Questions - Edexcel P2 - A-Level Maths

when $\mathrm { f } ( x )$ is divided by $( x - 1 )$ the remainder is $k$
when $\mathrm { f } ( x )$ is divided by $( x + 1 )$ the remainder is $- 10 k$
$k$ is a constant
1. show that

$$11 A + 9 B = 83$$ Given also that $( 3 x - 2 )$ is a factor of $\mathrm { f } ( x )$,

find the value of $A$ and the value of $B$.

Hence find the quadratic expression $\mathrm { g } ( x )$ such that $$f ( x ) = ( 3 x - 2 ) g ( x )$$

Edexcel P2 2022 January Q6

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{59c9f675-e7eb-47b9-b233-dfbe1844f792-18_579_620_219_667} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} The points $P ( 23,14 ) , Q ( 15 , - 30 )$ and $R ( - 7 , - 26 )$ lie on the circle $C$, as shown in Figure 1.

Show that angle $P Q R = 90 ^ { \circ }$
Hence, or otherwise, find
1. the centre of $C$,
2. the radius of $C$. Given that the point $S$ lies on $C$ such that the distance $Q S$ is greatest,
find an equation of the tangent to $C$ at $S$, giving your answer in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers to be found.

Edexcel P2 2022 January Q7

7. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

Solve, for $- 90 ^ { \circ } < x < 90 ^ { \circ }$, the equation $$3 \sin \left( 2 x - 15 ^ { \circ } \right) = \cos \left( 2 x - 15 ^ { \circ } \right)$$ giving your answers to one decimal place.
Solve, for $0 < \theta < 2 \pi$, the equation $$4 \sin ^ { 2 } \theta + 8 \cos \theta = 3$$ giving your answers to 3 significant figures.

Edexcel P2 2022 January Q8

8. A metal post is repeatedly hit in order to drive it into the ground. Given that

on the 1st hit, the post is driven 100 mm into the ground
on the 2nd hit, the post is driven an additional 98 mm into the ground
on the 3rd hit, the post is driven an additional 96 mm into the ground
the additional distances the post travels on each subsequent hit form an arithmetic sequence
1. show that the post is driven an additional 62 mm into the ground with the 20th hit.
2. Find the total distance that the post has been driven into the ground after 20 hits.

Given that for each subsequent hit after the 20th hit

the additional distances the post travels form a geometric sequence with common ratio $r$
on the 22 nd hit, the post is driven an additional 60 mm into the ground
find the value of $r$, giving your answer to 3 decimal places.

After a total of $N$ hits, the post will have been driven more than 3 m into the ground.

Find, showing all steps in your working, the smallest possible value of $N$.

Edexcel P2 2022 January Q9

9. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{59c9f675-e7eb-47b9-b233-dfbe1844f792-30_639_929_214_511} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows

the curve $C$ with equation $y = x - x ^ { 2 }$
the line $l$ with equation $y = m x$, where $m$ is a constant and $0 < m < 1$

The line and the curve intersect at the origin $O$ and at the point $P$.

Find, in terms of $m$, the coordinates of $P$. The region $R _ { 1 }$, shown shaded in Figure 2, is bounded by $C$ and $l$.
Show that the area of $R _ { 1 }$ is $$\frac { ( 1 - m ) ^ { 3 } } { 6 }$$ The region $R _ { 2 }$, also shown shaded in Figure 2, is bounded by $C$, the $x$-axis and $l$. Given that the area of $R _ { 1 }$ is equal to the area of $R _ { 2 }$
find the exact value of $m$.
\includegraphics[max width=\textwidth, alt={}, center]{59c9f675-e7eb-47b9-b233-dfbe1844f792-33_108_76_2613_1875}
\includegraphics[max width=\textwidth, alt={}, center]{59c9f675-e7eb-47b9-b233-dfbe1844f792-33_52_83_2722_1850}

Edexcel P2 2022 January Q10

10. (i) Prove by counter example that the statement
"if $p$ is a prime number then $2 p + 1$ is also a prime number" is not true.
(ii) Use proof by exhaustion to prove that if $n$ is an integer then $$5 n ^ { 2 } + n + 12$$ is always even.

