Questions - Edexcel - A-Level Maths

Edexcel S2 2024 October Q2

Standard +0.3

A multiple-choice test consists of 25 questions, each having 5 responses, only one of which is correct.

Each correct answer gains 4 marks but each incorrect answer loses 1 mark.
Sam answers all 25 questions by choosing at random one response for each question.
Let $X$ be the number of correct answers that Sam achieves.

State the distribution of $X$ Let $M$ be the number of marks that Sam achieves.
1. State the distribution of $M$ in terms of $X$
2. Hence, show clearly that the number of marks that Sam is expected to achieve is zero. In order to pass the test at least 30 marks are required.
Find the probability that Sam will pass the test. Past records show that when the test is done properly, the probability that a student answers the first question correctly is 0.5 A random sample of 50 students that did the test properly was taken.
Given that the probability that more than $n$ but at most 30 students answered the first question correctly was 0.9328 to 4 decimal places,
find the value of $n$

Edexcel S2 2024 October Q3

Standard +0.3

During Monday afternoons, customers are known to enter a certain shop at a mean rate of 7 customers every 10 minutes.

Suggest a suitable distribution to model the number of customers that enter this shop in a 10-minute interval on Monday afternoons.
State two assumptions necessary for this distribution to be a suitable model of this situation. A new shop manager wants to find out if the rate of customers has changed since they took over.
Write down suitable null and alternative hypotheses that the shop manager should use. The shop manager decides to monitor the number of customers entering the shop in a random 10-minute interval next Monday afternoon.
Using a $3 \%$ level of significance, find the critical region to test whether the rate of customers has changed.
Find the actual significance level of this test based on your critical region from part (d) During the random 10-minute interval that Monday afternoon, 12 customers entered the shop.
Comment on this finding, using the critical region in part (d)

Edexcel S2 2024 October Q4

Standard +0.3

The continuous random variable $X$ is uniformly distributed over the interval $[ a , b ]$ Given that
Given also that $$4 \times \mathrm { P } ( X < k - 10 ) = \mathrm { P } ( X > k + 20 )$$ (b) find the value of $k$
A piece of wire of length 42 cm is cut into 2 pieces at a random point. Each of the two pieces of the wire is bent to form the outline of a square.
Find the probability that the side length of the larger square minus the side length of the smaller square will be greater than 2 cm .

Edexcel S2 2024 October Q5

Moderate -0.3

The continuous random variable $X$ has a probability density function given by

$$f ( x ) = \begin{cases} \frac { 1 } { 4 } ( 3 - x ) & 1 \leqslant x \leqslant 2 \\ \frac { 1 } { 4 } & 2 < x \leqslant 3 \\ \frac { 1 } { 4 } ( x - 2 ) & 3 < x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$ The cumulative distribution function of $X$ is $\mathrm { F } ( x )$

Show that $\mathrm { F } ( x ) = \frac { 1 } { 4 } \left( 3 x - \frac { x ^ { 2 } } { 2 } \right) - \frac { 5 } { 8 }$ for $1 \leqslant x \leqslant 2$
Find $\mathrm { F } ( x )$ for all values of $x$
Find $\mathrm { P } ( 1.2 < X < 3.1 )$

Edexcel S2 2024 October Q6

Standard +0.3

Two boxes, A and B , each contain a large number of coins.

In box A

there are only 1 p coins and 2 p coins
the ratio of 1 p coins to 2 p coins is $1 : 3$

In box B

there are only 2 p coins and 5 p coins
the ratio of 2 p coins to 5 p coins is $1 : 4$

One coin is randomly selected from box A and two coins are randomly selected from box B The random variable $T$ represents the total of the values of the three coins selected.

