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OCR MEI Paper 3 2019 June Q7
4 marks Standard +0.3
7 In this question you must show detailed reasoning.
  1. Express \(\ln 3 \times \ln 9 \times \ln 27\) in terms of \(\ln 3\).
  2. Hence show that \(\ln 3 \times \ln 9 \times \ln 27 > 6\).
OCR MEI Paper 3 2019 June Q8
10 marks Standard +0.3
8 In this question you must show detailed reasoning. A is the point \(( 1,0 ) , B\) is the point \(( 1,1 )\) and \(D\) is the point where the tangent to the curve \(y = x ^ { 3 }\) at B crosses the \(x\)-axis, as shown in Fig. 8. The tangent meets the \(y\)-axis at E. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{99485c27-9ff8-4bdb-a7e6-49dfcaedc579-6_1154_832_450_242} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure}
  1. Find the area of triangle ODE.
  2. Find the area of the region bounded by the curve \(y = x ^ { 3 }\), the tangent at B and the \(y\)-axis.
OCR MEI Paper 3 2019 June Q9
6 marks Standard +0.8
9 In this question you must show detailed reasoning.
The curve \(x y + y ^ { 2 } = 8\) is shown in Fig. 9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{99485c27-9ff8-4bdb-a7e6-49dfcaedc579-7_734_750_397_244} \captionsetup{labelformat=empty} \caption{Fig. 9}
\end{figure} Find the coordinates of the points on the curve at which the normal has gradient 2.
OCR MEI Paper 3 2019 June Q10
4 marks Moderate -0.3
10 Show that \(\mathrm { f } ( x ) = \frac { \mathrm { e } ^ { x } } { 1 + \mathrm { e } ^ { x } }\) is an increasing function for all values of \(x\).
OCR MEI Paper 3 2019 June Q11
6 marks Standard +0.3
11 By using the substitution \(u = 1 + \sqrt { x }\), find \(\int \frac { x } { 1 + \sqrt { x } } \mathrm {~d} x\). Answer all the questions.
OCR MEI Paper 3 2019 June Q12
2 marks Moderate -0.5
12 Show that the equation of the line in Fig. C2 is \(r y + h x = h r\), as given in line 24.
OCR MEI Paper 3 2019 June Q13
6 marks Standard +0.3
13
    1. Show that the cross-sectional area in Fig. C3.2 is \(\pi x ( 2 r - x )\).
    2. Hence show that the cross-sectional area is \(\frac { \pi r ^ { 2 } } { h ^ { 2 } } \left( h ^ { 2 } - y ^ { 2 } \right)\), as given in line 37 .
  2. Verify that the formula \(\frac { \pi r ^ { 2 } } { h ^ { 2 } } \left( h ^ { 2 } - y ^ { 2 } \right)\) for the cross-sectional area is also valid for
    1. Fig. C3.1,
    2. Fig. C3.3.
OCR MEI Paper 3 2019 June Q14
4 marks Challenging +1.2
14
  1. Express \(\lim _ { \delta y \rightarrow 0 } \sum _ { 0 } ^ { h } \left( h ^ { 2 } - y ^ { 2 } \right) \delta y\) as an integral.
  2. Hence show that \(V = \frac { 2 } { 3 } \pi r ^ { 2 } h\), as given in line 41 .
OCR MEI Paper 3 2019 June Q15
3 marks Easy -1.8
15 A typical tube of toothpaste measures 5.4 cm across the straight edge at the top and is 12 cm high. It contains 75 ml of toothpaste so it needs to have an internal volume of \(75 \mathrm {~cm} ^ { 3 }\). Comment on the accuracy of the formula \(V = \frac { 2 } { 3 } \pi r ^ { 2 } h\), as given in line 41 , for the volume in this case. \section*{END OF QUESTION PAPER}
AQA Paper 3 2018 June Q1
1 marks Easy -2.0
A circle has equation \((x - 4)^2 + (y + 4)^2 = 9\) What is the area of the circle? Circle your answer. [1 mark] \(3\pi\) \quad \(9\pi\) \quad \(16\pi\) \quad \(81\pi\)
AQA Paper 3 2018 June Q2
1 marks Easy -1.8
A curve has equation \(y = x^5 + 4x^3 + 7x + q\) where \(q\) is a positive constant. Find the gradient of the curve at the point where \(x = 0\) Circle your answer. [1 mark] \(0\) \quad \(4\) \quad \(7\) \quad \(q\)
AQA Paper 3 2018 June Q3
1 marks Easy -1.8
The line \(L\) has equation \(2x + 3y = 7\) Which one of the following is perpendicular to \(L\)? Tick one box. [1 mark] \(2x - 3y = 7\) \(3x + 2y = -7\) \(2x + 3y = -\frac{1}{7}\) \(3x - 2y = 7\)
AQA Paper 3 2018 June Q4
3 marks Easy -1.2
Sketch the graph of \(y = |2x + a|\), where \(a\) is a positive constant. Show clearly where the graph intersects the axes. [3 marks] \includegraphics{figure_4}
AQA Paper 3 2018 June Q5
3 marks Moderate -0.3
Show that, for small values of \(x\), the graph of \(y = 5 + 4\sin\frac{x}{2} + 12\tan\frac{x}{3}\) can be approximated by a straight line. [3 marks]
AQA Paper 3 2018 June Q6
13 marks Standard +0.8
A function \(f\) is defined by \(f(x) = \frac{x}{\sqrt{2x - 2}}\)
  1. State the maximum possible domain of \(f\). [2 marks]
  2. Use the quotient rule to show that \(f'(x) = \frac{x - 2}{(2x - 2)^{\frac{3}{2}}}\). [3 marks]
  3. Show that the graph of \(y = f(x)\) has exactly one point of inflection. [7 marks]
  4. Write down the values of \(x\) for which the graph of \(y = f(x)\) is convex. [1 mark]
AQA Paper 3 2018 June Q7
5 marks Moderate -0.8
  1. Given that \(\log_a y = 2\log_a 7 + \log_a 4 + \frac{1}{2}\), find \(y\) in terms of \(a\). [4 marks]
  2. When asked to solve the equation $$2\log_a x = \log_a 9 - \log_a 4$$ a student gives the following solution: \(2\log_a x = \log_a 9 - \log_a 4\) \(\Rightarrow 2\log_a x = \log_a \frac{9}{4}\) \(\Rightarrow \log_a x^2 = \log_a \frac{9}{4}\) \(\Rightarrow x^2 = \frac{9}{4}\) \(\therefore x = \frac{3}{2}\) or \(-\frac{3}{2}\) Explain what is wrong with the student's solution. [1 mark]
AQA Paper 3 2018 June Q8
9 marks Standard +0.3
  1. Prove the identity \(\frac{\sin 2x}{1 + \tan^2 x} = 2\sin x \cos^3 x\) [3 marks]
  2. Hence find \(\int \frac{4\sin 4\theta}{1 + \tan^2 2\theta} d\theta\) [6 marks]
AQA Paper 3 2018 June Q9
7 marks Standard +0.3
Helen is creating a mosaic pattern by placing square tiles next to each other along a straight line. \includegraphics{figure_9} The area of each tile is half the area of the previous tile, and the sides of the largest tile have length \(w\) centimetres.
