Questions H240/02 (151 questions)

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OCR H240/02 2018 December Q4
10 marks Standard +0.3
In this question you must show detailed reasoning.
  1. Show that \(\cos A + \sin A \tan A = \sec A\). [3]
  2. Solve the equation \(\tan 2\theta = 3 \tan \theta\) for \(0° \leqslant \theta \leqslant 180°\). [7]
OCR H240/02 2018 December Q5
8 marks Moderate -0.3
Points \(A\) and \(B\) have position vectors \(\mathbf{a}\) and \(\mathbf{b}\). Point \(C\) lies on \(AB\) such that \(AC : CB = p : 1\).
  1. Show that the position vector of \(C\) is \(\frac{1}{p+1}(\mathbf{a} + p\mathbf{b})\). [3]
It is now given that \(\mathbf{a} = 2\mathbf{i} + 3\mathbf{j} - 4\mathbf{k}\) and \(\mathbf{b} = -6\mathbf{i} + 4\mathbf{j} + 12\mathbf{k}\), and that \(C\) lies on the \(y\)-axis.
  1. Find the value of \(p\). [4]
  2. Write down the position vector of \(C\). [1]
OCR H240/02 2018 December Q6
8 marks Moderate -0.8
The table shows information about three geometric series. The three geometric series have different common ratios.
First termCommon ratioNumber of termsLast term
Series 112\(n_1\)1024
Series 21\(r_2\)\(n_2\)1024
Series 31\(r_3\)\(n_3\)1024
  1. Find \(n_1\). [2]
  2. Given that \(r_2\) is an integer less than 10, find the value of \(r_2\) and the value of \(n_2\). [2]
  3. Given that \(r_3\) is not an integer, find a possible value for the sum of all the terms in Series 3. [4]
OCR H240/02 2018 December Q7
5 marks Standard +0.8
  1. Show that, if \(n\) is a positive integer, then \((x^n - 1)\) is divisible by \((x - 1)\). [1]
  2. Hence show that, if \(k\) is a positive integer, then \(2^{8k} - 1\) is divisible by 17. [4]
OCR H240/02 2018 December Q8
7 marks Challenging +1.8
Use a suitable trigonometric substitution to find \(\int \frac{x^2}{\sqrt{1-x^2}} \text{d}x\). [7]
OCR H240/02 2018 December Q9
7 marks Standard +0.3
Research has shown that drug A is effective in 32% of patients with a certain disease. In a trial, drug B is given to a random sample of 1000 patients with the disease, and it is found that the drug is effective in 290 of these patients. Test at the 2.5% significance level whether there is evidence that drug B is effective in a lower proportion of patients than drug A. [7]
OCR H240/02 2018 December Q10
6 marks Moderate -0.8
Using the 2001 UK census results and some software, Javid intended to calculate the mean number of people who travelled to work by underground, metro, light rail or tram (UMLT) for all 348 Local Authorities. However, Javid noticed that for one LA the entry in the UMLT column is a dash, rather than a 0. See the extract below.
Data extract for one LA in 2001
Work mainly at or from homeUMLTTrainBus, minibus or coach
295–44
Javid felt that it was not clear how this LA was to be treated so he decided to omit it from his calculation.
  1. Explain how the omission of this LA affects Javid's calculation of the mean. [1]
The value of the mean that Javid obtained was 2046.3.
  1. Calculate the value of the mean when this LA is not removed. [2]
Javid finds that the corresponding mean for all Local Authorities for 2011 is 2860.8. In order to compare the means for the two years, Javid also finds the total number of employees in each of these years. His results are given below.
Year20012011
Total number of employees23 627 75326 526 336
  1. Show that a higher proportion of employees used the metro to travel to work in 2011 than in 2001. [2]
  2. Suggest a reason for this increase. [1]
OCR H240/02 2018 December Q11
6 marks Moderate -0.8
Laxmi wishes to test whether there is linear correlation between the mass and the height of adult males.
  1. State, with a reason, whether Laxmi should use a 1-tail or a 2-tail test. [1]
Laxmi chooses a random sample of 40 adult males and calculates Pearson's product-moment correlation coefficient, \(r\). She finds that \(r = 0.2705\).
  1. Use the table below to carry out the test at the 5% significance level. [5]
Critical values of Pearson's product-moment correlation coefficient.