Find the sampling distribution of $T$ The random variable $M$ represents the median of the values of the three coins selected.
Find the sampling distribution of $M$

Edexcel S2 2024 October Q7

Standard +0.3

The continuous random variable $X$ has probability density function given by

$$f ( x ) = \begin{cases} a x & 0 \leqslant x \leqslant 4 \\ b x + c & 4 < x \leqslant 8 \\ 0 & \text { otherwise } \end{cases}$$ where $a$, $b$ and $c$ are constants. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{6e6f7a1a-b577-4f28-a7a9-557b9d325851-24_389_1013_630_529} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows the graph of the probability density function $\mathrm { f } ( x )$ The graph consists of two straight line segments of equal length joined at the point where $x = 4$

Show that $a = \frac { 1 } { 16 }$
Hence find
1. the value of $b$
2. the value of $c$
Using algebraic integration, show that $\operatorname { Var } ( X ) = \frac { 8 } { 3 }$
Find, to 2 decimal places, the lower quartile and the upper quartile of $X$ A statistician claims that $$\mathrm { P } ( - \sigma < X - \mu < \sigma ) > 0.5$$ where $\mu$ and $\sigma$ are the mean and standard deviation of $X$
Show that the statistician's claim is correct.

Edexcel S3 Q7

Standard +0.8

7. A set of scaffolding poles come in two sizes, long and short. The length $L$ of a long pole has the normal distribution $\mathrm { N } \left( 19.7,0.5 ^ { 2 } \right)$. The length $S$ of a short pole has the normal distribution $\mathrm { N } \left( 4.9,0.2 ^ { 2 } \right)$. The random variables $L$ and $S$ are independent. A long pole and a short pole are selected at random.

Find the probability that the length of the long pole is more than 4 times the length of the short pole. Four short poles are selected at random and placed end to end in a row. The random variable $T$ represents the length of the row.
Find the distribution of $T$.
Find $\mathrm { P } ( | L - T | < 0.1 )$.
\end{table}
1. Some biologists were studying a large group of wading birds. A random sample of 36 were measured and the wing length, $x \mathrm {~mm}$ of each wading bird was recorded. The results are summarised as follows
$$\sum x = 6046 \quad \sum x ^ { 2 } = 1016338$$

(a) Calculate unbiased estimates of the mean and the variance of the wing lengths of these birds. Given that the standard deviation of the wing lengths of this particular type of bird is actually 5.1 mm ,
(b) find a $99 \%$ confidence interval for the mean wing length of the birds from this group.
2. Students in a mixed sixth form college are classified as taking courses in either Arts, Science or Humanities. A random sample of students from the college gave the following results \end{table}

