Questions Paper 2 (402 questions)

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OCR MEI Paper 2 Specimen Q15
5 marks
15 A quality control department checks the lifetimes of batteries produced by a company. The lifetimes, \(x\) minutes, for a random sample of 80 'Superstrength' batteries are shown in the table below.
Lifetime\(160 \leq x < 165\)\(165 \leq x < 168\)\(168 \leq x < 170\)\(170 \leq x < 172\)\(172 \leq x < 175\)\(175 \leq x < 180\)
Frequency5142021164
  1. Estimate the proportion of these batteries which have a lifetime of at least 174.0 minutes.
  2. Use the data in the table to estimate
    • the sample mean,
    • the sample standard deviation.
    The data in the table on the previous page are represented in the following histogram, Fig 15: \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{e9f3a5f3-210b-453d-9ff5-8518340f5689-10_728_1577_312_315} \captionsetup{labelformat=empty} \caption{Fig. 15}
    \end{figure} A quality control manager models the data by a Normal distribution with the mean and standard deviation you calculated in part (b).
  3. Comment briefly on whether the histogram supports this choice of model.
    1. Use this model to estimate the probability that a randomly selected battery will have a lifetime of more than 174.0 minutes.
    2. Compare your answer with your answer to part (a). The company also manufactures 'Ultrapower' batteries, which are stated to have a mean lifetime of 210 minutes.
  4. A random sample of 8 Ultrapower batteries is selected. The mean lifetime of these batteries is 207.3 minutes. Carry out a hypothesis test at the \(5 \%\) level to investigate whether the mean lifetime is as high as stated. You should use the following hypotheses \(\mathrm { H } _ { 0 } : \mu = 210 , \mathrm { H } _ { 1 } : \mu < 210\), where \(\mu\) represents the population mean for Ultrapower batteries. You should assume that the population is Normally distributed with standard deviation 3.4.
    [0pt] [5]
OCR MEI Paper 2 Specimen Q16
16 Fig. 16.1, Fig. 16.2 and Fig. 16.3 show some data about life expectancy, including some from the pre-release data set. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Life expectancy at birth 1974 for 193 countries} \includegraphics[alt={},max width=\textwidth]{e9f3a5f3-210b-453d-9ff5-8518340f5689-11_520_1671_479_182}
\end{figure} Life expectancy at birth 2014 for 222 countries \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Fig. 16.1} \includegraphics[alt={},max width=\textwidth]{e9f3a5f3-210b-453d-9ff5-8518340f5689-11_589_1477_1324_349}
\end{figure} Fig. 16.2
\includegraphics[max width=\textwidth, alt={}, center]{e9f3a5f3-210b-453d-9ff5-8518340f5689-11_207_828_2213_310} Increase in life expectancy from 1974 to 2014 (years) \begin{table}[h]
Increase in life expectancy for
193 countries from 1974 to 2014
Number of values193
Minimum- 4.618
Lower quartile6.9576
Median9.986
Upper quartile15.873
Maximum30.742
\captionsetup{labelformat=empty} \caption{Fig. 16.3}
\end{table} Source: CIA World
Factbook and
  1. Comment on the shapes of the distributions of life expectancy at birth in 2014 and 1974.
    1. The minimum value shown in the box plot is negative. What does a negative value indicate?
    2. What feature of Fig 16.3 suggests that a Normal distribution would not be an appropriate model for increase in life expectancy from one year to another year?
    3. Software has been used to obtain the values in the table in Fig. 16.3. Decide whether the level of accuracy is appropriate. Justify your answer.
    4. John claims that for half the people in the world their life expectancy has improved by 10 years or more.
      Explain why Fig. 16.3 does not provide conclusive evidence for John's claim.
  2. Decide whether the maximum increase in life expectancy from 1974 to 2014 is an outlier. Justify your answer. Here is some further information from the pre-release data set.
    Country
    Life expectancy
    at birth in 2014
    Ethiopia60.8
    Sweden81.9
    1. Estimate the change in life expectancy at birth for Ethiopia between 1974 and 2014.
    2. Estimate the change in life expectancy at birth for Sweden between 1974 and 2014.
    3. Give one possible reason why the answers to parts (i) and (ii) are so different. \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Fig. 16.4 shows the relationship between life expectancy at birth in 2014 and 1974.} \includegraphics[alt={},max width=\textwidth]{e9f3a5f3-210b-453d-9ff5-8518340f5689-13_981_1520_333_292}
      \end{figure} Fig. 16.4 A spreadsheet gives the following linear model for all the data in Fig 16.4.
