Questions H240/02 - A-Level Maths

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OCR H240/02 2020 November Q8

9 marks Standard +0.3

The rate of change of a certain population $P$ at time $t$ is modelled by the equation $\frac{dP}{dt} = (100 - P)$. Initially $P = 2000$.

Determine an expression for $P$ in terms of $t$. [7]
Describe how the population changes over time. [2]

OCR H240/02 2020 November Q9

4 marks Easy -1.3

The histogram shows information about the numbers of cars in five different price ranges, sold in one year at a car showroom. \includegraphics{figure_9} It is given that 66 cars in the price range £10000 to £20000 were sold.

Find the number of cars sold in the price range £50000 to £90000. [1]
State the units of the frequency density. [1]
Suggest one change that the management could make to the diagram so that it would provide more information. [1]
Estimate the number of cars sold in the price range £50000 to £60000. [1]

OCR H240/02 2020 November Q10

7 marks Moderate -0.3

Pierre is a chef. He claims that 90% of his customers are satisfied with his cooking. Yvette suspects that Pierre is over-confident about the level of satisfaction amongst his customers. She talks to a random sample of 15 of Pierre's customers, and finds that 11 customers say that they are satisfied. She then performs a hypothesis test. Carry out the test at the 5% significance level. [7]

OCR H240/02 2020 November Q11

9 marks Moderate -0.3

As part of a research project, the masses, $m$ grams, of a random sample of 1000 pebbles from a certain beach were recorded. The results are summarised in the table.

Mass (g)	$50 \leq m < 150$	$150 \leq m < 200$	$200 \leq m < 250$	$250 \leq m < 350$
Frequency	162	318	355	165

Calculate estimates of the mean and standard deviation of these masses. [2]

The masses, $x$ grams, of a random sample of 1000 pebbles on a different beach were also found. It was proposed that the distribution of these masses should be modelled by the random variable $X \sim N(200, 3600)$.

Use the model to find $P(150 < X < 210)$. [1]
Use the model to determine $x_1$ such that $P(160 < X < x_1) = 0.6$, giving your answer correct to five significant figures. [3]

It was found that the smallest and largest masses of the pebbles in this second sample were 112 g and 288 g respectively.

Use these results to show that the model may not be appropriate. [1]
Suggest a different value of a parameter of the model in the light of these results. [2]

OCR H240/02 2020 November Q12

6 marks Moderate -0.3

In the past, the time for Jeff's journey to work had mean 45.7 minutes and standard deviation 5.6 minutes. This year he is trying a new route. In order to test whether the new route has reduced his journey time, Jeff finds the mean time for a random sample of 30 journeys using the new route. He carries out a hypothesis test at the 2.5% significance level. Jeff assumes that, for the new route, the journey time has a normal distribution with standard deviation 5.6 minutes.

State appropriate null and alternative hypotheses for the test. [2]
Determine the rejection region for the test. [4]

OCR H240/02 2020 November Q13

8 marks Standard +0.8

Andy and Bev are playing a game.

The game consists of three points.
On each point, P(Andy wins) = 0.4 and P(Bev wins) = 0.6.
If one player wins two consecutive points, then they win the game, otherwise neither player wins.

Determine the probability of the following events.
1. Andy wins the game. [2]
2. Neither player wins the game. [3]

Andy and Bev now decide to play a match which consists of a series of games.

In each game, if a player wins the game then they win the match.
If neither player wins the game then the players play another game.

Determine the probability that Andy wins the match. [3]

OCR H240/02 2020 November Q14

6 marks Easy -1.8

Table 1 shows the numbers of usual residents in the age range 0 to 4 in 15 Local Authorities (LAs) in 2001 and 2011. The table also shows the increase in the numbers in this age group, and the same increase as a percentage. \includegraphics{figure_14} Fig. 2 shows the increase in each LA in raw numbers, and Fig. 3 shows the percentage increase in each LA. \includegraphics{figure_14_2} \includegraphics{figure_14_3}

The Education Committees in these LAs need to plan for the provision of schools for pupils in their districts.
1. Explain why, in this context, the increase is more important than the actual numbers. [1]
2. In which of the following LAs was there likely to have been the greatest need for extra teachers in the years following 2011: Bolton, Sefton, Tameside or Wigan? Give a reason for your answer. [2]
3. State an assumption about the populations needed to make your answer in part (ii) valid. [1]
In two of the 15 LAs the proportion of young families is greater than in the other 13 LAs. Suggest, using only data from Fig. 2 and Fig. 3 and/or Table 1, which two LAs these are most likely to be. [2]

OCR H240/02 2020 November Q15

10 marks Challenging +1.2

In this question you must show detailed reasoning. The random variable $X$ has probability distribution defined as follows. $$P(X = x) = \begin{cases} \frac{15}{64} \times \frac{2^x}{x!} & x = 2, 3, 4, 5, \\ 0 & \text{otherwise}. \end{cases}$$

Show that $P(X = 2) = \frac{15}{32}$. [1]

The values of three independent observations of $X$ are denoted by $X_1$, $X_2$ and $X_3$.

