Questions — OCR MEI (4456 questions)

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OCR MEI Paper 2 2021 November Q12
5 marks Moderate -0.5
12 Fig. 12.1 shows an excerpt from the pre-release material. \begin{table}[h]
ABCDEFGH
1SexAgeMaritalWeightHeightBMIWaistPulse
2Female34Married60.3173.420.0582.574
3Female85Widowed64.7161.224.9\#N/A\#N/A
4Female48Divorced100.6171.434.24105.692
5Male61Married70.9169.524.6892.270
6Male68Divorced96.8181.629.35112.968
\captionsetup{labelformat=empty} \caption{Fig. 12.1}
\end{table} There was no data available for cell H3.
  1. Explain why \#N/A is used when no data is available. Fig. 12.2 shows a scatter diagram of pulse rate against BMI (Body Mass Index) for females. All the available data was used. Pulse rate against BMI for females \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{c9d14a4d-a1c8-42ad-9c0b-42cef6b3612f-08_659_1552_1363_233} \captionsetup{labelformat=empty} \caption{Fig. 12.2}
    \end{figure} There are two outliers on the diagram.
  2. On the copy of Fig. 12.2 in the Printed Answer Booklet, ring these outliers.
  3. Use your knowledge of the pre-release material to explain whether either of these outliers should be removed.
  4. State whether the diagram suggests there is any correlation between pulse rate and BMI. The product moment correlation coefficient between waist measurement, \(w\), in cm and BMI, \(b\), for females was found to be 0.912 . All the available data was used.
  5. Explain why a model of the form \(\mathrm { w } = \mathrm { mb } + \mathrm { c }\) for the relationship between waist measurement and BMI is likely to be appropriate. The LINEST function on a spreadsheet gives \(m = 2.16\) and \(c = 33.0\).
  6. Calculate an estimate of the value for cell G3 in Fig. 12.1.
OCR MEI Paper 2 2021 November Q13
7 marks Moderate -0.3
13 At a certain factory Christmas tree decorations are packed in boxes of 10 . The quality control manager collects a random sample of 100 boxes of decorations and records the number of decorations in each box which are damaged. His results are displayed in Fig. 13.1. \begin{table}[h]
Number of damaged decorations012345 or more
Number of boxes1935281350
\captionsetup{labelformat=empty} \caption{Fig. 13.1}
\end{table}
  1. Calculate
    It is believed that the number of damaged decorations in a box of 10, \(X\), may be modelled by a binomial distribution such that \(\mathrm { X } \sim \mathrm { B } ( \mathrm { n } , \mathrm { p } )\).
  2. State suitable values for \(n\) and \(p\).
  3. Use the binomial model to complete the copy of Fig. 13.2 in the Printed Answer Booklet, giving your answers correct to \(\mathbf { 1 }\) decimal place. \begin{table}[h]
    Number of damaged decorations012345 or more
    Observed number of boxes1935281350
    Expected number of boxes
    \captionsetup{labelformat=empty} \caption{Fig. 13.2}
    \end{table}
  4. Explain whether the model is a good fit for these data.
OCR MEI Paper 2 2021 November Q14
13 marks Moderate -0.3
14 The equation of a curve is \(y = x ^ { 2 } ( x - 2 ) ^ { 3 }\).
  1. Find \(\frac { \mathrm { dy } } { \mathrm { dx } }\), giving your answer in factorised form.
  2. Determine the coordinates of the stationary points on the curve. In part (c) you may use the result \(\frac { d ^ { 2 } y } { d x ^ { 2 } } = 4 ( x - 2 ) \left( 5 x ^ { 2 } - 8 x + 2 \right)\).
  3. Determine the nature of the stationary points on the curve.
  4. Sketch the curve.
OCR MEI Paper 2 2021 November Q15
11 marks Moderate -0.8
15
  1. Show that \(\sum _ { r = 1 } ^ { \infty } 0.99 ^ { r - 1 } \times 0.01 = 1\). Kofi is a very good table tennis player. Layla is determined to beat him.
    Every week they play one match of table tennis against each other. They will stop playing when Layla wins the match for the first time. \(X\) is the discrete random variable "the number of matches they play in total". Kofi models the situation using the probability function \(\mathrm { P } ( \mathrm { X } = \mathrm { r } ) = 0.99 ^ { \mathrm { r } - 1 } \times 0.01 \quad r = 1,2,3,4 , \ldots\) Kofi states that he is \(95 \%\) certain that Layla will not beat him within 6 years.
  2. Determine whether Kofi's statement is consistent with his model. In between matches, Layla practises, but Kofi does not.