      (Life expectancy at birth 2014) \(= 30.98 + 0.67 \times\) (Life expectancy at birth 1974) The life expectancy at birth in 1974 for the region that now constitutes the country of South Sudan was 37.4 years. The value for this country in 2014 is not available.
    1. Use the linear model to estimate the life expectancy at birth in 2014 for South Sudan.
    2. Give two reasons why your answer to part (i) is not likely to be an accurate estimate for the life expectancy at birth in 2014 for South Sudan.
      You should refer to both information from Fig 16.4 and your knowledge of the large data set.
  3. In how many of the countries represented in Fig. 16.4 did life expectancy drop between 1974 and 2014? Justify your answer.
OCR MEI Paper 2 2024 June Q5
  1. In the Printed Answer Booklet, complete the copy of the two-way table.
  2. Calculate the probability that an A-level student selected at random does not study chemistry given that they do not study mathematics.
OCR MEI Paper 2 2020 November Q12
  1. Given that \(q < 2 p\), determine the values of \(p\) and \(q\).
  2. The spinner is spun 10 times. Calculate the probability that exactly one 5 is obtained. Elaine's teacher believes that the probability that the spinner shows a 1 is greater than 0.2 . The spinner is spun 100 times and gives a score of 1 on 28 occasions.
  3. Conduct a hypothesis test at the \(5 \%\) level to determine whether there is any evidence to suggest that the probability of obtaining a score of 1 is greater than 0.2 .
OCR MEI Paper 2 2018 June Q12
12 You must show detailed reasoning in this question. In the summer of 2017 in England a large number of candidates sat GCSE examinations in both mathematics and English. 56\% of these candidates achieved at least level 4 in mathematics and \(80 \%\) of these candidates achieved at least level 4 in English. 14\% of these candidates did not achieve at least level 4 in either mathematics or English. Determine whether achieving level 4 or above in English and achieving level 4 or above in mathematics were independent events.
OCR MEI Paper 2 2019 June Q15
15 You must show detailed reasoning in this question. The screenshot in Fig. 15 shows the probability distribution for the continuous random variable \(X\), where \(X \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{95eb3bcc-6d3c-4f7e-9b27-5e046ab57ec5-11_387_954_1599_260} \captionsetup{labelformat=empty} \caption{Fig. 15}
\end{figure} The distribution is symmetrical about the line \(x = 35\) and there is a point of inflection at \(x = 31\).
Fifty independent readings of \(X\) are made. Show that the probability that at least 45 of these readings are between 30 and 40 is less than 0.05 .
OCR MEI Paper 2 2023 June Q15
15 In this question you must show detailed reasoning. The equation of a curve is $$\ln y + x ^ { 3 } y = 8$$ Find the equation of the normal to the curve at the point where \(y = 1\), giving your answer in the form \(\mathrm { ax } + \mathrm { by } + \mathrm { c } = 0\), where \(a , b\) and \(c\) are constants to be found.
OCR MEI Paper 2 2023 June Q17
17 In this question you must show detailed reasoning. Solve the equation \(2 \sin x + \sec x = 4 \cos x\), where \(- \pi < x < \pi\).
OCR MEI Paper 2 2024 June Q16
16 In this question you must show detailed reasoning. Find the particular solution of the differential equation $$\frac { d y } { d x } = \frac { 9 y } { ( x - 1 ) ( x + 2 ) }$$ given that \(x = 2\) when \(y = 16\). \section*{END OF QUESTION PAPER}
OCR MEI Paper 2 2020 November Q10
10 In this question you must show detailed reasoning. The equation of a curve is $$y = \frac { \sin 2 x - x } { x \sin x }$$
  1. Use the small angle approximation given in the list of formulae on pages 2-3 of this question paper to show that $$\int _ { 0.01 } ^ { 0.05 } \mathrm { ydx } \approx \ln 5$$
  2. Use the same small angle approximation to show that $$\frac { d y } { d x } \approx - 10000 \text { at the point where } x = 0.01 \text {. }$$ The equation \(y = 0\) has a root near \(x = 1\). Joan uses the Newton-Raphson method to find this root. The output from the spreadsheet she uses is shown in Fig. 10.1. \begin{table}[h]
    \(n\)01234567
    \(\mathrm { x } _ { \mathrm { n } }\)10.9585090.9500840.9482610.947860.9477720.9477530.947748
    \captionsetup{labelformat=empty} \caption{Fig. 10.1}
    \end{table} Joan carries out some analysis of this output. The results are shown in Fig. 10.2. \begin{table}[h]
    \(x\)\(y\)
    0.9477475\(- 7.79967 \mathrm { E } - 07\)
    0.9477485\(- 2.90821 \mathrm { E } - 06\)
    \(x\)\(y\)
    0.947745\(4.54066 \mathrm { E } - 06\)
    0.947755\(- 1.67417 \mathrm { E } - 05\)
    \captionsetup{labelformat=empty} \caption{Fig. 10.2}
    \end{table}
  3. Consider the information in Fig. 10.1 and Fig. 10.2.
    • Write 4.54066E-06 in standard mathematical notation.
