Questions P2 - A-Level Maths

Show that $a + b = - 1$ When $\mathrm { f } ( x )$ is divided by ( $x - 2$ ), the remainder is - 23
Find the value of $a$ and the value of $b$.

Edexcel P2 2021 January Q2

2. A curve has equation $$y = x ^ { 3 } - x ^ { 2 } - 16 x + 2$$

Using calculus, find the $x$ coordinates of the stationary points of the curve.
Justify, by further calculus, the nature of all of the stationary points of the curve.

Edexcel P2 2021 January Q3

3. (i) Solve $$7 ^ { x + 2 } = 3$$ giving your answer in the form $x = \log _ { 7 } a$ where $a$ is a rational number in its simplest form.
(ii) Using the laws of logarithms, solve $$1 + \log _ { 2 } y + \log _ { 2 } ( y + 4 ) = \log _ { 2 } ( 5 - y )$$

Edexcel P2 2021 January Q4

4. (a) Find the first three terms, in ascending powers of $x$, of the binomial expansion of $$( 2 + p x ) ^ { 6 }$$ where $p$ is a constant. Give each term in simplest form. Given that in the expansion of $$\left( 3 - \frac { 1 } { 2 } x \right) ( 2 + p x ) ^ { 6 }$$ the coefficient of $x ^ { 2 }$ is $- \frac { 3 } { 4 }$
(b) find the possible values of $p$.
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VIXV SIHIANI III IM IONOO

VIAV SIHI NI JYHAM ION OO

VI4V SIHI NI JLIYM ION OO

Edexcel P2 2021 January Q5

5. (i) Use algebra to prove that for all $x \geqslant 0$ $$3 x + 1 \geqslant 2 \sqrt { 3 x }$$ (ii) Show that the following statement is not true.
"The sum of three consecutive prime numbers is always a multiple of 5 "

Edexcel P2 2021 January Q6

(a) Show that the equation

$$\frac { 3 \sin \theta \cos \theta } { 2 \sin \theta - 1 } = 5 \tan \theta \quad \sin \theta \neq \frac { 1 } { 2 }$$ can be written in the form $$3 \sin ^ { 3 } \theta + 10 \sin ^ { 2 } \theta - 8 \sin \theta = 0$$ (b) Hence solve, for $- \frac { \pi } { 4 } < x < \frac { \pi } { 4 }$ $$\frac { 3 \sin 2 x \cos 2 x } { 2 \sin 2 x - 1 } = 5 \tan 2 x$$ giving your answers to 3 decimal places where appropriate.

Edexcel P2 2021 January Q7

7. Figure 1 Solar panels are installed on the roof of a building. The power, $P$, produced on a particular day, in kW , can be modelled by the equation $$P = 0.95 + 2 ^ { t - 12 } + 2 ^ { 12 - t } - ( t - 12 ) ^ { 2 } \quad 8.5 \leqslant t \leqslant 15.2$$ where $t$ is the time in hours after midnight. The graph of $P$ against $t$ is shown in Figure 1. A table of values of $t$ and $P$ is shown below, with the values of $P$ given to 4 significant figures where appropriate.

Use the given equation to complete the table, giving the values of $P$ to 4 significant figures where appropriate. The amount of energy, in kWh , produced between 10:00 and 12:00 can be found by calculating the area of region $R$, shown shaded in Figure 1.
Use the trapezium rule, with all the values of $P$ in the completed table, to find an estimate for the amount of energy produced between 10:00 and 12:00. Give your answer to 2 decimal places.
7.
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Edexcel P2 2021 January Q8

8. A sequence $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots$ is defined by $$\begin{gathered} a _ { n + 1 } = 2 \left( a _ { n } + 3 \right) ^ { 2 } - 7
a _ { 1 } = p - 3 \end{gathered}$$ where $p$ is a constant.

