Questions - Edexcel - A-Level Maths

Edexcel AEA 2024 June Q3

14 marks Challenging +1.8

3.(i)Determine the value of $k$ such that $$\arctan \frac { 1 } { 2 } - \arctan \frac { 1 } { 3 } = \arctan k$$ (ii)(a)Prove that $$\cos 3 A \equiv 4 \cos ^ { 3 } A - 3 \cos A$$ Given that $a = \cos 20 ^ { \circ }$ (b)write down,in terms of $a$ ,an expression for $\cos 40 ^ { \circ }$ (c)determine,in terms of $a$ ,a simplified expression for $\cos 80 ^ { \circ }$ (d)Use part(a)to show that $$4 a ^ { 3 } - 3 a = \frac { 1 } { 2 }$$ (e)Hence,or otherwise,show that $$\cos 20 ^ { \circ } \cos 40 ^ { \circ } \cos 80 ^ { \circ } = \frac { 1 } { 8 }$$

Edexcel AEA 2024 June Q4

16 marks Challenging +1.8

4.(a)Use the substitution $x = \sqrt { 3 } \tan u$ to show that $$\int \frac { 1 } { 3 + x ^ { 2 } } \mathrm {~d} x = p \arctan ( p x ) + c$$ where $p$ is a real constant to be determined and $c$ is an arbitrary constant.
(b)Use the substitution $x = \frac { 3 u + 3 } { u - 3 }$ to determine the exact value of $I$ where $$I = \int _ { - 3 } ^ { 1 } \frac { \ln ( 3 - x ) } { 3 + x ^ { 2 } } \mathrm {~d} x$$ giving your answer in simplest form. \includegraphics[max width=\textwidth, alt={}, center]{a8e9db6b-dfad-4278-82d8-a8fa5ba61008-10_2264_47_314_1984}

Edexcel AEA 2024 June Q5

15 marks Challenging +1.8

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{a8e9db6b-dfad-4278-82d8-a8fa5ba61008-14_300_1043_251_513} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows a sketch of a hexagon $O A B C D E$ where
-the interior angle at $O$ and at $C$ are each $60 ^ { \circ }$ -the interior angle at each of the other vertices is $150 ^ { \circ }$ -$O A = O E = B C = C D$ -$A B = E D = 3 \times O A$ Given that $\overrightarrow { O A } = \mathbf { a }$ and $\overrightarrow { O E } = \mathbf { e }$ determine as simplified expressions in terms of $\mathbf { a }$ and $\mathbf { e }$

$\overrightarrow { A B }$
$\overrightarrow { O D }$ The point $R$ divides $A B$ internally in the ratio $1 : 2$
Determine $\overrightarrow { R C }$ as a simplified expression in terms of $\mathbf { a }$ and $\mathbf { e }$ The line through the points $R$ and $C$ meets the line through the points $O$ and $D$ at the point $X$ .
Determine $\overrightarrow { O X }$ in the form $\lambda \mathbf { a } + \mu \mathbf { e }$ ,where $\lambda$ and $\mu$ are real values in simplest form.

Edexcel AEA 2024 June Q6

18 marks Hard +2.3

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{a8e9db6b-dfad-4278-82d8-a8fa5ba61008-20_234_1357_244_354} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} Figure 3 shows a block $A$ with mass $4 m$ and a block $B$ with mass $5 m$.
Block $A$ is at rest on a rough plane inclined at an angle $\alpha$ to the horizontal.
Block $B$ is at rest on a rough plane inclined at an angle $\beta$ to the horizontal.
The blocks are connected by a light inextensible string which passes over a small smooth pulley at the top of each plane. A small smooth ring $C$, of mass $8 m$, is threaded on the string between the pulleys so that $A , B$ and $C$ all lie in the same vertical plane. The part of the string between $A$ and its pulley lies along a line of greatest slope of the plane of angle $\alpha$. The part of the string between $B$ and its pulley lies along a line of greatest slope of the plane of angle $\beta$. The angle between the vertical and the string between each pulley and the ring $C$ is $\gamma$.
The two blocks, $A$ and $B$, are modelled as particles.
Given that