Edexcel P2 2023 January Q1

1. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{f6af51c1-5f85-4952-b3c4-9dca42b2a309-02_614_739_248_664} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of part of the curve with equation $y = \mathrm { f } ( x )$
The table below shows some corresponding values of $x$ and $y$ for this curve.
The values of $y$ are given to 3 decimal places.

$x$	- 1	- 0.5	0	0.5	1
$y$	2.287	4.470	6.719	7.291	2.834

Using the trapezium rule with all the values of $y$ in the given table,

obtain an estimate for $$\int _ { - 1 } ^ { 1 } \mathrm { f } ( x ) \mathrm { d } x$$ giving your answer to 2 decimal places.
Use your answer to part (a) to estimate
1. $\int _ { - 1 } ^ { 1 } ( \mathrm { f } ( x ) - 2 ) \mathrm { d } x$
2. $\int _ { 1 } ^ { 3 } \mathrm { f } ( x - 2 ) \mathrm { d } x$

Edexcel P2 2023 January Q2

In this question you must show all stages of your working.

\section*{Solutions based entirely on calculator technology are not acceptable.} \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{f6af51c1-5f85-4952-b3c4-9dca42b2a309-04_629_995_411_534} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A brick is in the shape of a cuboid with width $x \mathrm {~cm}$ ,length $3 x \mathrm {~cm}$ and height $h \mathrm {~cm}$ ,as shown in Figure 2. The volume of the brick is $972 \mathrm {~cm} ^ { 3 }$

Show that the surface area of the brick,$S \mathrm {~cm} ^ { 2 }$ ,is given by $$S = 6 x ^ { 2 } + \frac { 2592 } { x }$$
Find $\frac { \mathrm { d } S } { \mathrm {~d} x }$
Hence find the value of $x$ for which $S$ is stationary.
Find $\frac { \mathrm { d } ^ { 2 } S } { \mathrm {~d} x ^ { 2 } }$ and hence show that the value of $x$ found in part(c)gives the minimum value of $S$ .
Hence find the minimum surface area of the brick.

Edexcel P2 2023 January Q3

$\mathrm { f } ( x ) = \left( 2 + \frac { k x } { 8 } \right) ^ { 7 }$ where $k$ is a non-zero constant
1. Find the first 4 terms, in ascending powers of $x$, of the binomial expansion of $\mathrm { f } ( x )$. Give each term in simplest form.
Given that, in the binomial expansion of $\mathrm { f } ( x )$, the coefficients of $x , x ^ { 2 }$ and $x ^ { 3 }$ are the first 3 terms of an arithmetic progression,
find, using algebra, the possible values of $k$.
(Solutions relying entirely on calculator technology are not acceptable.)

Edexcel P2 2023 January Q4

(i) Using the laws of logarithms, solve

$$\log _ { 3 } ( 4 x ) + 2 = \log _ { 3 } ( 5 x + 7 )$$ (ii) Given that $$\sum _ { r = 1 } ^ { 2 } \log _ { a } \left( y ^ { r } \right) = \sum _ { r = 1 } ^ { 2 } \left( \log _ { a } y \right) ^ { r } \quad y > 1 , a > 1 , y \neq a$$ find $y$ in terms of $a$, giving your answer in simplest form.

Edexcel P2 2023 January Q5

5. $$f ( x ) = x ^ { 3 } + ( p + 3 ) x ^ { 2 } - x + q$$ where $p$ and $q$ are constants and $p > 0$
Given that ( $x - 3$ ) is a factor of $\mathrm { f } ( x )$

show that $$9 p + q = - 51$$ Given also that when $\mathrm { f } ( x )$ is divided by ( $x + p$ ) the remainder is 9
show that $$3 p ^ { 2 } + p + q - 9 = 0$$
Hence find the value of $p$ and the value of $q$.
Hence find a quadratic expression $\mathrm { g } ( x )$ such that $$f ( x ) = ( x - 3 ) g ( x )$$

Edexcel P2 2023 January Q6

The circle $C$ has equation

$$x ^ { 2 } + y ^ { 2 } + 8 x - 4 y = 0$$

Find
1. the coordinates of the centre of $C$,
2. the exact radius of $C$. The point $P$ lies on $C$.
  Given that the tangent to $C$ at $P$ has equation $x + 2 y + 10 = 0$
find the coordinates of $P$
Find the equation of the normal to $C$ at $P$, giving your answer in the form $y = m x + c$ where $m$ and $c$ are integers to be found.