    • State the value of the root as accurately as you can, justifying your answer.
OCR MEI Paper 2 2020 November Q14
14 In this question you must show detailed reasoning. Fig. 14 shows the graphs of \(y = \sin x \cos 2 x\) and \(y = \frac { 1 } { 2 } - \sin 2 x \cos x\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cea67565-8074-4703-8e1a-09b98e380baf-16_647_898_404_233} \captionsetup{labelformat=empty} \caption{Fig. 14}
\end{figure} Use integration to find the area between the two curves, giving your answer in an exact form.
OCR MEI Paper 2 Specimen Q1
1 In this question you must show detailed reasoning. Find the coordinates of the points of intersection of the curve \(y = x ^ { 2 } + x\) and the line \(2 x + y = 4\).
AQA Paper 2 2018 June Q1
1 Which of these statements is correct? Tick one box. $$\begin{aligned} & x = 2 \Rightarrow x ^ { 2 } = 4
& x ^ { 2 } = 4 \Rightarrow x = 2
& x ^ { 2 } = 4 \Leftrightarrow x = 2
& x ^ { 2 } = 4 \Rightarrow x = - 2 \end{aligned}$$
AQA Paper 2 2018 June Q2
2 Find the coefficient of \(x ^ { 2 }\) in the expansion of \(( 1 + 2 x ) ^ { 7 }\)
Circle your answer. 4242184
\includegraphics[max width=\textwidth, alt={}, center]{5e475c96-6bbb-44fb-a411-706dac20e2d6-03_1442_1168_219_365} Find the total shaded area. Circle your answer.
-68 60686128
AQA Paper 2 2018 June Q4
4 A curve, \(C\), has equation \(y = x ^ { 2 } - 6 x + k\), where \(k\) is a constant. The equation \(x ^ { 2 } - 6 x + k = 0\) has two distinct positive roots. 4
  1. Sketch \(C\) on the axes below.
    \includegraphics[max width=\textwidth, alt={}, center]{5e475c96-6bbb-44fb-a411-706dac20e2d6-04_1013_1016_534_513} 4
  2. Find the range of possible values for \(k\). Fully justify your answer.
    \begin{center} \begin{tabular}{ | l | } \hline
AQA Paper 2 2018 June Q5
2 marks
5 Prove that 23 is a prime number.
[0pt] [2 marks]
\end{tabular} \end{center}
AQA Paper 2 2018 June Q6
6 Find the coordinates of the stationary point of the curve with equation $$( x + y - 2 ) ^ { 2 } = \mathrm { e } ^ { y } - 1$$ \(7 \quad\) A function f has domain \(\mathbb { R }\) and range \(\{ y \in \mathbb { R } : y \geq \mathrm { e } \}\) The graph of \(y = \mathrm { f } ( x )\) is shown.
\includegraphics[max width=\textwidth, alt={}, center]{5e475c96-6bbb-44fb-a411-706dac20e2d6-08_922_1108_447_466} The gradient of the curve at the point \(( x , y )\) is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = ( x - 1 ) \mathrm { e } ^ { x }\)
Find an expression for \(\mathrm { f } ( x )\).
Fully justify your answer.
AQA Paper 2 2018 June Q8
8
  1. Determine a sequence of transformations which maps the graph of \(y = \sin x\) onto the graph of \(y = \sqrt { 3 } \sin x - 3 \cos x + 4\) Fully justify your answer.
    8
  2. (ii) Find the greatest value of \(\frac { 1 } { \sqrt { 3 } \sin x - 3 \cos x + 4 }\)
AQA Paper 2 2018 June Q9
9 A market trader notices that daily sales are dependent on two variables:
number of hours, \(t\), after the stall opens
total sales, \(x\), in pounds since the stall opened.
The trader models the rate of sales as directly proportional to \(\frac { 8 - t } { x }\)
After two hours the rate of sales is \(\pounds 72\) per hour and total sales are \(\pounds 336\)
9
  1. Show that $$x \frac { \mathrm {~d} x } { \mathrm {~d} t } = 4032 ( 8 - t )$$ 9
  2. Hence, show that $$x ^ { 2 } = 4032 t ( 16 - t )$$ \(\mathbf { 9 }\) (c) The stall opens at 09.30. 9
    1. The trader closes the stall when the rate of sales falls below \(\pounds 24\) per hour.
      Using the results in parts (a) and (b), calculate the earliest time that the trader closes the stall.