$\tan \alpha = \frac { 5 } { 12 }$ and $\tan \beta = \frac { 7 } { 24 }$ and $\tan \gamma = \frac { 3 } { 4 }$
the coefficient of friction, $\mu$, is the same between each block and its plane
one of the blocks is on the point of sliding up its plane
the tension in the string is $T$
1. determine, in terms of $m$ and $g$, an expression for $T$,
2. draw a diagram showing the forces on block $A$, clearly labelling each of the forces acting on the block,
3. determine the value of $\mu$, giving a justification for your answer. \includegraphics[max width=\textwidth, alt={}, center]{a8e9db6b-dfad-4278-82d8-a8fa5ba61008-20_2266_50_312_1978}

Edexcel AEA 2024 June Q7

24 marks Hard +2.3

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{a8e9db6b-dfad-4278-82d8-a8fa5ba61008-26_725_1773_242_146} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} Figure 4 shows a circle with radius $r _ { 1 }$ and a circle with radius $r _ { 2 }$ The circles touch externally at a single point above the $x$-axis.
Both circles also have the $x$-axis as a tangent.

Show that the horizontal distance between the centres of the circles, $d$, is given by $$d ^ { 2 } = 4 r _ { 1 } r _ { 2 }$$ The finite region $R$, shown shaded in Figure 4, is bounded by the $x$-axis and minor arcs of the two circles. Given that $r _ { 1 } \geqslant r _ { 2 }$
show that the area of $R$ is given by $$\left( r _ { 1 } + r _ { 2 } \right) \sqrt { r _ { 1 } r _ { 2 } } - \frac { 1 } { 2 } \left( r _ { 1 } ^ { 2 } - r _ { 2 } ^ { 2 } \right) \theta - \frac { 1 } { 2 } \pi r _ { 2 } ^ { 2 }$$ where $\cos \theta = \frac { r _ { 1 } - r _ { 2 } } { r _ { 1 } + r _ { 2 } }$ Question 7 continues on the next page.