A telephone directory contains 50000 names. A researcher wishes to select a systematic sample of 100 names from the directory.
1. Explain in detail how the researcher should obtain such a sample.
2. Give one advantage and one disadvantage of
1. quota sampling,
2. systematic sampling.
3. The heights of a random sample of 10 imported orchids are measured. The mean height of the sample is found to be 20.1 cm . The heights of the orchids are normally distributed.
Given that the population standard deviation is 0.5 cm ,
1. estimate limits between which $95 \%$ of the heights of the orchids lie,
2. find a 98\% confidence interval for the mean height of the orchids. A grower claims that the mean height of this type of orchid is 19.5 cm .
3. Comment on the grower's claim. Give a reason for your answer.
  3. A doctor is interested in the relationship between a person's Body Mass Index (BMI) and their level of fitness. She believes that a lower BMI leads to a greater level of fitness. She randomly selects 10 female 18 year-olds and calculates each individual's BMI. The females then run a race and the doctor records their finishing positions. The results are shown in the table.
  Individual $A$ $B$ $C$ $D$ $E$ $F$ $G$ $H$ $I$ $J$
  BMI 17.4 21.4 18.9 24.4 19.4 20.1 22.6 18.4 25.8 28.1
  Finishing position 3 5 1 9 6 4 10 2 7 8
1. Calculate Spearman's rank correlation coefficient for these data.
2. Stating your hypotheses clearly and using a one tailed test with a $5 \%$ level of significance, interpret your rank correlation coefficient.
3. Give a reason to support the use of the rank correlation coefficient rather than the product moment correlation coefficient with these data.
  4. A sample of size 8 is to be taken from a population that is normally distributed with mean 55 and standard deviation 3. Find the probability that the sample mean will be greater than 57.
  5. The number of goals scored by a football team is recorded for 100 games. The results are summarised in Table 1 below. \begin{table}[h]
  Number of goals Frequency
  0 40
  1 33
  2 14
  3 8
  4 5
  \captionsetup{labelformat=empty} \caption{Table 1}
  \end{table}
(a) Calculate the mean number of goals scored per game. The manager claimed that the number of goals scored per match follows a Poisson distribution. He used the answer in part (a) to calculate the expected frequencies given in Table 2. \begin{table}[h]
Number of goals Expected Frequency
0 34.994
1 $r$
2 $s$
3 6.752
$\geqslant 4$ 2.221
\captionsetup{labelformat=empty} \caption{Table 2}
\end{table} (b) Find the value of $r$ and the value of $s$ giving your answers to 3 decimal places.
(c) Stating your hypotheses clearly, use a $5 \%$ level of significance to test the manager's claim.
1. The lengths of a random sample of 120 limpets taken from the upper shore of a beach had a mean of 4.97 cm and a standard deviation of 0.42 cm . The lengths of a second random sample of 150 limpets taken from the lower shore of the same beach had a mean of 5.05 cm and a standard deviation of 0.67 cm .
  1. Test, using a $5 \%$ level of significance, whether or not the mean length of limpets from the upper shore is less than the mean length of limpets from the lower shore. State your hypotheses clearly.
  2. State two assumptions you made in carrying out the test in part (a).
  3. A company produces climbing ropes. The lengths of the climbing ropes are normally distributed. A random sample of 5 ropes is taken and the length, in metres, of each rope is measured. The results are given below.
    119.9
    120.3
    120.1
    120.4
    120.2
  (a) Calculate unbiased estimates for the mean and the variance of the lengths of the climbing ropes produced by the company.
The lengths of climbing rope are known to have a standard deviation of 0.2 m . The company wants to make sure that there is a probability of at least 0.90 that the estimate of the population mean, based on a random sample size of $n$, lies within 0.05 m of its true value.
(b) Find the minimum sample size required.

Edexcel S3 Q8

Moderate -0.3

The random variable $A$ is defined as

$$A = 4 X - 3 Y$$ where $X \sim \mathrm {~N} \left( 30,3 ^ { 2 } \right) , Y \sim \mathrm {~N} \left( 20,2 ^ { 2 } \right)$ and $X$ and $Y$ are independent. Find

$\mathrm { E } ( A )$,
$\operatorname { Var } ( A )$. The random variables $Y _ { 1 } , Y _ { 2 } , Y _ { 3 }$ and $Y _ { 4 }$ are independent and each has the same distribution as $Y$. The random variable $B$ is defined as $$B = \sum _ { i = 1 } ^ { 4 } Y _ { i }$$
Find $\mathrm { P } ( B > A )$.
advancing learning, changing lives
1. A report states that employees spend, on average, 80 minutes every working day on personal use of the Internet. A company takes a random sample of 100 employees and finds their mean personal Internet use is 83 minutes with a standard deviation of 15 minutes. The company's managing director claims that his employees spend more time on average on personal use of the Internet than the report states.
Test, at the $5 \%$ level of significance, the managing director's claim. State your hypotheses clearly.
2. Philip and James are racing car drivers. Philip's lap times, in seconds, are normally distributed with mean 90 and variance 9. James' lap times, in seconds, are normally distributed with mean 91 and variance 12. The lap times of Philip and James are independent. Before a race, they each take a qualifying lap.

Find the probability that James' time for the qualifying lap is less than Philip's. The race is made up of 60 laps. Assuming that they both start from the same starting line and lap times are independent,
find the probability that Philip beats James in the race by more than 2 minutes.
3. A woodwork teacher measures the width, $w \mathrm {~mm}$, of a board. The measured width, $X \mathrm {~mm}$, is normally distributed with mean $w \mathrm {~mm}$ and standard deviation 0.5 mm .

