Questions - Edexcel - A-Level Maths

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Edexcel AS Paper 2 2024 June Q2

3 marks Easy -1.8

Keith is studying the variable Daily Mean Wind Direction, in degrees, from the large data set.

Keith summarised the data for Camborne from 1987 into 4 directions $A , B , C$ and $D$ representing North, South, East and West in some order.

Direction	$A$	$B$	$C$	$D$
Frequency	22	48	56	58

Using your knowledge of the large data set state, giving a reason, which direction $A$ represents. The entry for Hurn on 27th September 1987 was 999
State, giving a reason, what Keith should do with this value.

Edexcel AS Paper 2 2024 June Q3

7 marks Moderate -0.3

Customers in a shop have to queue to pay.

The partially completed table below and partially completed histogram opposite, give information about the time, $x$ minutes, spent in the queue by each of 112 customers one day.

Time in queue ( $\boldsymbol { x }$ minutes)	Frequency
$1 - 2$	64
$2 - 3$
$3 - 4$	13
$4 - 6$
$6 - 8$	3

No customer spent less than 1 minute or longer than 8 minutes in the queue.

Complete the table.
Complete the histogram. Ting decides to model the frequency density for these 112 customers by a curve with equation $$y = \frac { k } { x ^ { 2 } } \quad 1 \leqslant x \leqslant 8$$ where $k$ is a constant.
Find the value of $k$
\includegraphics[max width=\textwidth, alt={}]{6a0b46f8-7a6a-4ed8-8c7a-9772787f155a-07_1584_1189_285_443}
Only use this grid if you need to redraw your histogram. \includegraphics[max width=\textwidth, alt={}, center]{6a0b46f8-7a6a-4ed8-8c7a-9772787f155a-09_1582_1192_367_440} \includegraphics[max width=\textwidth, alt={}, center]{6a0b46f8-7a6a-4ed8-8c7a-9772787f155a-09_2267_51_307_36}

Edexcel AS Paper 2 2024 June Q4

8 marks Moderate -0.3

4. The random variable $X \sim \mathrm {~B} ( 27,0.35 )$

Find
1. $\mathrm { P } ( X = 10 )$
2. $\mathrm { P } ( 12 \leqslant X < 15 )$ Historical records show that the proportion of defective items produced by a machine is 0.12 Following a maintenance service of the machine, a random sample of 60 items is taken and 3 defective items are found.
Carry out a suitable test to determine whether the proportion of defective items produced by the machine has decreased following the maintenance service. You should state your hypotheses clearly and use a $5 \%$ level of significance.
Write down the $p$-value for your test in part (b)

Edexcel AS Paper 2 2024 June Q5

8 marks Standard +0.3

A biased 4 -sided spinner has the numbers $6,7,8$ and 10 on it.

The discrete random variable $X$ represents the score when the spinner is spun once and has the following probability distribution,

$x$	6	7	8	10
$\mathrm { P } ( X = x )$	0.5	0.2	$q$	$q$

where $q$ is a probability.

Find the value of $q$ Karen spins the spinner repeatedly until she either gets a 7 or she has taken 4 spins.
Show that the probability that Karen stops after taking her 3rd spin is 0.128 The random variable $S$ represents the number of spins Karen takes.
Find the probability distribution for $S$ The random variable $N$ represents the number of times Karen gets a 7
Find $\mathrm { P } ( S > N )$

Edexcel AS Paper 2 2021 November Q1

7 marks Moderate -0.8

At time $t = 0$, a small stone is thrown vertically upwards with speed $14.7 \mathrm {~ms} ^ { - 1 }$ from a point $A$.

At time $t = T$ seconds, the stone passes through $A$, moving downwards.
The stone is modelled as a particle moving freely under gravity throughout its motion.
Using the model,

find the value of $T$,
find the total distance travelled by the stone in the first 4 seconds of its motion.
State one refinement that could be made to the model, apart from air resistance, that would make the model more realistic.

Edexcel AS Paper 2 2021 November Q2

10 marks Moderate -0.8

A particle $P$ moves along a straight line.

At time $t$ seconds, the velocity $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ of $P$ is modelled as $$v = 10 t - t ^ { 2 } - k \quad t \geqslant 0$$ where $k$ is a constant.