\includegraphics[max width=\textwidth, alt={}]{a8e9db6b-dfad-4278-82d8-a8fa5ba61008-27_2269_53_306_36}
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a8e9db6b-dfad-4278-82d8-a8fa5ba61008-27_759_1378_269_347} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} A sequence of circles, $C _ { 1 } , C _ { 2 } , C _ { 3 } , \ldots$ with radii $r _ { 1 } , r _ { 2 } , r _ { 3 } , \ldots$ respectively, is constructed such that
- each circle is tangential to and above the $x$-axis
- the first circle, $C _ { 1 }$, has centre $( 0,1 )$
- each successive circle touches the preceding one externally at a single point
- the horizontal distances between the centres of successive circles form a geometric sequence with first term 2 and common ratio $\frac { 1 } { \sqrt { 3 } }$
The first few circles in the sequence are shown in Figure 5.
1. Determine the value of $r _ { 3 }$
2. Show that, for $n \geqslant 1 , r _ { n + 2 } = k r _ { n }$ where $k$ is a constant to be determined.
3. Hence show that, for $n \geqslant 1 , r _ { 2 n } = r _ { 2 n - 1 }$ The region bounded between $C _ { n } , C _ { n + 1 }$ and the $x$-axis is $R _ { n }$ The total area, $A$, bounded above the $x$-axis and under all the circles is the sum of the areas of all these regions.
Determine the value of $A$, giving the answer in simplest form. \section*{Paper reference} \section*{Advanced Extension Award Mathematics} Insert for questions 5, 6 and 7
Do not write on this insert.
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a8e9db6b-dfad-4278-82d8-a8fa5ba61008-34_298_1040_212_516} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of a hexagon $O A B C D E$ where
Given that $\overrightarrow { O A } = \mathbf { a }$ and $\overrightarrow { O E } = \mathbf { e }$ determine as simplified expressions in terms of $\mathbf { a }$ and $\mathbf { e }$
1. $\overrightarrow { A B }$
2. $\overrightarrow { O D }$ The point $R$ divides $A B$ internally in the ratio $1 : 2$
3. Determine $\overrightarrow { R C }$ as a simplified expression in terms of $\mathbf { a }$ and $\mathbf { e }$ The line through the points $R$ and $C$ meets the line through the points $O$ and $D$ at the point $X$.
4. Determine $\overrightarrow { O X }$ in the form $\lambda \mathbf { a } + \mu \mathbf { e }$, where $\lambda$ and $\mu$ are real values in simplest form.
  6. \begin{figure}[h]
  \includegraphics[alt={},max width=\textwidth]{a8e9db6b-dfad-4278-82d8-a8fa5ba61008-35_236_1363_205_351} \captionsetup{labelformat=empty} \caption{Figure 3}
  \end{figure} Figure 3 shows a block $A$ with mass $4 m$ and a block $B$ with mass $5 m$.
  Block $A$ is at rest on a rough plane inclined at an angle $\alpha$ to the horizontal.
  Block $B$ is at rest on a rough plane inclined at an angle $\beta$ to the horizontal.
  The blocks are connected by a light inextensible string which passes over a small smooth pulley at the top of each plane. A small smooth ring $C$, of mass $8 m$, is threaded on the string between the pulleys so that $A , B$ and $C$ all lie in the same vertical plane. The part of the string between $A$ and its pulley lies along a line of greatest slope of the plane of angle $\alpha$. The part of the string between $B$ and its pulley lies along a line of greatest slope of the plane of angle $\beta$. The angle between the vertical and the string between each pulley and the ring $C$ is $\gamma$.
  The two blocks, $A$ and $B$, are modelled as particles.
  Given that
  7. \begin{figure}[h]
  \includegraphics[alt={},max width=\textwidth]{a8e9db6b-dfad-4278-82d8-a8fa5ba61008-36_721_1771_205_146} \captionsetup{labelformat=empty} \caption{Figure 4}
  \end{figure} Figure 4 shows a circle with radius $r _ { 1 }$ and a circle with radius $r _ { 2 }$ The circles touch externally at a single point above the $x$-axis.
  Both circles also have the $x$-axis as a tangent.
5. Show that the horizontal distance between the centres of the circles, $d$, is given by $$d ^ { 2 } = 4 r _ { 1 } r _ { 2 }$$ The finite region $R$, shown shaded in Figure 4, is bounded by the $x$-axis and minor arcs of the two circles. Given that $r _ { 1 } \geqslant r _ { 2 }$
6. show that the area of $R$ is given by $$\left( r _ { 1 } + r _ { 2 } \right) \sqrt { r _ { 1 } r _ { 2 } } - \frac { 1 } { 2 } \left( r _ { 1 } ^ { 2 } - r _ { 2 } ^ { 2 } \right) \theta - \frac { 1 } { 2 } \pi r _ { 2 } ^ { 2 }$$ where $\cos \theta = \frac { r _ { 1 } - r _ { 2 } } { r _ { 1 } + r _ { 2 } }$ Question 7 continues on the next page. \begin{figure}[h]
  \includegraphics[alt={},max width=\textwidth]{a8e9db6b-dfad-4278-82d8-a8fa5ba61008-37_761_1376_210_349} \captionsetup{labelformat=empty} \caption{Figure 5}
  \end{figure} A sequence of circles, $C _ { 1 } , C _ { 2 } , C _ { 3 } , \ldots$ with radii $r _ { 1 } , r _ { 2 } , r _ { 3 } , \ldots$ respectively, is constructed such that
  The first few circles in the sequence are shown in Figure 5.
7. 1. Determine the value of $r _ { 3 }$
  2. Show that, for $n \geqslant 1 , r _ { n + 2 } = k r _ { n }$ where $k$ is a constant to be determined.
  3. Hence show that, for $n \geqslant 1 , r _ { 2 n } = r _ { 2 n - 1 }$ The region bounded between $C _ { n } , C _ { n + 1 }$ and the $x$-axis is $R _ { n }$ The total area, $A$, bounded above the $x$-axis and under all the circles is the sum of the areas of all these regions.
8. Determine the value of $A$, giving the answer in simplest form.

Edexcel AEA 2018 June Q1

5 marks Challenging +1.2

1.(a)Show that $\sqrt { \frac { 1 + x } { 1 - x } }$ can be written in the form $\frac { 1 + x } { \sqrt { 1 - x ^ { 2 } } }$ for $| x | < 1$ (b)Hence,or otherwise,find the expansion,in ascending powers of $x$ up to and including the term in $x ^ { 5 }$ ,of $\sqrt { \frac { 1 + x } { 1 - x } }$

Edexcel AEA 2018 June Q2

7 marks Challenging +1.8

2.Solve,for $0 \leqslant x \leqslant 360 ^ { \circ }$ $$\sin 47 ^ { \circ } \cos ^ { 3 } x + \cos 47 ^ { \circ } \sin x \cos ^ { 2 } x = \frac { 1 } { 2 } \cos ^ { 2 } x$$