Find the probability that $X$ is within 0.6 mm of $w$. The same board is measured 16 times and the results are recorded.
Find the probability that the mean of these results is within 0.3 mm of $w$. Given that the mean of these 16 measurements is 35.6 mm ,
find a $98 \%$ confidence interval for $w$.
1. A researcher claims that, at a river bend, the water gradually gets deeper as the distance from the inner bank increases. He measures the distance from the inner bank, $b \mathrm {~cm}$, and the depth of a river, $s \mathrm {~cm}$, at seven positions. The results are shown in the table below.
advancing learning, changing lives \includegraphics[max width=\textwidth, alt={}, center]{fb233c8c-e1b7-4ba5-aa4d-c23d5382dc84-055_2632_1828_123_121}
2. A county councillor is investigating the level of hardship, h , of a town and the number of calls per 100 people to the emergency services, c. He collects data for 7 randomly selected towns in the county. The results are shown in the table below.
1. Interviews for a job are carried out by two managers. Candidates are given a score by each manager and the results for a random sample of 8 candidates are shown in the table below.
\includegraphics[max width=\textwidth, alt={}, center]{fb233c8c-e1b7-4ba5-aa4d-c23d5382dc84-081_2642_1833_118_118}
2. A random sample of size n is to be taken from a population that is normally distributed with mean 40 and standard deviation 3 . Find the minimum sample size such that the probability of the sample mean being greater than 42 is less than $5 \%$.
(5)
3. The table below shows the population and the number of council employees for different towns and villages. \end{table} A nswers without working may not gain full credit. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{ $0 - 3$ & 8
\hline $3 - 5$ & 12
\hline $5 - 6$ & 13
\hline $6 - 8$ & 9
\hline $8 - 12$ & 8
\hline \captionsetup{labelformat=empty} \caption{Table 1}
\end{table}
1. Show that an estimate of $\bar { X } = 5.49$ and an estimate of $S _ { X } ^ { 2 } = 6.88$ The post office manager believes that the customers' waiting times can be modelled by a normal distribution.
  Assuming the data is normally distributed, she calculates the expected frequencies for these data and some of these frequencies are shown in Table 2. \begin{table}[h]
  Waiting Time $\mathrm { x } < 3$ $3 - 5$ $5 - 6$ $6 - 8$ $\mathrm { x } > 8$
  Expected Frequency 8.56 12.73 7.56 a b
  \captionsetup{labelformat=empty} \caption{Table 2}
  \end{table}
2. Find the value of a and the value of b .
3. Test, at the $5 \%$ level of significance, the manager's belief. State your hypotheses clearly.
  \section*{Q uestion 4 continued}
  1. Blumen is a perfume sold in bottles. The amount of perfume in each bottle is normally distributed. The amount of perfume in a large bottle has mean 50 ml and standard deviation 5 ml . The amount of perfume in a small bottle has mean 15 ml and standard deviation 3 ml .
  One large and 3 small bottles of Blumen are chosen at random.
1. Find the probability that the amount in the large bottle is less than the total amount in the 3 small bottles. A large bottle and a small bottle of Blumen are chosen at random.
2. Find the probability that the large bottle contains more than 3 times the amount in the small bottle.
  \section*{Q uestion 5 continued} 6. Fruit-n-Veg4U M arket Gardens grow tomatoes. They want to improve their yield of tomatoes by at least 1 kg per plant by buying a new variety. The variance of the yield of the old variety of plant is $0.5 \mathrm {~kg} ^ { 2 }$ and the variance of the yield for the new variety of plant is $0.75 \mathrm {~kg} ^ { 2 }$. A random sample of 60 plants of the old variety has a mean yield of 5.5 kg . A random sample of 70 of the new variety has a mean yield of 7 kg .
1. Stating your hypotheses clearly test, at the $5 \%$ level of significance, whether or not there is evidence that the mean yield of the new variety is more than 1 kg greater than the mean yield of the old variety.
2. Explain the relevance of the Central Limit Theorem to the test in part (a). \section*{Q uestion 6 continued} \includegraphics[max width=\textwidth, alt={}, center]{fb233c8c-e1b7-4ba5-aa4d-c23d5382dc84-102_46_79_2620_1818}
  7. Lambs are born in a shed on M ill Farm. The birth weights, $x \mathrm {~kg}$, of a random sample of 8 newborn lambs are given below. $$\begin{array} { l l l l l l l l } 4.12 & 5.12 & 4.84 & 4.65 & 3.55 & 3.65 & 3.96 & 3.40 \end{array}$$
(a) Calculate unbiased estimates of the mean and variance of the birth weight of lambs born on Mill Farm. A further random sample of 32 lambs is chosen and the unbiased estimates of the mean and variance of the birth weight of lambs from this sample are 4.55 and 0.25 respectively.
Treating the combined sample of 40 lambs as a single sample, estimate the standard error of the mean. The owner of M ill Farm researches the breed of lamb and discovers that the population of birth weights is normally distributed with standard deviation 0.67 kg .
Calculate a $95 \%$ confidence interval for the mean birth weight of this breed of lamb using your combined sample mean.
\section*{Q uestion 7 continued} \end{figure}