Find the acceleration of $P$ at time $t$ seconds. The particle $P$ is instantaneously at rest when $t = 6$
Find the other value of $t$ when $P$ is instantaneously at rest.
Find the total distance travelled by $P$ in the interval $0 \leqslant t \leqslant 6$

Edexcel AS Paper 2 2021 November Q3

13 marks Moderate -0.3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{4a022ec0-7640-4664-87a6-1963309cad6a-08_761_595_210_735} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A ball $P$ of mass $2 m$ is attached to one end of a string.
The other end of the string is attached to a ball $Q$ of mass $5 m$.
The string passes over a fixed pulley.
The system is held at rest with the balls hanging freely and the string taut.
The hanging parts of the string are vertical with $P$ at a height $2 h$ above horizontal ground and with $Q$ at a height $h$ above the ground, as shown in Figure 1. The system is released from rest.
In the subsequent motion, $Q$ does not rebound when it hits the ground and $P$ does not hit the pulley. The balls are modelled as particles.
The string is modelled as being light and inextensible.
The pulley is modelled as being small and smooth.
Air resistance is modelled as being negligible.
Using this model,

1. write down an equation of motion for $P$,
2. write down an equation of motion for $Q$,
find, in terms of $h$ only, the height above the ground at which $P$ first comes to instantaneous rest.
State one limitation of modelling the balls as particles that could affect your answer to part (b). In reality, the string will not be inextensible.
State how this would affect the accelerations of the particles.
VIAV SIHI NI III IM ION OC VIIN SIHI NI III M M O N OO VIIV SIHI NI IIIYM ION OC

Edexcel AS Paper 2 2021 November Q1

2 marks Easy -1.2

1. \includegraphics[max width=\textwidth, alt={}, center]{6dfefd72-338f-40be-ac37-aef56bfaccaa-02_399_743_248_662} The Venn diagram, where $p$ is a probability, shows the 3 events $A , B$ and $C$ with their associated probabilities.

Find the value of $p$.
Write down a pair of mutually exclusive events from $A , B$ and $C$.

Edexcel AS Paper 2 2021 November Q2

9 marks Moderate -0.3

The partially completed table and partially completed histogram give information about the ages of passengers on an airline.

There were no passengers aged 90 or over.

Age ( $x$ years)	$0 \leqslant x < 5$	$5 \leqslant x < 20$	$20 \leqslant x < 40$	$40 \leqslant x < 65$	$65 \leqslant x < 80$	$80 \leqslant x < 90$
Frequency	5	45	90			1

\includegraphics[max width=\textwidth, alt={}, center]{6dfefd72-338f-40be-ac37-aef56bfaccaa-04_1173_1792_721_139}

Complete the histogram.
Use linear interpolation to estimate the median age. An outlier is defined as a value greater than $Q _ { 3 } + 1.5 \times$ interquartile range.
Given that $Q _ { 1 } = 27.3$ and $Q _ { 3 } = 58.9$
determine, giving a reason, whether or not the oldest passenger could be considered as an outlier.
(2)

Edexcel AS Paper 2 2021 November Q3

4 marks Easy -1.8

Helen is studying one of the qualitative variables from the large data set for Heathrow from 2015.

She started with the data from 3rd May and then took every 10th reading.
There were only 3 different outcomes with the following frequencies

Outcome	$A$	$B$	$C$
Frequency	16	2	1

State the sampling technique Helen used.
From your knowledge of the large data set
1. suggest which variable was being studied,
2. state the name of outcome $A$. George is also studying the same variable from the large data set for Heathrow from 2015. He started with the data from 5th May and then took every 10th reading and obtained the following
  Outcome $A$ $B$ $C$
  Frequency 16 1 1
  Helen and George decided they should examine all of the data for this variable for Heathrow from 2015 and obtained the following
  Outcome $A$ $B$ $C$
  Frequency 155 26 3
(c) State what inference Helen and George could reliably make from their original samples about the outcomes of this variable at Heathrow, for the period covered by the large data set in 2015.

Edexcel AS Paper 2 2021 November Q4

10 marks Standard +0.3

A nursery has a sack containing a large number of coloured beads of which $14 \%$ are coloured red.

Aliya takes a random sample of 18 beads from the sack to make a bracelet.

State a suitable binomial distribution to model the number of red beads in Aliya's bracelet.
Use this binomial distribution to find the probability that
1. Aliya has just 1 red bead in her bracelet,
2. there are at least 4 red beads in Aliya's bracelet.
Comment on the suitability of a binomial distribution to model this situation. After several children have used beads from the sack, the nursery teacher decides to test whether or not the proportion of red beads in the sack has changed. She takes a random sample of 75 beads and finds 4 red beads.
Stating your hypotheses clearly, use a 5\% significance level to carry out a suitable test for the teacher.
Find the $p$-value in this case.

Edexcel AS Paper 2 2021 November Q5

5 marks Standard +0.8

Two bags, $\mathbf { A }$ and $\mathbf { B }$, each contain balls which are either red or yellow or green.