Edexcel AEA 2018 June Q3

10 marks Challenging +1.2

3.The lines $L _ { 1 }$ and $L _ { 2 }$ have the equations $$L _ { 1 } : \mathbf { r } = \left( \begin{array} { l } 1 \\ 0 \\ 9 \end{array} \right) + s \left( \begin{array} { l } 2 \\ p \\ 6 \end{array} \right) \quad \text { and } \quad L _ { 2 } : \mathbf { r } = \left( \begin{array} { r } - 15 \\ 12 \\ - 9 \end{array} \right) + t \left( \begin{array} { r } 4 \\ - 5 \\ 2 \end{array} \right)$$ where $p$ is a constant.
The acute angle between $L _ { 1 }$ and $L _ { 2 }$ is $\theta$ where $\cos \theta = \frac { \sqrt { 5 } } { 3 }$

Find the value of $p$ . The line $L _ { 3 }$ has equation $\mathbf { r } = \left( \begin{array} { r } - 15 \\ 12 \\ - 9 \end{array} \right) + u \left( \begin{array} { r } 8 \\ - 6 \\ - 5 \end{array} \right)$ and the lines $L _ { 3 }$ and $L _ { 2 }$ intersect at the point $A$ .
The point $B$ on $L _ { 2 }$ has position vector $\left( \begin{array} { r } 5 \\ - 13 \\ 1 \end{array} \right)$ and point $C$ lies on $L _ { 3 }$ such that $A B D C$ is a rhombus.
Find the two possible position vectors of $D$ .

Edexcel AEA 2018 June Q4

13 marks Challenging +1.2

4.A curve $C$ has equation $y = \mathrm { f } ( x )$ where $x \in \mathbb { R }$ and f is a one-one function.

Describe a single transformation that transforms $C$ to the curve with equation $y = - \mathrm { f } ( - x )$ . The curve $C$ is reflected in the line with equation $y = - x$ to give the curve $V$ . The equation of $V$ is $y = \mathrm { g } ( x )$ .
Explain why $\mathrm { g } ^ { - 1 } ( x ) = - \mathrm { f } ( - x )$ . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2a7c2530-a93c-4a26-bc37-c20c0f40c8f2-3_793_979_819_633} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve $C$ with equation $y = \mathrm { f } ( x )$ where $$\mathrm { f } ( x ) = \frac { 3 ( x - 1 ) } { x - 2 } \quad x \in \mathbb { R } , x \neq 2$$ The curve has asymptotes with equations $x = p$ and $y = q$ and $C$ crosses the $x$-axis at the point $A$ and the $y$-axis at the point $B$ .
Write down the value of $p$ and the value of $q$ .
Write down the coordinates of the point $A$ and the coordinates of the point $B$ . Given the definition of $\mathrm { g } ( x )$ in part(b),
find the function g .
Solve $\mathrm { g } ^ { - 1 } \mathrm { f } ( x ) = x$

Edexcel AEA 2018 June Q5

14 marks Challenging +1.8

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{2a7c2530-a93c-4a26-bc37-c20c0f40c8f2-4_484_581_287_843} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows part of the curve $T$ with equation $y = \cos 2 x$ and the circle $C _ { 1 }$ that touches $T$ at $\left( \frac { \pi } { 4 } , 0 \right)$ and $\left( \frac { 3 \pi } { 4 } , 0 \right)$ .

Find the radius of $C _ { 1 }$ \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2a7c2530-a93c-4a26-bc37-c20c0f40c8f2-4_486_586_1199_841} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a sketch of part of $T$ and part of a circle $C _ { 2 }$ that touches $T$ at the point $P$ with coordinates $\left( \frac { \pi } { 2 } , - 1 \right)$ .For values of $x$ close to $\frac { \pi } { 2 }$ the curve $T$ lies inside $C _ { 2 }$ as shown in Figure 3.
Without doing any calculation,explain why the value of $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ for $C _ { 2 }$ at $P$ is less than the value of $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ for $T$ at $P$ . The radius of $C _ { 2 }$ is $r$ .
Use the result from part(b)to find a value of $k$ such that $r > k$ . Given that $C _ { 2 }$ cuts $T$ at the point $( 0,1 )$ ,
find the value of $r$ .