  3. Explain why Layla might disagree with Kofi's model. Layla models the situation using the probability function \(\mathrm { P } ( \mathrm { X } = \mathrm { r } ) = \mathrm { kr } ^ { 2 } \quad r = 1,2,3,4,5,6,7,8\).
  4. Explain how Layla's model takes into account the fact that she practises between matches, but Kofi's does not. Layla states that she is \(95 \%\) certain that she will beat Kofi within the first 6 matches.
  5. Determine whether Layla's statement is consistent with her model.
OCR MEI Paper 2 2021 November Q16
8 marks Standard +0.8
16 In this question you must show detailed reasoning.
Find \(\int \frac { x } { 1 + \sqrt { x } } d x\). END OF QUESTION PAPER
OCR MEI Paper 3 2018 June Q1
3 marks Easy -1.2
1 Triangle ABC is shown in Fig. 1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{31bc8bde-8d37-4e97-94e2-e3e73aab55e9-4_451_565_520_744} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} Find the perimeter of triangle ABC .
OCR MEI Paper 3 2018 June Q2
2 marks Easy -1.2
2 The curve \(y = x ^ { 3 } - 2 x\) is translated by the vector \(\binom { 1 } { - 4 }\). Write down the equation of the translated curve. [2]
OCR MEI Paper 3 2018 June Q3
2 marks Challenging +1.2
3 Fig. 3 shows a circle with centre O and radius 1 unit. Points A and B lie on the circle with angle \(\mathrm { AOB } = \theta\) radians. C lies on AO , and BC is perpendicular to AO . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{31bc8bde-8d37-4e97-94e2-e3e73aab55e9-4_648_627_1507_717} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} Show that, when \(\theta\) is small, \(\mathrm { AC } \approx \frac { 1 } { 2 } \theta ^ { 2 }\).
OCR MEI Paper 3 2018 June Q4
10 marks Standard +0.3
4 In this question you must show detailed reasoning.
A curve has equation \(y = x - 5 + \frac { 1 } { x - 2 }\). The curve is shown in Fig. 4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{31bc8bde-8d37-4e97-94e2-e3e73aab55e9-5_723_844_424_612} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure}
  1. Determine the coordinates of the stationary points on the curve.
  2. Determine the nature of each stationary point.
  3. Write down the equation of the vertical asymptote.
  4. Deduce the set of values of \(x\) for which the curve is concave upwards.
OCR MEI Paper 3 2018 June Q5
11 marks Moderate -0.3
5 A social media website launched on 1 January 2017. The owners of the website report the number of users the site has at the start of each month. They believe that the relationship between the number of users, \(n\), and the number of months after launch, \(t\), can be modelled by \(n = a \times 2 ^ { k t }\) where \(a\) and \(k\) are constants.
  1. Show that, according to the model, the graph of \(\log _ { 10 } n\) against \(t\) is a straight line.
  2. Fig. 5 shows a plot of the values of \(t\) and \(\log _ { 10 } n\) for the first seven months. The point at \(t = 1\) is for 1 February 2017, and so on. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{31bc8bde-8d37-4e97-94e2-e3e73aab55e9-6_831_1442_609_388} \captionsetup{labelformat=empty} \caption{Fig. 5}
    \end{figure} Find estimates of the values of \(a\) and \(k\).
  3. The owners of the website wanted to know the date on which they would report that the website had half a million users. Use the model to estimate this date.
  4. Give a reason why the model may not be appropriate for large values of \(t\).
OCR MEI Paper 3 2018 June Q6
2 marks Moderate -0.3
6 Find the constant term in the expansion of \(\left( x ^ { 2 } + \frac { 1 } { x } \right) ^ { 15 }\).
OCR MEI Paper 3 2018 June Q7
8 marks Standard +0.3
7 In this question you must show detailed reasoning.
Fig. 7 shows the curve \(y = 5 x - x ^ { 2 }\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{31bc8bde-8d37-4e97-94e2-e3e73aab55e9-7_511_684_383_694} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure} The line \(y = 4 - k x\) crosses the curve \(y = 5 x - x ^ { 2 }\) on the \(x\)-axis and at one other point.
Determine the coordinates of this other point.
OCR MEI Paper 3 2018 June Q8
8 marks Standard +0.8
8 A curve has parametric equations \(x = \frac { t } { 1 + t ^ { 3 } } , y = \frac { t ^ { 2 } } { 1 + t ^ { 3 } }\), where \(t \neq - 1\).
  1. In this question you must show detailed reasoning. Determine the gradient of the curve at the point where \(t = 1\).
  2. Verify that the cartesian equation of the curve is \(x ^ { 3 } + y ^ { 3 } = x y\).
OCR MEI Paper 3 2018 June Q9
4 marks Standard +0.3
9 The function \(\mathrm { f } ( x ) = \frac { \mathrm { e } ^ { x } } { 1 - \mathrm { e } ^ { x } }\) is defined on the domain \(x \in \mathbb { R } , x \neq 0\).