Edexcel P2 2023 January Q7

A geometric sequence has first term $a$ and common ratio $r$, where $r > 0$

Given that

the 3rd term is 20
the 5th term is 12.8
1. show that $r = 0.8$
2. Hence find the value of $a$.

Given that the sum of the first $n$ terms of this sequence is greater than 156

find the smallest possible value of $n$.
(Solutions based entirely on graphical or numerical methods are not acceptable.)

Edexcel P2 2023 January Q8

In this question you must show all stages of your working.

Solutions based entirely on calculator technology are not acceptable.

Solve, for $- \frac { \pi } { 2 } < x < \pi$, the equation $$5 \sin ( 3 x + 0.1 ) + 2 = 0$$ giving your answers, in radians, to 2 decimal places.
Solve, for $0 < \theta < 360 ^ { \circ }$, the equation $$2 \tan \theta \sin \theta = 5 + \cos \theta$$ giving your answers, in degrees, to one decimal place.

Edexcel P2 2023 January Q9

In this question you must show all stages of your working.

\section*{Solutions based entirely on calculator technology are not acceptable.} \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{f6af51c1-5f85-4952-b3c4-9dca42b2a309-26_761_940_411_566} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} Figure 3 shows

the curve $C$ with equation $y = x ^ { 2 } - 4 x + 5$
the line $l$ with equation $y = 2$

The curve $C$ intersects the $y$-axis at the point $D$.

Write down the coordinates of $D$. The curve $C$ intersects the line $l$ at the points $E$ and $F$, as shown in Figure 3.
Find the $x$ coordinate of $E$ and the $x$ coordinate of $F$. Shown shaded in Figure 3 is
- the region $R _ { 1 }$ which is bounded by $C , l$ and the $y$-axis
- the region $R _ { 2 }$ which is bounded by $C$ and the line segments $E F$ and $D F$
Given that $\frac { \text { area of } R _ { 1 } } { \text { area of } R _ { 2 } } = k$, where $k$ is a constant,
use algebraic integration to find the exact value of $k$, giving your answer as a simplified fraction.

Edexcel P2 2023 January Q10

A student was asked to prove by exhaustion that
if $n$ is an integer then $2 n ^ { 2 } + n + 1$ is not divisible by 3

The start of the student's proof is shown in the box below. Consider the case when $n = 3 k$ $$2 n ^ { 2 } + n + 1 = 18 k ^ { 2 } + 3 k + 1 = 3 \left( 6 k ^ { 2 } + k \right) + 1$$ which is not divisible by 3 Complete this proof.

Edexcel P2 2024 January Q1

1. $$f ( x ) = a x ^ { 3 } + 3 x ^ { 2 } - 8 x + 2 \quad \text { where } a \text { is a constant }$$ Given that when $\mathrm { f } ( x )$ is divided by $( x - 2 )$ the remainder is 3 , find the value of $a$.

Edexcel P2 2024 January Q2

Find the coefficient of the term in $x ^ { 7 }$ of the binomial expansion of

$$\left( \frac { 3 } { 8 } + 4 x \right) ^ { 12 }$$ giving your answer in simplest form.

Edexcel P2 2024 January Q3

The circle $C$

has centre $A ( 3,5 )$
passes through the point $B ( 8 , - 7 )$
1. Find an equation for $C$.

The points $M$ and $N$ lie on $C$ such that $M N$ is a chord of $C$.
Given that $M N$

lies above the $x$-axis
is parallel to the $x$-axis
has length $4 \sqrt { 22 }$
find an equation for the line passing through points $M$ and $N$.

Edexcel P2 2024 January Q4

(a) Sketch the curve with equation

$$y = a ^ { - x } + 4$$ where $a$ is a constant and $a > 1$
On your sketch show

the coordinates of the point of intersection of the curve with the $y$-axis
the equation of the asymptote to the curve.