Edexcel AEA 2018 June Q6

17 marks Challenging +1.8

6. (a) Use the substitution $u = \sqrt { t }$ to show that $$\int _ { 1 } ^ { x } \frac { \ln t } { \sqrt { t } } \mathrm {~d} t = 4 - 4 \sqrt { x } + 2 \sqrt { x } \ln x \quad x \geqslant 1$$ (b) The function g is such that $$\int _ { 1 } ^ { x } \mathrm {~g} ( t ) \mathrm { d } t = x - \sqrt { x } \ln x - 1 \quad x \geqslant 1$$

Use differentiation to find the function g .
Evaluate $\int _ { 4 } ^ { 16 } \mathrm {~g} ( t ) \mathrm { d } t$ and simplify your answer.
(c) Find the value of $x$ (where $x > 1$ ) that gives the maximum value of $$\int _ { x } ^ { x + 1 } \frac { \ln t } { 2 ^ { t } } \mathrm {~d} t$$

Edexcel AEA 2018 June Q7

27 marks Hard +2.3

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{2a7c2530-a93c-4a26-bc37-c20c0f40c8f2-6_559_923_292_670} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} Figure 4 shows a shape $S ( \theta )$ made up of five line segments $A B , B C , C D , D E$ and $E A$ . The lengths of the sides are $A B = B C = 5 \mathrm {~cm} , C D = E A = 3 \mathrm {~cm}$ and $D E = 7 \mathrm {~cm}$ . Angle $B A E =$ angle $B C D = \theta$ radians. The length of each line segment always remains the same but the value of $\theta$ can be varied so that different symmetrical shapes can be formed,with the added restriction that none of the line segments cross.

Sketch $S ( \pi )$ ,labelling the vertices clearly. The shape $S ( \phi )$ is a trapezium.
Sketch $S ( \phi )$ and calculate the value of $\phi$ . The smallest possible value for $\theta$ is $\alpha$ ,where $\alpha > 0$ ,and the largest possible value for $\theta$ is $\beta$ , where $\beta > \pi$ .
Show that $\alpha = \arccos \left( \frac { 29 } { 40 } \right) \cdot \left[ \arccos ( x ) \right.$ is an alternative notation for $\left. \cos ^ { - 1 } ( x ) \right]$
Find the value of $\beta$ . The area,in $\mathrm { cm } ^ { 2 }$ ,of shape $S ( \theta )$ is $R ( \theta )$ .
Show that for $\alpha \leqslant \theta < \pi$ $$R ( \theta ) = 15 \sin \theta + \frac { 7 } { 4 } \sqrt { 87 - 120 \cos \theta }$$ Given that this formula for $R ( \theta )$ holds for $\alpha \leqslant \theta \leqslant \beta$
show that $R ( \theta )$ has only one stationary point and that this occurs when $\theta = \frac { 2 \pi } { 3 }$
find the maximum and minimum values of $R ( \theta )$. FOR STYLE, CLARITY AND PRESENTATION: 7 MARKS TOTAL FOR PAPER: 100 MARKS
END

Edexcel AS Paper 2 2018 June Q1

3 marks Moderate -0.8

A company is introducing a job evaluation scheme. Points ( $x$ ) will be awarded to each job based on the qualifications and skills needed and the level of responsibility. Pay ( $\pounds y$ ) will then be allocated to each job according to the number of points awarded.

Before the scheme is introduced, a random sample of 8 employees was taken and the linear regression equation of pay on points was $y = 4.5 x - 47$

Describe the correlation between points and pay.
Give an interpretation of the gradient of this regression line.
Explain why this model might not be appropriate for all jobs in the company.

Edexcel AS Paper 2 2018 June Q2

4 marks Moderate -0.3

A factory buys $10 \%$ of its components from supplier $A , 30 \%$ from supplier $B$ and the rest from supplier $C$. It is known that $6 \%$ of the components it buys are faulty.

Of the components bought from supplier $A , 9 \%$ are faulty and of the components bought from supplier $B , 3 \%$ are faulty.

Find the percentage of components bought from supplier $C$ that are faulty. A component is selected at random.
Explain why the event "the component was bought from supplier $B$ " is not statistically independent from the event "the component is faulty".

Edexcel AS Paper 2 2018 June Q3

7 marks Moderate -0.3

Naasir is playing a game with two friends. The game is designed to be a game of chance so that the probability of Naasir winning each game is $\frac { 1 } { 3 }$ Naasir and his friends play the game 15 times.

