatorname { arc } B D$,
the perimeter of $R$, giving your answer to 3 significant figures,
the area of $R$, giving your answer to 3 significant figures.

Edexcel C2 2005 January Q8

12 marks Moderate -0.3

8. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{13bca882-27da-40f2-99d8-4fdeb6629c4e-14_1102_1317_308_340}

\end{figure} The line with equation $y = 3 x + 20$ cuts the curve with equation $y = x ^ { 2 } + 6 x + 10$ at the points $A$ and $B$, as shown in Figure 2.

Use algebra to find the coordinates of $A$ and the coordinates of $B$. The shaded region $S$ is bounded by the line and the curve, as shown in Figure 2.
Use calculus to find the exact area of $S$.

Edexcel C2 2005 January Q9

12 marks Standard +0.3

9. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{13bca882-27da-40f2-99d8-4fdeb6629c4e-16_821_958_301_516}

\end{figure} Figure 3 shows the plan of a stage in the shape of a rectangle joined to a semicircle. The length of the rectangular part is $2 x$ metres and the width is $y$ metres. The diameter of the semicircular part is $2 x$ metres. The perimeter of the stage is 80 m .

Show that the area, $A \mathrm {~m} ^ { 2 }$, of the stage is given by $$A = 80 x - \left( 2 + \frac { \pi } { 2 } \right) x ^ { 2 } .$$
Use calculus to find the value of $x$ at which $A$ has a stationary value.
Prove that the value of $x$ you found in part (b) gives the maximum value of $A$.
Calculate, to the nearest $\mathrm { m } ^ { 2 }$, the maximum area of the stage.

Edexcel C2 2006 January Q1

8 marks Moderate -0.8

$\mathrm { f } ( x ) = 2 x ^ { 3 } + x ^ { 2 } - 5 x + c$, where $c$ is a constant.

Given that $\mathrm { f } ( 1 ) = 0$,

find the value of $c$,
factorise $\mathrm { f } ( x )$ completely,
find the remainder when $\mathrm { f } ( x )$ is divided by ( $2 x - 3$ ).

Edexcel C2 2006 January Q2

6 marks Moderate -0.8

2. (a) Find the first 3 terms, in ascending powers of $x$, of the binomial expansion of $$( 1 + p x ) ^ { 9 }$$ where $p$ is a constant. These first 3 terms are $1,36 x$ and $q x ^ { 2 }$, where $q$ is a constant.
(b) Find the value of $p$ and the value of $q$.

Edexcel C2 2006 January Q3

7 marks Easy -1.2

3. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{84b2d36b-c112-4d35-84a1-bc2b707f162d-04_675_792_287_568}

\end{figure} In Figure $1 , A ( 4,0 )$ and $B ( 3,5 )$ are the end points of a diameter of the circle $C$. Find

the exact length of $A B$,
the coordinates of the midpoint $P$ of $A B$,
an equation for the circle $C$.

Edexcel C2 2006 January Q4

11 marks Moderate -0.3

The first term of a geometric series is 120 . The sum to infinity of the series is 480 .
1. Show that the common ratio, $r$, is $\frac { 3 } { 4 }$.
2. Find, to 2 decimal places, the difference between the 5th and 6th term.
3. Calculate the sum of the first 7 terms.
The sum of the first $n$ terms of the series is greater than 300 .
Calculate the smallest possible value of $n$.

Edexcel C2 2006 January Q5

8 marks Moderate -0.3

5. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{84b2d36b-c112-4d35-84a1-bc2b707f162d-07_538_611_301_680}

\end{figure} In Figure $2 O A B$ is a sector of a circle radius 5 m . The chord $A B$ is 6 m long.

Show that $\cos A \hat { O } B = \frac { 7 } { 25 }$.
Hence find the angle $A \hat { O } B$ in radians, giving your answer to 3 decimal places.
Calculate the area of the sector $O A B$.
Hence calculate the shaded area.

Edexcel C2 2006 January Q6

6 marks Moderate -0.8

The speed, $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, of a train at time $t$ seconds is given by

$$v = \sqrt { } \left( 1.2 ^ { t } - 1 \right) , \quad 0 \leqslant t \leqslant 30$$ The following table shows the speed of the train at 5 second intervals.

$t$	0	5	10	15	20	25	30
$v$	0	1.22	2.28		6.11

Complete the table, giving the values of $v$ to 2 decimal places. The distance, $s$ metres, travelled by the train in 30 seconds is given by $$s = \int _ { 0 } ^ { 30 } \sqrt { } \left( 1.2 ^ { t } - 1 \right) \mathrm { d } t$$
Use the trapezium rule, with all the values from your table, to estimate the value of $s$.
(3)

Edexcel C2 2006 January Q7

10 marks Moderate -0.8

7. The curve $C$ has equation $$y = 2 x ^ { 3 } - 5 x ^ { 2 } - 4 x + 2$$

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.
Using the result from part (a), find the coordinates of the turning points of $C$.
Find $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$.
Hence, or otherwise, determine the nature of the turning points of $C$.

\(x\)	0	0.25	0.5	0.75	1
\(y\)	1	1.251			2

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