Edexcel C4 2014 June Q1

7 marks Standard +0.3

A curve $C$ has the equation $$x^3 + 2xy - x - y^3 - 20 = 0$$

[(a)] Find $\frac{dy}{dx}$ in terms of $x$ and $y$. \hfill [5]
[(b)] Find an equation of the tangent to $C$ at the point $(3, -2)$, giving your answer in the form $ax + by + c = 0$, where $a$, $b$ and $c$ are integers. \hfill [2]

Edexcel C4 2014 June Q2

5 marks Moderate -0.3

Given that the binomial expansion of $(1 + kx)^{-4}$, $|kx| < 1$, is $$1 - 6x + Ax^2 + \ldots$$

[(a)] find the value of the constant $k$, \hfill [2]
[(b)] find the value of the constant $A$, giving your answer in its simplest form. \hfill [3]

Edexcel C4 2014 June Q3

11 marks Standard +0.3

\includegraphics{figure_1} Figure 1 shows a sketch of part of the curve with equation $y = \frac{10}{2x + 5\sqrt{x}}$, $x > 0$ The finite region $R$, shown shaded in Figure 1, is bounded by the curve, the $x$-axis, and the lines with equations $x = 1$ and $x = 4$ The table below shows corresponding values of $x$ and $y$ for $y = \frac{10}{2x + 5\sqrt{x}}$

$x$	1	2	3	4
$y$	1.42857	0.90326		0.55556

[(a)] Complete the table above by giving the missing value of $y$ to 5 decimal places. \hfill [1]
[(b)] Use the trapezium rule, with all the values of $y$ in the completed table, to find an estimate for the area of $R$, giving your answer to 4 decimal places. \hfill [3]
[(c)] By reference to the curve in Figure 1, state, giving a reason, whether your estimate in part (b) is an overestimate or an underestimate for the area of $R$. \hfill [1]
[(d)] Use the substitution $u = \sqrt{x}$, or otherwise, to find the exact value of $$\int_1^4 \frac{10}{2x + 5\sqrt{x}} dx$$ \hfill [6]

Edexcel C4 2014 June Q4

5 marks Moderate -0.3

\includegraphics{figure_2} A vase with a circular cross-section is shown in Figure 2. Water is flowing into the vase. When the depth of the water is $h$ cm, the volume of water $V$ cm$^3$ is given by $$V = 4\pi h(h + 4), \quad 0 \leq h \leq 25$$ Water flows into the vase at a constant rate of $80\pi$ cm$^3$s$^{-1}$ Find the rate of change of the depth of the water, in cm s$^{-1}$, when $h = 6$ \hfill [5]