Find the probability that Naasir wins
1. exactly 2 games,
2. more than 5 games. Naasir claims he has a method to help him win more than $\frac { 1 } { 3 }$ of the games. To test this claim, the three of them played the game again 32 times and Naasir won 16 of these games.
Stating your hypotheses clearly, test Naasir's claim at the $5 \%$ level of significance.

Edexcel AS Paper 2 2018 June Q4

8 marks Moderate -0.8

Helen is studying the daily mean wind speed for Camborne using the large data set from 1987. The data for one month are summarised in Table 1 below.

\begin{table}[h]

Windspeed	$\mathrm { n } / \mathrm { a }$	6	7	8	9	11	12	13	14	16
Frequency	13	2	3	2	2	3	1	2	1	2

\captionsetup{labelformat=empty} \caption{Table 1}

\end{table}

Calculate the mean for these data.
Calculate the standard deviation for these data and state the units. The means and standard deviations of the daily mean wind speed for the other months from the large data set for Camborne in 1987 are given in Table 2 below. The data are not in month order. \begin{table}[h]
Month $A$ $B$ $C$ $D$ $E$
Mean 7.58 8.26 8.57 8.57 11.57
Standard Deviation 2.93 3.89 3.46 3.87 4.64
\captionsetup{labelformat=empty} \caption{Table 2}
\end{table}
Using your knowledge of the large data set, suggest, giving a reason, which month had a mean of 11.57 The data for these months are summarised in the box plots on the opposite page. They are not in month order or the same order as in Table 2.
1. State the meaning of the * symbol on some of the box plots.
2. Suggest, giving your reasons, which of the months in Table 2 is most likely to be summarised in the box plot marked $Y$. \includegraphics[max width=\textwidth, alt={}, center]{2edcf965-9c93-4a9b-9395-2d3c023801af-11_1177_1216_324_427}

Edexcel AS Paper 2 2018 June Q5

8 marks Moderate -0.3

5. A biased spinner can only land on one of the numbers $1,2,3$ or 4 . The random variable $X$ represents the number that the spinner lands on after a single spin and $\mathrm { P } ( X = r ) = \mathrm { P } ( X = r + 2 )$ for $r = 1,2$ Given that $\mathrm { P } ( X = 2 ) = 0.35$

find the complete probability distribution of $X$. Ambroh spins the spinner 60 times.
Find the probability that more than half of the spins land on the number 4 Give your answer to 3 significant figures. The random variable $Y = \frac { 12 } { X }$
Find $\mathrm { P } ( Y - X \leqslant 4 )$

Edexcel AS Paper 2 2018 June Q6

4 marks Moderate -0.8

A man throws a tennis ball into the air so that, at the instant when the ball leaves his hand, the ball is 2 m above the ground and is moving vertically upwards with speed $9 \mathrm {~m} \mathrm {~s} ^ { - 1 }$

The motion of the ball is modelled as that of a particle moving freely under gravity and the acceleration due to gravity is modelled as being of constant magnitude $10 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ The ball hits the ground $T$ seconds after leaving the man's hand.
Using the model, find the value of $T$.

Edexcel AS Paper 2 2018 June Q7

7 marks Moderate -0.3

A train travels along a straight horizontal track between two stations, $A$ and $B$.

In a model of the motion, the train starts from rest at $A$ and moves with constant acceleration $0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ for 80 s .
The train then moves at constant velocity before it moves with a constant deceleration of $0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }$, coming to rest at $B$.

For this model of the motion of the train between $A$ and $B$,
1. state the value of the constant velocity of the train,
2. state the time for which the train is decelerating,
3. sketch a velocity-time graph. The total distance between the two stations is 4800 m .
Using the model, find the total time taken by the train to travel from $A$ to $B$.
Suggest one improvement that could be made to the model of the motion of the train from $A$ to $B$ in order to make the model more realistic.

Edexcel AS Paper 2 2018 June Q8

10 marks Standard +0.3

A particle, $P$, moves along the $x$-axis. At time $t$ seconds, $t \geqslant 0$, the displacement, $x$ metres, of $P$ from the origin $O$, is given by $x = \frac { 1 } { 2 } t ^ { 2 } \left( t ^ { 2 } - 2 t + 1 \right)$

Find the times when $P$ is instantaneously at rest.
Find the total distance travelled by $P$ in the time interval $0 \leqslant t \leqslant 2$
Show that $P$ will never move along the negative $x$-axis.