  1. Find \(\mathrm { f } ^ { - 1 } ( x )\).
  2. Write down the range of \(\mathrm { f } ^ { - 1 } ( x )\).
OCR MEI Paper 3 2018 June Q10
10 marks Challenging +1.2
10 Point A has position vector \(\left( \begin{array} { l } a \\ b \\ 0 \end{array} \right)\) where \(a\) and \(b\) can vary, point B has position vector \(\left( \begin{array} { l } 4 \\ 2 \\ 0 \end{array} \right)\) and point C has position vector \(\left( \begin{array} { l } 2 \\ 4 \\ 2 \end{array} \right)\). ABC is an isosceles triangle with \(\mathrm { AC } = \mathrm { AB }\).
  1. Show that \(a - b + 1 = 0\).
  2. Determine the position vector of A such that triangle ABC has minimum area. Answer all the questions.
    Section B (15 marks) The questions in this section refer to the article on the Insert. You should read the article before attempting the questions.
OCR MEI Paper 3 2018 June Q11
2 marks Moderate -0.8
11 Line 8 states that \(\frac { a + b } { 2 } \geqslant \sqrt { a b }\) for \(a\), \(b \geqslant 0\). Explain why the result cannot be extended to apply in each of the following cases.
  1. One of the numbers \(a\) and \(b\) is positive and the other is negative.
  2. Both numbers \(a\) and \(b\) are negative.
OCR MEI Paper 3 2018 June Q12
3 marks Standard +0.3
12 Lines 5 and 6 outline the stages in a proof that \(\frac { a + b } { 2 } \geqslant \sqrt { a b }\). Starting from \(( a - b ) ^ { 2 } \geqslant 0\), give a detailed proof of the inequality of arithmetic and geometric means.
OCR MEI Paper 3 2018 June Q13
3 marks Moderate -0.5
13 Consider a geometric sequence in which all the terms are positive real numbers. Show that, for any three consecutive terms of this sequence, the middle one is the geometric mean of the other two.
OCR MEI Paper 3 2018 June Q14
4 marks Standard +0.8
14
  1. In Fig. C1.3, angle CBD \(= \theta\). Show that angle CDA is also \(\theta\), as given in line 23 .
  2. Prove that \(h = \sqrt { a b }\), as given in line 24 .
OCR MEI Paper 3 2018 June Q15
3 marks Challenging +1.2
15 It is given in lines \(31 - 32\) that the square has the smallest perimeter of all rectangles with the same area. Using this fact, prove by contradiction that among rectangles of a given perimeter, \(4 L\), the square with side \(L\) has the largest area. \section*{END OF QUESTION PAPER}
OCR MEI Paper 3 2022 June Q1
2 marks Easy -1.8
1 A curve for which \(y\) is inversely proportional to \(x\) is shown below. \includegraphics[max width=\textwidth, alt={}, center]{c30a926b-d832-46f5-aa65-0066ef482c3d-4_824_1125_561_242} Find the equation of the curve.
OCR MEI Paper 3 2022 June Q2
6 marks Moderate -0.3
2 The function \(\mathrm { f } ( x ) = \sqrt { x }\) is defined on the domain \(x \geqslant 0\).
The function \(\mathrm { g } ( x ) = 25 - x ^ { 2 }\) is defined on the domain \(\mathbb { R }\).
  1. Write down an expression for \(\mathrm { fg } ( x )\).
    1. Find the domain of \(\mathrm { fg } ( x )\).
    2. Find the range of \(\mathrm { fg } ( x )\).
OCR MEI Paper 3 2022 June Q3
4 marks Moderate -0.3
3 An infinite sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by \(a _ { \mathrm { n } } = \frac { \mathrm { n } } { \mathrm { n } + 1 }\), for all positive integers \(n\).
  1. Find the limit of the sequence.
  2. Prove that this is an increasing sequence.
OCR MEI Paper 3 2022 June Q4
5 marks Standard +0.3
4 In this question you must show detailed reasoning.
Determine the exact solutions of the equation \(2 \cos ^ { 2 } x = 3 \sin x\) for \(0 \leqslant x \leqslant 2 \pi\).
OCR MEI Paper 3 2022 June Q5
7 marks Standard +0.8
5 A curve is defined implicitly by the equation \(2 x ^ { 2 } + 3 x y + y ^ { 2 } + 2 = 0\).
  1. Show that \(\frac { d y } { d x } = - \frac { 4 x + 3 y } { 3 x + 2 y }\).
  2. In this question you must show detailed reasoning. Find the coordinates of the stationary points of the curve.