Edexcel C4 2014 June Q5

5 marks Moderate -0.3

\includegraphics{figure_3} Figure 3 shows a sketch of the curve $C$ with parametric equations $$x = 4\cos\left(t + \frac{\pi}{6}\right), \quad y = 2\sin t, \quad 0 \leq t < 2\pi$$

[(a)] Show that $$x + y = 2\sqrt{3}\cos t$$ \hfill [3]
[(b)] Show that a cartesian equation of $C$ is $$(x + y)^2 + ay^2 = b$$ where $a$ and $b$ are integers to be determined. \hfill [2]

Edexcel C4 2014 June Q6

12 marks Standard +0.3

[(i)] Find $$\int xe^{4x} dx$$ \hfill [3]
[(ii)] Find $$\int \frac{8}{(2x - 1)^3} dx, \quad x > \frac{1}{2}$$ \hfill [2]
[(iii)] Given that $y = \frac{\pi}{6}$ at $x = 0$, solve the differential equation $$\frac{dy}{dx} = e^x \cosec 2y \cosec y$$ \hfill [7] \end{enumerate}

Edexcel C4 2014 June Q7

15 marks Challenging +1.2

\includegraphics{figure_4} Figure 4 shows a sketch of part of the curve $C$ with parametric equations $$x = 3\tan\theta, \quad y = 4\cos^2\theta, \quad 0 \leq \theta < \frac{\pi}{2}$$ The point $P$ lies on $C$ and has coordinates $(3, 2)$. The line $l$ is the normal to $C$ at $P$. The normal cuts the $x$-axis at the point $Q$.

[(a)] Find the $x$ coordinate of the point $Q$. \hfill [6]

The finite region $S$, shown shaded in Figure 4, is bounded by the curve $C$, the $x$-axis, the $y$-axis and the line $l$. This shaded region is rotated $2\pi$ radians about the $x$-axis to form a solid of revolution.

[(b)] Find the exact value of the volume of the solid of revolution, giving your answer in the form $p\pi + q\pi^2$, where $p$ and $q$ are rational numbers to be determined. [You may use the formula $V = \frac{1}{3}\pi r^2 h$ for the volume of a cone.] \hfill [9] \end{enumerate} \end{enumerate}

Edexcel C4 2014 June Q8

15 marks Standard +0.3

Relative to a fixed origin $O$, the point $A$ has position vector $\begin{pmatrix} -2 \\ 4 \\ 7 \end{pmatrix}$ and the point $B$ has position vector $\begin{pmatrix} -1 \\ 3 \\ 8 \end{pmatrix}$ The line $l_1$ passes through the points $A$ and $B$.

[(a)] Find the vector $\overrightarrow{AB}$. \hfill [2]
[(b)] Hence find a vector equation for the line $l_1$ \hfill [1]

The point $P$ has position vector $\begin{pmatrix} 0 \\ 2 \\ 3 \end{pmatrix}$ Given that angle $PBA$ is $\theta$,

[(c)] show that $\cos\theta = \frac{1}{3}$ \hfill [3]

The line $l_2$ passes through the point $P$ and is parallel to the line $l_1$

[(d)] Find a vector equation for the line $l_2$ \hfill [2]

The points $C$ and $D$ both lie on the line $l_2$ Given that $AB = PC = DP$ and the $x$ coordinate of $C$ is positive,

[(e)] find the coordinates of $C$ and the coordinates of $D$. \hfill [3]
[(f)] find the exact area of the trapezium $ABCD$, giving your answer as a simplified surd. \hfill [4] \end{enumerate} \end{enumerate} \end{enumerate} \end{enumerate}

Edexcel P1 2018 Specimen Q1

6 marks Easy -1.2

Given that $y = 4x^3 - \frac{5}{x^2}$, $x \neq 0$, find in their simplest form

$\frac{dy}{dx}$. [3]
$\int y \, dx$ [3]