Find an expression for $a _ { 2 }$ in terms of $p$, giving your answer in simplest form. Given that $\sum _ { n = 1 } ^ { 3 } a _ { n } = p + 15$
find the possible values of $a _ { 2 }$

VIXV SIHIANI III IM IONOO VIAV SIHI NI JYHAM ION OO VI4V SIHI NI JLIYM ION OO

Edexcel P2 2021 January Q9

9. A circle $C$ has equation $$( x - k ) ^ { 2 } + ( y - 2 k ) ^ { 2 } = k + 7$$ where $k$ is a positive constant.

Write down, in terms of $k$,
1. the coordinates of the centre of $C$,
2. the radius of $C$. Given that the point $P ( 2,3 )$ lies on $C$
1. show that $5 k ^ { 2 } - 17 k + 6 = 0$
2. hence find the possible values of $k$. The tangent to the circle at $P$ intersects the $x$-axis at point $T$.
  Given that $k < 2$
calculate the exact area of triangle $O P T$.

Edexcel P2 2021 January Q10

10. In this question you must show detailed reasoning. Owen wants to train for 12 weeks in preparation for running a marathon. During the 12-week period he will run every Sunday and every Wednesday.

On Sunday in week 1 he will run 15 km
On Sunday in week 12 he will run 37 km

He considers two different 12-week training plans. In training plan $A$, he will increase the distance he runs each Sunday by the same amount.

Calculate the distance he will run on Sunday in week 5 under training plan $A$. In training plan $B$, he will increase the distance he runs each Sunday by the same percentage.
Calculate the distance he will run on Sunday in week 5 under training plan $B$. Give your answer in km to one decimal place. Owen will also run a fixed distance, $x \mathrm {~km}$, each Wednesday over the 12-week period. Given that
- $x$ is an integer
- the total distance that Owen will run on Sundays and Wednesdays over the 12 weeks will not exceed 360 km
- 1. find the maximum value of $x$, if he uses training plan $A$,
  2. find the maximum value of $x$, if he uses training plan $B$.
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Edexcel P2 2022 January Q1

The table below shows corresponding values of $x$ and $y$ for

$$y = 2 ^ { 5 - \sqrt { x } }$$ The values of $y$ are given to 3 decimal places.

$x$	5	5.5	6	6.5	7
$y$	6.792	6.298	5.858	5.466	5.113

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

obtain an estimate for $$\int _ { 5 } ^ { 7 } 2 ^ { 5 - \sqrt { x } } \mathrm {~d} x$$ giving your answer to 2 decimal places.
Using your answer to part (a) and making your method clear, estimate
1. $\quad \int _ { 5 } ^ { 7 } 2 ^ { 6 - \sqrt { x } } \mathrm {~d} x$
2. $\int _ { 5 } ^ { 7 } \left( 3 + 2 ^ { 5 - \sqrt { x } } \right) \mathrm { d } x$

Edexcel P2 2022 January Q2

2. In this question you must show all stages of your working. \section*{Solutions relying entirely on calculator technology are not acceptable.} The curve $C$ has equation $$y = 27 x ^ { \frac { 1 } { 2 } } - x ^ { \frac { 3 } { 2 } } - 20 \quad x > 0$$

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$, giving each term in simplest form.
Hence find the coordinates of the stationary point of $C$.
Find $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ and hence determine the nature of the stationary point of $C$.

Edexcel P2 2022 January Q3

3. (a) Find the first 4 terms, in ascending powers of $x$, of the binomial expansion of $$\left( 2 - \frac { k x } { 4 } \right) ^ { 8 }$$ where $k$ is a non-zero constant. Give each term in simplest form. $$f ( x ) = ( 5 - 3 x ) \left( 2 - \frac { k x } { 4 } \right) ^ { 8 }$$ In the expansion of $\mathrm { f } ( x )$, the constant term is 3 times the coefficient of $x$.
(b) Find the value of $k$.
T

Edexcel P2 2022 January Q4

4. Using the laws of logarithms, solve $$\log _ { 3 } ( 32 - 12 x ) = 2 \log _ { 3 } ( 1 - x ) + 3$$

Edexcel P2 2022 January Q5

5. $$f ( x ) = 3 x ^ { 3 } + A x ^ { 2 } + B x - 10$$ where $A$ and $B$ are integers.
Given that

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 $$f ( x ) = ( x - 3 ) g ( x )$$

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