  1. Find, in terms of \(w\), the length of the sides of the second largest tile. [1 mark]
  2. Assume the tiles are in contact with adjacent tiles, but do not overlap. Show that, no matter how many tiles are in the pattern, the total length of the series of tiles will be less than \(3.5w\). [4 marks]
  3. Helen decides the pattern will look better if she leaves a 3 millimetre gap between adjacent tiles. Explain how you could refine the model used in part (b) to account for the 3 millimetre gap, and state how the total length of the series of tiles will be affected. [2 marks]
AQA Paper 3 2018 June Q10
10 marks Standard +0.8
Prove by contradiction that \(\sqrt[3]{2}\) is an irrational number. [7 marks]
AQA Paper 3 2018 June Q11
1 marks Easy -1.8
The table below shows the probability distribution for a discrete random variable \(X\).
\(x\)12345
P(\(X = x\))\(k\)\(2k\)\(4k\)\(2k\)\(k\)
Find the value of \(k\). Circle your answer. [1 mark] \(\frac{1}{2}\) \quad \(\frac{1}{4}\) \quad \(\frac{1}{10}\) \quad \(1\)
AQA Paper 3 2018 June Q12
1 marks Easy -1.8
The histogram below shows the heights, in cm, of male A-level students at a particular school. \includegraphics{figure_12} Which class interval contains the median height? Circle your answer. [1 mark] \([155, 160)\) \quad \([160, 170)\) \quad \([170, 180)\) \quad \([180, 190]\)
AQA Paper 3 2018 June Q13
3 marks Easy -1.8
The table below shows an extract from the Large Data Set.
Year2011201220132014\% change since 2011
Other takeaway food brought home0000\(-29\)
Sarah claims that the \(-29\%\) change since 2011 is incorrect, as there is no change between 2011 and 2014. Using your knowledge of the Large Data Set to justify your answer, explain whether Sarah's claim is correct. [3 marks]
AQA Paper 3 2018 June Q14
6 marks Moderate -0.8
A teacher in a college asks her mathematics students what other subjects they are studying. She finds that, of her 24 students: 12 study physics 8 study geography 4 study geography and physics
  1. A student is chosen at random from the class. Determine whether the event 'the student studies physics' and the event 'the student studies geography' are independent. [2 marks]
  2. It is known that for the whole college: the probability of a student studying mathematics is \(\frac{1}{5}\) the probability of a student studying biology is \(\frac{1}{6}\) the probability of a student studying biology given that they study mathematics is \(\frac{3}{8}\) Calculate the probability that a student studies mathematics or biology or both. [4 marks]
AQA Paper 3 2018 June Q15
7 marks Easy -1.3
Abu visits his local hardware store to buy six light bulbs. He knows that 15% of all bulbs at this store are faulty.
  1. State a distribution which can be used to model the number of faulty bulbs he buys. [1 mark]
  2. Find the probability that all of the bulbs he buys are faulty. [1 mark]
  3. Find the probability that at least two of the bulbs he buys are faulty. [2 marks]
  4. Find the mean of the distribution stated in part (a). [1 mark]
  5. State two necessary assumptions in context so that the distribution stated in part (a) is valid. [2 marks]
AQA Paper 3 2018 June Q16
12 marks Moderate -0.3
A survey of 120 adults found that the volume, \(X\) litres per person, of carbonated drinks they consumed in a week had the following results: $$\sum x = 165.6 \quad \sum x^2 = 261.8$$
    1. Calculate the mean of \(X\). [1 mark]
    2. Calculate the standard deviation of \(X\). [2 marks]
  2. Assuming that \(X\) can be modelled by a normal distribution find
    1. P\((0.5 < X < 1.5)\) [2 marks]
    2. P\((X = 1)\) [1 mark]
  3. Determine with a reason, whether a normal distribution is suitable to model this data. [2 marks]
  4. It is known that the volume, \(Y\) litres per person, of energy drinks consumed in a week may be modelled by a normal distribution with standard deviation 0.21 Given that P\((Y > 0.75) = 0.10\), find the value of \(\mu\), correct to three significant figures. [4 marks]