1-tail test2-tail test
5%2.5%1%0.5%
10%5%2.5%1%
380.27090.32020.37600.4128
390.26730.31600.37120.4076
\(n\) 400.26380.31200.36650.4026
410.26050.30810.36210.3978
OCR H240/02 2018 December Q12
7 marks Moderate -0.8
Paul drew a cumulative frequency graph showing information about the numbers of people in various age-groups in a certain region X. He forgot to include the scale on the cumulative frequency axis, as shown below. \includegraphics{figure_12}
  1. Find an estimate of the median age of the population of region X. [1]
  2. Find an estimate of the proportion of people aged over 60 in region X. [2]
Sonika drew similar cumulative graphs for another two regions, Y and Z, but she included the scales on the cumulative frequency axes, as shown below. \includegraphics{figure_12b}
  1. Find an age group, of width 20 years, in which region Z has approximately 3 times as many people as region Y. [1]
  2. State one advantage and one disadvantage of using Sonika's two diagrams to compare the populations in Regions Y and Z. [2]
  3. Without calculation state, with a reason, which of regions Y or Z has the greater proportion of people aged under 40. [1]
OCR H240/02 2018 December Q13
3 marks Moderate -0.8
The marks of 24 students in a test had mean \(m\) and standard deviation \(\sqrt{6}\). Two new students took the same test. Their marks were \(m - 4\) and \(m + 4\). Show that the standard deviation of the marks of all 26 students is 2.60, correct to 3 significant figures. [3]
OCR H240/02 2018 December Q14
11 marks Standard +0.8
Mr Jones has 3 tins of beans and 2 tins of pears. His daughter has removed the labels for a school project, and the tins are identical in appearance. Mr Jones opens tins in turn until he has opened at least 1 tin of beans and at least 1 tin of pears. He does not open any remaining tins.
  1. Draw a tree diagram to illustrate this situation, labelling each branch with its associated probability. [3]
  2. Find the probability that Mr Jones opens exactly 3 tins. [3]
  3. It is given that the last tin Mr Jones opens is a tin of pears. Find the probability that he opens exactly 3 tins. [5]
OCR H240/02 2018 December Q15
9 marks Moderate -0.3
A fair dice is thrown 1000 times and the number, \(X\), of throws on which the score is 6 is noted.
    1. State the distribution of \(X\). [1]
    2. Explain why a normal distribution would be an appropriate approximation to the distribution of \(X\). [1]
  2. Use a normal distribution to find two positive integer values, \(a\) and \(b\), such that \(\text{P}(a < X < b) \approx 0.4\). [5]
  3. For your two values of \(a\) and \(b\), use the distribution of part (a)(i) to find the value of \(\text{P}(a < X < b)\), correct to 3 significant figures. [2]
OCR H240/02 2017 Specimen Q1
4 marks Easy -1.8
Simplify fully.
  1. \(\sqrt{a^3} \times \sqrt{16a}\) [2]
  2. \((4b^6)^{\frac{3}{2}}\) [2]
OCR H240/02 2017 Specimen Q2
7 marks Moderate -0.8
A curve has equation \(y = x^5 - 5x^4\).
  1. Find \(\frac{dy}{dx}\) and \(\frac{d^2y}{dx^2}\). [3]
  2. Verify that the curve has a stationary point when \(x = 4\). [2]
  3. Determine the nature of this stationary point. [2]
OCR H240/02 2017 Specimen Q3
9 marks Moderate -0.8
A publisher has to choose the price at which to sell a certain new book. The total profit, \(£t\), that the publisher will make depends on the price, \(£p\). He decides to use a model that includes the following assumptions. • If the price is low, many copies will be sold, but the profit on each copy sold will be small, and the total profit will be small. • If the price is high, the profit on each copy sold will be high, but few copies will be sold, and the total profit will be small. The graphs below show two possible models. \includegraphics{figure_3}
  1. Explain how model A is inconsistent with one of the assumptions given above. [1]
  2. Given that the equation of the curve in model B is quadratic, show that this equation is of the form \(t = k(12p - p^2)\), and find the value of the constant \(k\). [2]
  3. The publisher needs to make a total profit of at least £6400. Use the equation found in part (b) to find the range of values within which model B suggests that the price of the book must lie. [4]
  4. Comment briefly on how realistic model B may be in the following cases. • \(p = 0\) • \(p = 12.1\) [2]
OCR H240/02 2017 Specimen Q4
7 marks Moderate -0.3
  1. Express \(\frac{1}{(x-1)(x+2)}\) in partial fractions [2]
  2. In this question you must show detailed reasoning. Hence find \(\int_2^3 \frac{1}{(x-1)(x+2)} dx\). Give your answer in its simplest form. [5]
OCR H240/02 2017 Specimen Q5
11 marks Challenging +1.2
The diagram shows the circle with centre O and radius 2, and the parabola \(y = \frac{1}{\sqrt{3}}(4 - x^2)\). \includegraphics{figure_5} The circle meets the parabola at points \(P\) and \(Q\), as shown in the diagram.
  1. Verify that the coordinates of \(Q\) are \((1, \sqrt{3})\). [3]
  2. Find the exact area of the shaded region enclosed by the arc \(PQ\) of the circle and the parabola. [8]
OCR H240/02 2017 Specimen Q6
12 marks Standard +0.3
Helga invests £4000 in a savings account. After \(t\) days, her investment is worth \(£y\). The rate of increase of \(y\) is \(ky\), where \(k\) is a constant.
  1. Write down a differential equation in terms of \(t\), \(y\) and \(k\). [1]
  2. Solve your differential equation to find the value of Helga's investment after \(t\) days. Give your answer in terms of \(k\) and \(t\). [4]
It is given that \(k = \frac{r}{365}\ln\left(1 + \frac{r}{100}\right)\) where \(r\%\) is the rate of interest per annum. During the first year the rate of interest is 6% per annum.