Given that $X_1 + X_2 + X_3 = 9$, determine the probability that at least one of these three values is equal to 2. [6]

Freda chooses values of $X$ at random until she has obtained $X = 2$ exactly three times. She then stops.

Determine the probability that she chooses exactly 10 values of $X$. [3]

OCR H240/02 2023 June Q1

5 marks Easy -1.3

1. Express $x^2 - 8x + 11$ in the form $(x - a)^2 + b$ where $a$ and $b$ are constants. [2]
2. Hence write down the minimum value of $x^2 - 8x + 11$. [1]
Determine the value of the constant $k$ for which the equation $x^2 - 8x + 11 = k$ has two equal roots. [2]

OCR H240/02 2023 June Q2

5 marks Moderate -0.3

The points $O$ and $A$ have position vectors $\begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}$ and $\begin{pmatrix} 6 \\ 0 \\ 8 \end{pmatrix}$ respectively. The point $P$ is such that $\overrightarrow{OP} = k\overrightarrow{OA}$, where $k$ is a non-zero constant.

Find, in terms of $k$, the length of $OP$. [1] Point $B$ has position vector $\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}$ and angle $OPB$ is a right angle.
Determine the value of $k$. [4]

OCR H240/02 2023 June Q3

3 marks Moderate -0.8

In this question you must show detailed reasoning. Find the exact area of the region enclosed by the curve $y = \frac{1}{x+2}$, the two axes and the line $x = 2.5$. [3]

OCR H240/02 2023 June Q4

9 marks Standard +0.3

The diagram shows part of the graph of $y = x^2$. The normal to the curve at the point $A(1, 1)$ meets the curve again at $B$. Angle $AOB$ is denoted by $\alpha$. \includegraphics{figure_4}

Determine the coordinates of $B$. [6]
Hence determine the exact value of $\tan \alpha$. [3]

OCR H240/02 2023 June Q5

12 marks Standard +0.3

In this question you must show detailed reasoning. The function f is defined by $\text{f}(x) = \cos x + \sqrt{3} \sin x$ with domain $0 \leqslant x \leqslant 2\pi$.

Solve the following equations.
1. $\text{f}'(x) = 0$ [4]
2. $\text{f}''(x) = 0$ [3]
The diagram shows the graph of the gradient function $y = \text{f}'(x)$ for the domain $0 \leqslant x \leqslant 2\pi$. \includegraphics{figure_5}
Use your answers to parts (a)(i) and (a)(ii) to find the coordinates of points $A$, $B$, $C$ and $D$. [2]
1. Explain how to use the graph of the gradient function to find the values of $x$ for which f(x) is increasing. [1]
2. Using set notation, write down the set of values of $x$ for which f(x) is increasing in the domain $0 \leqslant x \leqslant 2\pi$. [2]

OCR H240/02 2023 June Q6

10 marks Standard +0.3

A circle has centre $C$ which lies on the $x$-axis, as shown in the diagram. The line $y = x$ meets the circle at $A$ and $B$. The midpoint of $AB$ is $M$. \includegraphics{figure_6} The equation of the circle is $x^2 - 6x + y^2 + a = 0$, where $a$ is a constant.

In this question you must show detailed reasoning. Show that the area of triangle $ABC$ is $\frac{5}{2}\sqrt{9 - 2a}$. [7]
1. Find the value of $a$ when the area of triangle $ABC$ is zero. [1]
2. Give a geometrical interpretation of the case in part (b)(i). [1]
Give a geometrical interpretation of the case where $a = 5$. [1]

OCR H240/02 2023 June Q7

5 marks Standard +0.8

A student wishes to prove that, for all positive integers $a$ and $b$, $a^2 - 4b \neq 2$.

Prove that $a^2 - 4b = 2 \Rightarrow a$ is even. [2]
Hence or otherwise prove that, for all positive integers $a$ and $b$, $a^2 - 4b \neq 2$. [3]

OCR H240/02 2023 June Q8

7 marks Easy -1.2

The stem-and-leaf diagram shows the heights, in centimetres, of 15 plants. $\begin{array}{l|l} 0 & 2
1 & 0
2 & 4
3 & 0\ 2\ 4\ 9
4 & 1\ 2\ 4\ 7\ 9
5 & 3\ 7
6 & 2 \end{array}$ Key: $2 | 5$ means 25 cm.

Draw a box-and-whisker plot to illustrate the data. [4]

A statistician intends to analyse the data, but wants to ignore any outliers before doing so.