      9
  3. (ii) Explain why the model used by the trader is not valid at 09.30.
AQA Paper 2 2018 June Q10
10 A garden snail moves in a straight line from rest to \(1.28 \mathrm {~cm} \mathrm {~s} ^ { - 1 }\), with a constant acceleration in 1.8 seconds. Find the acceleration of the snail. Circle your answer.
\(2.30 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
\(0.71 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
\(0.0071 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
\(0.023 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
AQA Paper 2 2018 June Q11
1 marks
11 A uniform rod, \(A B\), has length 4 metres.
The rod is resting on a support at its midpoint \(C\).
A particle of mass 4 kg is placed 0.6 metres to the left of \(C\).
Another particle of mass 1.5 kg is placed \(x\) metres to the right of \(C\), as shown.
\includegraphics[max width=\textwidth, alt={}, center]{5e475c96-6bbb-44fb-a411-706dac20e2d6-16_277_908_1521_568} The rod is balanced in equilibrium at \(C\).
Find \(x\). Circle your answer.
[0pt] [1 mark]
\(1.8 \mathrm {~m} \quad 1.5 \mathrm {~m} \quad 1.75 \mathrm {~m} \quad 1.6 \mathrm {~m}\)
AQA Paper 2 2018 June Q12
12 The graph below shows the velocity of an object moving in a straight line over a 20 second journey.
\includegraphics[max width=\textwidth, alt={}, center]{5e475c96-6bbb-44fb-a411-706dac20e2d6-17_579_1682_406_169} 12
  1. Find the maximum magnitude of the acceleration of the object. 12
  2. The object is at its starting position at times \(0 , t _ { 1 }\) and \(t _ { 2 }\) seconds.
    Find \(t _ { 1 }\) and \(t _ { 2 }\)
AQA Paper 2 2018 June Q13
13 In this question use \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) A boy attempts to move a wooden crate of mass 20 kg along horizontal ground. The coefficient of friction between the crate and the ground is 0.85 13
  1. The boy applies a horizontal force of 150 N . Show that the crate remains stationary.
    13
  2. Instead, the boy uses a handle to pull the crate forward. He exerts a force of 150 N , at an angle of \(15 ^ { \circ }\) above the horizontal, as shown in the diagram.
    \includegraphics[max width=\textwidth, alt={}, center]{5e475c96-6bbb-44fb-a411-706dac20e2d6-19_244_915_408_561} Determine whether the crate remains stationary.
    Fully justify your answer.
AQA Paper 2 2018 June Q14
14 A quadrilateral has vertices \(A , B , C\) and \(D\) with position vectors given by $$\overrightarrow { O A } = \left[ \begin{array} { l } 3
5
1 \end{array} \right] , \overrightarrow { O B } = \left[ \begin{array} { r } - 1
2
7 \end{array} \right] , \overrightarrow { O C } = \left[ \begin{array} { l } 0
7
6 \end{array} \right] \text { and } \overrightarrow { O D } = \left[ \begin{array} { r } 4
10
0 \end{array} \right]$$ 14
  1. Write down the vector \(\overrightarrow { A B }\) 14
  2. Show that \(A B C D\) is a parallelogram, but not a rhombus.
AQA Paper 2 2018 June Q15
15 A driver is road-testing two minibuses, \(A\) and \(B\), for a taxi company. The performance of each minibus along a straight track is compared.
A flag is dropped to indicate the start of the test.
Each minibus starts from rest.
The acceleration in \(\mathrm { ms } ^ { - 2 }\) of each minibus is modelled as a function of time, \(t\) seconds, after the flag is dropped: The acceleration of \(\mathrm { A } = 0.138 t ^ { 2 }\)
The acceleration of \(\mathrm { B } = 0.024 t ^ { 3 }\)
15
  1. Find the time taken for A to travel 100 metres.
    Give your answer to four significant figures.
    15
  2. The company decides to buy the minibus which travels 100 metres in the shortest time. Determine which minibus should be bought.
    15
  3. The models assume that both minibuses start moving immediately when \(t = 0\) In light of this, explain why the company may, in reality, make the wrong decision.
    A particle is projected with an initial speed \(u\), at an angle of \(35 ^ { \circ }\) above the horizontal.
    It lands at a point 10 metres vertically below its starting position.
    The particle takes 1.5 seconds to reach the highest point of its trajectory.