  1. Find the value of Helga's investment after 90 days. [2]
After one year (365 days), the rate of interest drops to 5% per annum.
  1. Find the total time that it will take for Helga's investment to double in value. [5]
OCR H240/02 2017 Specimen Q7
6 marks Moderate -0.8
  1. The heights of English men aged 25 to 34 are normally distributed with mean 178 cm and standard deviation 8 cm. Three English men aged 25 to 34 are chosen at random. Find the probability that all three men have a height less than 194 cm. [3]
  2. The diagram shows the distribution of heights of Scottish women aged 25 to 34. \includegraphics{figure_7} The distribution is approximately normal. Use the diagram in the Printed Answer Booklet to estimate the standard deviation of these heights, explaining your method. [3]
OCR H240/02 2017 Specimen Q8
7 marks Moderate -0.8
A market gardener records the masses of a random sample of 100 of this year's crop of plums. The table shows his results.
Mass, \(m\) grams\(m < 25\)\(25 \leq m < 35\)\(35 \leq m < 45\)\(45 \leq m < 55\)\(55 \leq m < 65\)\(65 \leq m < 75\)\(m \geq 75\)
Number of plums0329363020
  1. Explain why the normal distribution might be a reasonable model for this distribution. [1]
The market gardener models the distribution of masses by \(N(47.5, 10^2)\).
  1. Find the number of plums in the sample that this model would predict to have masses in the range:
    1. \(35 \leq m < 45\) [2]
    2. \(m < 25\) [2]
  2. Use your answers to parts (b)(i) and (b)(ii) to comment on the suitability of this model. [1]
The market gardener plans to use this model to predict the distribution of the masses of next year's crop of plums.
  1. Comment on this plan. [1]
OCR H240/02 2017 Specimen Q9
4 marks Easy -1.8
The diagram below shows some "Cycle to work" data taken from the 2001 and 2011 UK censuses. The diagram shows the percentages, by age group, of male and female workers in England and Wales, excluding London, who cycled to work in 2001 and 2011. \includegraphics{figure_9} The following questions refer to the workers represented by the graphs in the diagram.
  1. A researcher is going to take a sample of men and a sample of women and ask them whether or not they cycle to work. Why would it be more important to stratify the sample of men? [1]
A research project followed a randomly chosen large sample of the group of male workers who were aged 30-34 in 2001.
  1. Does the diagram suggest that the proportion of this group who cycled to work has increased or decreased from 2001 to 2011? Justify your answer. [2]
  2. Write down one assumption that you have to make about these workers in order to draw this conclusion. [1]
OCR H240/02 2017 Specimen Q10
7 marks Standard +0.3
In the past, the time spent in minutes, by customers in a certain library had mean 32.5 and standard deviation 8.2. Following a change of layout in the library, the mean time spent in the library by a random sample of 50 customers is found to be 34.5 minutes. Assuming that the standard deviation remains at 8.2, test at the 5% significance level whether the mean time spent by customers in the library has changed. [7]
OCR H240/02 2017 Specimen Q11
8 marks Moderate -0.3
Each of the 30 students in a class plays at least one of squash, hockey and tennis. • 18 students play squash • 19 students play hockey • 17 students play tennis • 8 students play squash and hockey • 9 students play hockey and tennis • 11 students play squash and tennis
  1. Find the number of students who play all three sports. [3]
A student is picked at random from the class.
  1. Given that this student plays squash, find the probability that this student does not play hockey. [1]
Two different students are picked at random from the class, one after the other, without replacement.
  1. Given that the first student plays squash, find the probability that the second student plays hockey. [4]
OCR H240/02 2017 Specimen Q12
5 marks Challenging +1.2
The table shows information for England and Wales, taken from the UK 2011 census.
Total populationNumber of children aged 5-17
56 075 9128 473 617
A random sample of 10 000 people in another country was chosen in 2011, and the number, \(m\), of children aged 5-17 was noted. It was found that there was evidence at the 2.5% level that the proportion of children aged 5-17 in the same year was higher than in the UK. Unfortunately, when the results were recorded the value of \(m\) was omitted. Use an appropriate normal distribution to find an estimate of the smallest possible value of \(m\). [5]
OCR H240/02 2017 Specimen Q13
5 marks Moderate -0.8
The table and the four scatter diagrams below show data taken from the 2011 UK census for four regions. On the scatter diagrams the names have been replaced by letters. The table shows, for each region, the mean and standard deviation of the proportion of workers in each Local Authority who travel to work by driving a car or van and the proportion of workers in each Local Authority who travel to work as a passenger in a car or van. Each scatter diagram shows, for each of the Local Authorities in a particular region, the proportion of workers who travel to work by driving a car or van and the proportion of workers who travel to work as a passenger in a car or van. \includegraphics{figure_13}
  1. Using the values given in the table, match each region to its corresponding scatter diagram, explaining your reasoning. [3]
  2. Steven claims that the outlier in the scatter diagram for Region C consists of a group of small islands. Explain whether or not the data given above support his claim. [1]
  3. One of the Local Authorities in Region B consists of a single large island. Explain whether or not you would expect this Local Authority to appear as an outlier in the scatter diagram for Region B. [1]