Edexcel P1 2018 Specimen Q2

5 marks Easy -1.2

Given that $3^{-1.5} = a\sqrt{3}$ find the exact value of $a$ [2]
Simplify fully $\frac{(2x^{\frac{1}{2}})^3}{4x^2}$ [3]

Edexcel P1 2018 Specimen Q3

6 marks Moderate -0.3

Solve the simultaneous equations $$y + 4x + 1 = 0$$ $$y^2 + 5x^2 + 2y = 0$$ [6]

Edexcel P1 2018 Specimen Q4

5 marks Standard +0.3

The straight line with equation $y = 4x + c$, where $c$ is a constant, is a tangent to the curve with equation $y = 2x^2 + 8x + 3$ Calculate the value of $c$ [5]

Edexcel P1 2018 Specimen Q5

8 marks Moderate -0.8

On the same axes, sketch the graphs of $y = x + 2$ and $y = x^2 - x - 6$ showing the coordinates of all points at which each graph crosses the coordinate axes. [4]
On your sketch, show, by shading, the region $R$ defined by the inequalities $$y < x + 2 \text{ and } y > x^2 - x - 6$$ [1]
Hence, or otherwise, find the set of values of $x$ for which $x^2 - 2x - 8 < 0$ [3]

Edexcel P1 2018 Specimen Q6

7 marks Moderate -0.3

\includegraphics{figure_1} Figure 1 shows a sketch of the curve $C$ with equation $y = \text{f}(x)$ The curve $C$ passes through the origin and through $(6, 0)$ The curve $C$ has a minimum at the point $(3, -1)$ On separate diagrams, sketch the curve with equation

$y = \text{f}(2x)$ [3]
$y = \text{f}(x + p)$, where $p$ is a constant and $0 < p < 3$ [4]

On each diagram show the coordinates of any points where the curve intersects the $x$-axis and of any minimum or maximum points.

Edexcel P1 2018 Specimen Q7

5 marks Moderate -0.8

A curve with equation $y = \text{f}(x)$ passes through the point $(4, 25)$ Given that $$\text{f}'(x) = \frac{3}{8}x^2 - 10x^{-\frac{1}{2}} + 1, \quad x > 0$$ find $\text{f}(x)$, simplifying each term. [5]

Edexcel P1 2018 Specimen Q8

10 marks Moderate -0.3

\includegraphics{figure_2} The line $l_1$ shown in Figure 2 has equation $2x + 3y = 26$ The line $l_2$ passes through the origin $O$ and is perpendicular to $l_1$

Find an equation for the line $l_2$ [4]

The line $l_1$ intersects the line $l_1$ at the point $C$. Line $l_1$ crosses the $y$-axis at the point $B$ as shown in Figure 2.

Find the area of triangle $OBC$. Give your answer in the form $\frac{a}{b}$, where $a$ and $b$ are integers to be found. [6]

Edexcel P1 2018 Specimen Q9

11 marks Standard +0.3

\includegraphics{figure_3} A sketch of part of the curve $C$ with equation $$y = 20 - 4x - \frac{18}{x}, \quad x > 0$$ is shown in Figure 3. Point $A$ lies on $C$ and has $x$ coordinate equal to 2

Show that the equation of the normal to $C$ at $A$ is $y = -2x + 7$. [6]

The normal to $C$ at $A$ meets $C$ again at the point $B$, as shown in Figure 3.

Use algebra to find the coordinates of $B$. [5]

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Individual	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)	\(H\)	\(I\)	\(J\)
BMI	17.4	21.4	18.9	24.4	19.4	20.1	22.6	18.4	25.8	28.1
Finishing position	3	5	1	9	6	4	10	2	7	8

Number of goals	Expected Frequency
0	34.994
1	\(r\)
2	\(s\)
3	6.752
\(\geqslant 4\)	2.221

Waiting Time	\(\mathrm { x } < 3\)	\(3 - 5\)	\(5 - 6\)	\(6 - 8\)	\(\mathrm { x } > 8\)
Expected Frequency	8.56	12.73	7.56	a	b