Bag A contains 4 red, 3 yellow and $n$ green balls.
Bag $\mathbf { B }$ contains 5 red, 3 yellow and 1 green ball.
A ball is selected at random from bag $\mathbf { A }$ and placed into bag $\mathbf { B }$.
A ball is then selected at random from bag $\mathbf { B }$ and placed into bag $\mathbf { A }$.
The probability that bag $\mathbf { A }$ now contains an equal number of red, yellow and green balls is $p$. Given that $p > 0$, find the possible values of $n$ and $p$.

Edexcel PMT Mocks Q1

6 marks Standard +0.3

a. Find the first four terms, in ascending powers of $x$, of the binomial expansion of

$$\left( \frac { 1 } { 9 } - 2 x \right) ^ { \frac { 1 } { 2 } }$$ giving each coefficient in its simplest form.
b. Explain how you could use $x = \frac { 1 } { 36 }$ in the expansion to find an approximation for $\sqrt { 2 }$. There is no need to carry out the calculation.

Edexcel PMT Mocks Q2

3 marks Moderate -0.8

2. The curves $C _ { 1 }$ and $C _ { 2 }$ have equations $$\begin{aligned} & C _ { 1 } : \quad y = 2 ^ { 3 x + 2 } \\ & C _ { 2 } : \quad y = 4 ^ { - x } \end{aligned}$$ Show that the $x$-coordinate of the point where $C _ { 1 }$ and $C _ { 2 }$ intersect is $\frac { - 2 } { 5 }$.

Edexcel PMT Mocks Q3

5 marks Moderate -0.8

3. Relative to a fixed origin,

point $A$ has position vector $- 2 \mathbf { i } + 4 \mathbf { j } + 7 \mathbf { k }$
point $B$ has position vector $- \mathbf { i } + 3 \mathbf { j } + 8 \mathbf { k }$
point $C$ has position vector $\mathbf { i } + \mathbf { j } + 4 \mathbf { k }$
point $D$ has position vector $- \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k }$ a. Show that $\overrightarrow { A B }$ and $\overrightarrow { C D }$ are parallel and the ratio $\overrightarrow { A B } : \overrightarrow { C D }$ in its simplest form.
b. Hence describe the quadrilateral $A B C D$.

Edexcel PMT Mocks Q4

6 marks Moderate -0.8

Ben starts a new company.

In year 1 his profits will be $\pounds 24000$.
In year 11 his profit is predicted to be $\pounds 64000$.

Model $\boldsymbol { P }$ assumes that his profit will increase by the same amount each year.
a. According to model $\boldsymbol { P }$, determine Ben's profit in year 5. Model $\boldsymbol { Q }$ assumes that his profit will increase by the same percentage each year.
b. According to model $\boldsymbol { Q }$, determine Ben's profit in year 5 . Give your answer to the nearest £10.

A sequence $a _ { 1 } , a _ { 2 } a _ { 3 } , \ldots$ is defined by

$$a _ { n + 1 } = 5 - p a _ { n } \quad n \geq 1$$ where $p$ is a constant.
Given that

$a _ { 1 } = 4$
the sequence is a periodic sequence of order 2.
a. Write down an expression for $a _ { 2 }$ and $a _ { 3 }$.
b. Find the value of $p$.
c. Find $\sum _ { r = 1 } ^ { 21 } a _ { r }$

Time in queue ( \(\boldsymbol { x }\) minutes)	Frequency
\(1 - 2\)	64
\(2 - 3\)
\(3 - 4\)	13
\(4 - 6\)
\(6 - 8\)	3

\(x\)	6	7	8	10
\(\mathrm { P } ( X = x )\)	0.5	0.2	\(q\)	\(q\)

Age ( \(x\) years)	\(0 \leqslant x < 5\)	\(5 \leqslant x < 20\)	\(20 \leqslant x < 40\)	\(40 \leqslant x < 65\)	\(65 \leqslant x < 80\)	\(80 \leqslant x < 90\)
Frequency	5	45	90			1

Direction	\(A\)	\(B\)	\(C\)	\(D\)
Frequency	22	48	56	58

Outcome	\(A\)	\(B\)	\(C\)
Frequency	16	2	1

Outcome	\(A\)	\(B\)	\(C\)
Frequency	16	1	1

Outcome	\(A\)	\(B\)	\(C\)
Frequency	155	26	3

Questions — Edexcel (10514 questions)

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Edexcel AS Paper 2 2024 June Q2

Edexcel AS Paper 2 2024 June Q3

Edexcel AS Paper 2 2024 June Q4

Edexcel AS Paper 2 2024 June Q5

Edexcel AS Paper 2 2021 November Q1

Edexcel AS Paper 2 2021 November Q2

Edexcel AS Paper 2 2021 November Q3

Edexcel AS Paper 2 2021 November Q1

Edexcel AS Paper 2 2021 November Q2

Edexcel AS Paper 2 2021 November Q3

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Edexcel AS Paper 2 2021 November Q5

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