Edexcel AS Paper 2 2018 June Q9

9 marks Standard +0.3

9. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{2edcf965-9c93-4a9b-9395-2d3c023801af-26_551_276_210_890} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Two small balls, $P$ and $Q$, have masses $2 m$ and $k m$ respectively, where $k < 2$.
The balls are attached to the ends of a string that passes over a fixed pulley.
The system is held at rest with the string taut and the hanging parts of the string vertical, as shown in Figure 1. The system is released from rest and, in the subsequent motion, $P$ moves downwards with an acceleration of magnitude $\frac { 5 g } { 7 }$ The balls are modelled as particles moving freely.
The string is modelled as being light and inextensible.
The pulley is modelled as being small and smooth.
Using the model,

find, in terms of $m$ and $g$, the tension in the string,
explain why the acceleration of $Q$ also has magnitude $\frac { 5 g } { 7 }$
find the value of $k$.
Identify one limitation of the model that will affect the accuracy of your answer to part (c).

Edexcel AS Paper 2 Specimen Q1

4 marks Easy -1.8

Sara is investigating the variation in daily maximum gust, $t \mathrm { kn }$, for Camborne in June and July 1987.

She used the large data set to select a sample of size 20 from the June and July data for 1987. Sara selected the first value using a random number from 1 to 4 and then selected every third value after that.

State the sampling technique Sara used.
From your knowledge of the large data set explain why this process may not generate a sample of size 20 . The data Sara collected are summarised as follows $$n = 20 \quad \sum t = 374 \quad \sum t ^ { 2 } = 7600$$
Calculate the standard deviation.

Edexcel AS Paper 2 Specimen Q2

5 marks Moderate -0.8

The partially completed histogram and the partially completed table show the time, to the nearest minute, that a random sample of motorists was delayed by roadworks on a stretch of motorway. \includegraphics[max width=\textwidth, alt={}, center]{8f3dbcb4-3260-4493-a230-12577b4ed691-04_1227_1465_354_301}

Delay (minutes)	Number of motorists
4-6	6
7-8
9	17
10-12	45
13-15	9
16-20

Estimate the percentage of these motorists who were delayed by the roadworks for between 8.5 and 13.5 minutes.

Edexcel AS Paper 2 Specimen Q3

5 marks Easy -1.2

The Venn diagram shows the probabilities for students at a college taking part in various sports. $A$ represents the event that a student takes part in Athletics. $T$ represents the event that a student takes part in Tennis. $C$ represents the event that a student takes part in Cricket. $p$ and $q$ are probabilities. \includegraphics[max width=\textwidth, alt={}, center]{8f3dbcb4-3260-4493-a230-12577b4ed691-06_668_935_596_566}

The probability that a student selected at random takes part in Athletics or Tennis is 0.75

Find the value of $p$.
State, giving a reason, whether or not the events $A$ and $T$ are statistically independent. Show your working clearly.
Find the probability that a student selected at random does not take part in Athletics or Cricket.

Edexcel AS Paper 2 Specimen Q4

7 marks Moderate -0.8

Sara was studying the relationship between rainfall, $r \mathrm {~mm}$, and humidity, $h \%$, in the UK. She takes a random sample of 11 days from May 1987 for Leuchars from the large data set.

She obtained the following results.

$h$	93	86	95	97	86	94	97	97	87	97	86
$r$	1.1	0.3	3.7	20.6	0	0	2.4	1.1	0.1	0.9	0.1

Sara examined the rainfall figures and found $$Q _ { 1 } = 0.1 \quad Q _ { 2 } = 0.9 \quad Q _ { 3 } = 2.4$$ A value that is more than 1.5 times the interquartile range (IQR) above $Q _ { 3 }$ is called an outlier.

Show that $r = 20.6$ is an outlier.
Give a reason why Sara might:
1. include
2. exclude
  this day's reading. Sara decided to exclude this day's reading and drew the following scatter diagram for the remaining 10 days' values of $r$ and $h$. \includegraphics[max width=\textwidth, alt={}, center]{8f3dbcb4-3260-4493-a230-12577b4ed691-08_988_1081_1555_420}
(c) Give an interpretation of the correlation between rainfall and humidity. The equation of the regression line of $r$ on $h$ for these 10 days is $r = - 12.8 + 0.15 h$

(d) Give an interpretation of the gradient of this regression line.
(e)

Comment on the suitability of Sara's sampling method for this study.
Suggest how Sara could make better use of the large data set for her study.

Questions — Edexcel (10514 questions)

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