Discuss briefly whether there are any heights in the diagram which the statistician should ignore. [3]

OCR H240/02 2023 June Q9

6 marks Easy -1.2

A school contains 500 students in years 7 to 11 and 250 students in years 12 and 13. A random sample of 20 students is selected to represent the school at a parents' evening. The number of students in the sample who are from years 12 and 13 is denoted by $X$.

State a suitable binomial model for $X$. [1]

Use your model to answer the following.

1. Write down an expression for $\text{P}(X = x)$. [1]
2. State, in set notation, the values of $x$ for which your expression is valid. [1]
Find $\text{P}(5 \leqslant X \leqslant 9)$. [2]
State one disadvantage of using a random sample in this context. [1]

OCR H240/02 2023 June Q10

8 marks Standard +0.3

The mass, in kilograms, of a species of fish in the UK has population mean 4.2 and standard deviation 0.25. An environmentalist believes that the fish in a particular river are smaller, on average, than those in other rivers in the UK. A random sample of 100 fish of this species, taken from the river, has sample mean 4.16 kg. Stating a necessary assumption, test at the 5% significance level whether the environmentalist is correct. [8]

OCR H240/02 2023 June Q11

9 marks Standard +0.3

The random variable $Y$ has the distribution $\text{N}(\mu, \sigma^2)$.

Find $\text{P}(Y > \mu - \sigma)$. [1]
Given that $\text{P}(Y > 45) = 0.2$ and $\text{P}(Y < 25) = 0.3$, determine the values of $\mu$ and $\sigma$. [6]

The random variables $U$ and $V$ have the distributions $\text{N}(10, 4)$ and $\text{N}(12, 9)$ respectively.

It is given that $\text{P}(U < b) = \text{P}(V > c)$, where $b > 10$ and $c < 12$. Determine $b$ in terms of $c$. [2]

OCR H240/02 2023 June Q12

4 marks Standard +0.3

A student has an ordinary six-sided dice. The student suspects that it is biased against six, so that when it is thrown, it is less likely to show a six than if it were fair. In order to test this suspicion, the student plans to carry out a hypothesis test at the 5% significance level. The student throws the dice 100 times and notes the number of times, $X$, that it shows a six.

Determine the largest value of $X$ that would provide evidence at the 5% significance level that the dice is biased against six. [3]

Later another student carries out a similar test, at the 5% significance level. This student also throws the dice 100 times.

It is given that the dice is fair. Find the probability that the conclusion of the test is that there is significant evidence that the dice is biased against six. [1]

OCR H240/02 2023 June Q13

10 marks Easy -1.8

The scatter diagram uses information about all the Local Authorities (LAs) in the UK, taken from the 2011 census. For each LA it shows the percentage ($x$) of employees who used public transport to travel to work and the percentage ($y$) who used motorised private transport. "Public transport" includes train, bus, minibus, coach, underground, metro and light rail. "Motorised private transport" includes car, van, motorcycle, scooter, moped, taxi and passenger in a car or van. \includegraphics{figure_13}

Most of the points in the diagram lie on or near the line with equation $x + y = k$, where $k$ is a constant.
1. Give a possible value for $k$. [1]
2. Hence give an approximate value for the percentage of employees who either worked from home or walked or cycled to work. [1]
The average amount of fuel used per person per day for travelling to work in any LA is denoted by F. Consider the two groups of LAs where the percentages using motorised private transport are highest and lowest.
1. Using only the information in the diagram, suggest, with a reason, which of these two groups will have greater values of F than the other group. [1]
A student says that it is not possible to give a reliable answer to part (b)(i) without some further information.
1. Suggest two kinds of further information which would enable a more reliable answer to be given. [2]
Points $A$ and $B$ in the diagram are the most extreme outliers. Use their positions on the diagram to answer the following questions about the two LAs represented by these two points.
1. The two LAs share a certain characteristic. Describe, with a justification, this characteristic. [2]
2. The environments in these two LAs are very different. Describe, with a justification, this difference. [2]
A student says that it is difficult to extract detailed information from the scatter diagram. Explain whether you agree with this criticism. [1]

OCR H240/02 2023 June Q14

7 marks Standard +0.3

In this question you must show detailed reasoning. A disease that affects trees shows no visible evidence for the first few years after the tree is infected. A test has been developed to determine whether a particular tree has the disease. A positive result to the test suggests that the tree has the disease. However, the test is not 100% reliable, and a researcher uses the following model. • If the tree has the disease, the probability of a positive result is 0.95. • If the tree does not have the disease, the probability of a positive result is 0.1.

It is known that in a certain county, $A$, 35% of the trees have the disease. A tree in county $A$ is chosen at random and is tested. Given that the result is positive, determine the probability that this tree has the disease. [3]

A forestry company wants to determine what proportion of trees in another county, $B$, have the disease. They choose a large random sample of trees in county $B$. Each tree in the sample is tested and it is found that the result is positive for 43% of these trees.