Questions - Edexcel P2 - A-Level Maths

the common ratio is $r$
$u _ { 2 } + u _ { 3 } = 6$
$u _ { 4 } = 8$
1. Show that $r$ satisfies

$$3 r ^ { 2 } - 4 r - 4 = 0$$ Given that the geometric sequence has a sum to infinity,

find $u _ { 1 }$

find $S _ { \infty }$ 7. $$f ( x ) = A x ^ { 3 } + 6 x ^ { 2 } - 4 x + B$$ where $A$ and $B$ are constants. Given that

( $x + 2$ ) is a factor of $\mathrm { f } ( x )$
$\int _ { 3 } ^ { 5 } \mathrm { f } ( x ) \mathrm { d } x = 176$
find the value of $A$ and the value of $B$.

Edexcel P2 2020 January Q1

The table below shows corresponding values of $x$ and $y$ for $y = \log _ { 2 } ( 2 x )$

The values of $y$ are given to 2 decimal places as appropriate. Using the trapezium rule with all the values of $y$ in the given table,

obtain an estimate for $\int _ { 2 } ^ { 14 } \log _ { 2 } ( 2 x ) \mathrm { d } x$, giving your answer to one decimal place. Using your answer to part (a) and making your method clear, estimate
1. $\quad \int _ { 2 } ^ { 14 } \frac { \log _ { 2 } \left( 4 x ^ { 2 } \right) } { 5 } \mathrm {~d} x$
2. $\int _ { 2 } ^ { 14 } \log _ { 2 } \left( \frac { 2 } { x } \right) \mathrm { d } x$
  $x$ 2 5 8 11 14
  $y$ 2 3.32 4 4.46 4.81

Edexcel P2 2020 January Q2

2. One of the terms in the binomial expansion of $( 3 + a x ) ^ { 6 }$, where $a$ is a constant, is $540 x ^ { 4 }$

Find the possible values of $a$.
Hence find the term independent of $x$ in the expansion of $$\left( \frac { 1 } { 81 } + \frac { 1 } { x ^ { 6 } } \right) ( 3 + a x ) ^ { 6 }$$

Edexcel P2 2020 January Q3

3. $$f ( x ) = 6 x ^ { 3 } + 17 x ^ { 2 } + 4 x - 12$$

Use the factor theorem to show that ( $2 x + 3$ ) is a factor of $\mathrm { f } ( x )$.
Hence, using algebra, write $\mathrm { f } ( x )$ as a product of three linear factors.
Solve, for $\frac { \pi } { 2 } < \theta < \pi$, the equation $$6 \tan ^ { 3 } \theta + 17 \tan ^ { 2 } \theta + 4 \tan \theta - 12 = 0$$ giving your answers to 3 significant figures.

Edexcel P2 2020 January Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{08aac50c-7317-4510-927a-7f5f2e00f485-08_858_654_118_671} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of the curve with equation $$y = 2 x ^ { 2 } + 7 \quad x \geqslant 0$$ The finite region $R$, shown shaded in Figure 1, is bounded by the curve, the $y$-axis and the line with equation $y = 17$ Find the exact area of $R$.

Edexcel P2 2020 January Q5

5. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. A colony of bees is being studied. The number of bees in the colony at the start of the study was 30000 Three years after the start of the study, the number of bees in the colony is 34000 A model predicts that the number of bees in the colony will increase by $p \%$ each year, so that the number of bees in the colony at the end of each year of study forms a geometric sequence. Assuming the model,

find the value of $p$, giving your answer to 2 decimal places. According to the model, at the end of $N$ years of study the number of bees in the colony exceeds 75000
Find, showing all steps in your working, the smallest integer value of $N$.

Edexcel P2 2020 January Q6

6. The circle $C$ has equation $$x ^ { 2 } + y ^ { 2 } + 6 x - 4 y - 14 = 0$$

Find
1. the coordinates of the centre of $C$,
2. the exact radius of $C$. The line with equation $y = k$, where $k$ is a constant, is a tangent to $C$.
Find the possible values of $k$. The line with equation $y = p$, where $p$ is a negative constant, is a chord of $C$.
Given that the length of this chord is 4 units,
find the value of $p$.

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

Edexcel P2 2020 January Q7

7. (a) Show that the equation $$8 \tan \theta = 3 \cos \theta$$ may be rewritten in the form $$3 \sin ^ { 2 } \theta + 8 \sin \theta - 3 = 0$$ (b) Hence solve, for $0 \leqslant x \leqslant 90 ^ { \circ }$, the equation $$8 \tan 2 x = 3 \cos 2 x$$ giving your answers to 2 decimal places.

Edexcel P2 2020 January Q8

8. (i) An arithmetic series has first term $a$ and common difference $d$. Prove that the sum to $n$ terms of this series is $$\frac { n } { 2 } \{ 2 a + ( n - 1 ) d \}$$ (ii) A sequence $u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots$ is given by $$u _ { n } = 5 n + 3 ( - 1 ) ^ { n }$$ Find the value of

$u _ { 5 }$
$\sum _ { n = 1 } ^ { 59 } u _ { n }$

Edexcel P2 2020 January Q9

9. (a) Sketch the curve with equation $$y = 3 \times 4 ^ { x }$$ showing the coordinates of any points of intersection with the coordinate axes. The curve with equation $y = 6 ^ { 1 - x }$ meets the curve with equation $y = 3 \times 4 ^ { x }$ at the point $P$.
(b) Show that the $x$ coordinate of $P$ is $\frac { \log _ { 10 } 2 } { \log _ { 10 } 24 }$

VIXV SIHIANI III IM IONOO

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Edexcel P2 2020 January Q10

10. A curve $C$ has equation $$y = 4 x ^ { 3 } - 9 x + \frac { k } { x } \quad x > 0$$ where $k$ is a constant.
The point $P$ with $x$ coordinate $\frac { 1 } { 2 }$ lies on $C$.
Given that $P$ is a stationary point of $C$,

show that $k = - \frac { 3 } { 2 }$
Determine the nature of the stationary point at $P$, justifying your answer. The curve $C$ has a second stationary point.
Using algebra, find the $x$ coordinate of this second stationary point.
\includegraphics[max width=\textwidth, alt={}, center]{08aac50c-7317-4510-927a-7f5f2e00f485-26_2255_50_312_1980}

Edexcel P2 2021 January Q1

1. $$f ( x ) = x ^ { 4 } + a x ^ { 3 } - 3 x ^ { 2 } + b x + 5$$ where $a$ and $b$ are constants.
When $\mathrm { f } ( x )$ is divided by ( $x + 1$ ), the remainder is 4

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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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.
\includegraphics[max width=\textwidth, alt={}, center]{52c90d0e-a5e4-45fa-95a4-9523287e7588-20_769_1038_116_450}

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 $$\log _ { 3 } ( 32 - 12 x ) = 2 \log _ { 3 } ( 1 - x ) + 3$$

\(x\)	2	5	8	11	14
\(y\)	2	3.32	4	4.46	4.81

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