$x$	- 4	- 1.5	1	3.5	6	8.5
$y$	13	6.280	4.577	4.146	4.037	4.009

The table above shows corresponding values of $x$ and $y$ for $y = 3 ^ { - \frac { 1 } { 2 } x } + 4$
The values of $y$ are given to four significant figures, as appropriate.
Using the trapezium rule with all the values of $y$ in the table,
(b) find an approximate value for $$\int _ { - 4 } ^ { 8.5 } \left( 3 ^ { - \frac { 1 } { 2 } x } + 4 \right) d x$$ giving your answer to two significant figures.
(c) Using the answer to part (b), find an approximate value for

$\int _ { - 4 } ^ { 8.5 } \left( 3 ^ { - \frac { 1 } { 2 } x } \right) \mathrm { d } x$
$\int _ { - 4 } ^ { 8.5 } \left( 3 ^ { - \frac { 1 } { 2 } x } + 4 \right) \mathrm { d } x + \int _ { - 8.5 } ^ { 4 } \left( 3 ^ { \frac { 1 } { 2 } x } + 4 \right) \mathrm { d } x$

Edexcel P2 2024 January Q5

1. Find the value of

$$\sum _ { r = 1 } ^ { \infty } 6 \times ( 0.25 ) ^ { r }$$ (3)
(ii) A sequence $u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots$ is defined by $$\begin{aligned} u _ { 1 } & = 3
u _ { n + 1 } & = \frac { u _ { n } - 3 } { u _ { n } - 2 } \quad n \in \mathbb { N } \end{aligned}$$

Show that this sequence is periodic.
State the order of this sequence.
Hence find $$\sum _ { n = 1 } ^ { 70 } u _ { n }$$

Edexcel P2 2024 January Q6

(a) Given that

$$2 \log _ { 4 } ( x + 3 ) + \log _ { 4 } x = \log _ { 4 } ( 4 x + 2 ) + \frac { 1 } { 2 }$$ show that $$x ^ { 3 } + 6 x ^ { 2 } + x - 4 = 0$$ (b) Given also that - 1 is a root of the equation $$x ^ { 3 } + 6 x ^ { 2 } + x - 4 = 0$$

use algebra to find the other two roots of the equation.
Hence solve $$2 \log _ { 4 } ( x + 3 ) + \log _ { 4 } x = \log _ { 4 } ( 4 x + 2 ) + \frac { 1 } { 2 }$$

Edexcel P2 2024 January Q7

Wheat is grown on a farm.

In year 1 , the farm produced 300 tonnes of wheat.
In year 12 , the farm is predicted to produce 4000 tonnes of wheat.

Model $A$ assumes that the amount of wheat produced on the farm will increase by the same amount each year.

Using model $A$, find the amount of wheat produced on the farm in year 4. Give your answer to the nearest 10 tonnes. Model $B$ assumes that the amount of wheat produced on the farm will increase by the same percentage each year.
Using model $B$, find the amount of wheat produced on the farm in year 2. Give your answer to the nearest 10 tonnes.
Calculate, according to the two models, the difference between the total amounts of wheat predicted to be produced on the farm from year 1 to year 12 inclusive. Give your answer to the nearest 10 tonnes.

Edexcel P2 2024 January Q8

(i) Use a counter example to show that the following statement is false

$$\text { " } n ^ { 2 } + 3 n + 1 \text { is prime for all } n \in \mathbb { N } \text { " }$$ (ii) Use algebra to prove by exhaustion that for all $n \in \mathbb { N }$ $$\text { " } n ^ { 2 } - 2 \text { is not a multiple of } 4 \text { " }$$

Edexcel P2 2024 January Q9

In this question you must show detailed reasoning.

Solutions relying entirely on calculator technology are not acceptable.

Solve, for $0 \leqslant x < 360 ^ { \circ }$, the equation $$\sin x \tan x = 5$$ giving your answers to one decimal place.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0e08d931-aa1c-48a8-8b39-47096f981950-26_643_736_721_660} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of part of the curve with equation $$y = A \sin \left( 2 \theta - \frac { 3 \pi } { 8 } \right) + 2$$ where $A$ is a constant and $\theta$ is measured in radians.
The points $P , Q$ and $R$ lie on the curve and are shown in Figure 1.
Given that the $y$ coordinate of $P$ is 7
(a) state the value of $A$,
(b) find the exact coordinates of $Q$,
(c) find the value of $\theta$ at $R$, giving your answer to 3 significant figures.

Questions — Edexcel P2 (157 questions)

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