Questions — OCR MEI (4455 questions)

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OCR MEI S1 Q2
7 marks Moderate -0.8
2 The total annual emissions of carbon dioxide, \(x\) tonnes per person, for 13 European countries are given below.
6.26.76.88.18.18.58.69.09.910.111.011.822.8
  1. Find the mean, median and midrange of these data.
  2. Comment on how useful each of these is as a measure of central tendency for these data, giving a brief reason for each of your answers.
OCR MEI S1 Q3
8 marks Easy -1.2
3 Every day, George attempts the quiz in a national newspaper. The quiz always consists of 7 questions. In the first 25 days of January, the numbers of questions George answers correctly each day are summarised in the table below.
  1. On the insert, draw a cumulative frequency diagram to illustrate the data.
  2. Use your graph to estimate the median length of journey and the quartiles. Hence find the interquartile range.
  3. State the type of skewness of the distribution of the data.
OCR MEI S1 Q5
20 marks Moderate -0.3
5 A pear grower collects a random sample of 120 pears from his orchard. The histogram below shows the lengths, in mm , of these pears. \includegraphics[max width=\textwidth, alt={}, center]{056d3e9a-088d-4c97-9546-7cecb59b8727-3_815_1628_505_304}
  1. Calculate the number of pears which are between 90 and 100 mm long.
  2. Calculate an estimate of the mean length of the pears. Explain why your answer is only an estimate.
  3. Calculate an estimate of the standard deviation.
  4. Use your answers to parts (ii) and (iii) to investigate whether there are any outliers.
  5. Name the type of skewness of the distribution.
  6. Illustrate the data using a cumulative frequency diagram.
OCR MEI S1 Q6
6 marks Easy -1.2
6 The times taken for 480 university students to travel from their accommodation to lectures are summarised below.
Time \(( t\) minutes \()\)\(0 \leqslant t < 5\)\(5 \leqslant t < 10\)\(10 \leqslant t < 20\)\(20 \leqslant t < 30\)\(30 \leqslant t < 40\)\(40 \leqslant t < 60\)
Frequency3415318873275
  1. Illustrate these data by means of a histogram.
  2. Identify the type of skewness of the distribution.
OCR MEI S1 Q5
22 marks Moderate -0.3
5 A pear grower collects a random sample of 120 pears from his orchard. The histogram below shows the lengths, in mm , of these pears. \includegraphics[max width=\textwidth, alt={}, center]{99c502aa-2c9f-461d-9dc0-ed55e3df32a2-3_815_1628_505_304}
  1. Calculate the number of pears which are between 90 and 100 mm long.
  2. Calculate an estimate of the mean length of the pears. Explain why your answer is only an estimate.
  3. Calculate an estimate of the standard deviation.
  4. Use your answers to parts (ii) and (iii) to investigate whether there are any outliers.
  5. Name the type of skewness of the distribution.
  6. Illustrate the data using a cumulative frequency diagram.
OCR MEI C1 2008 January Q1
3 marks Easy -1.8
1 Make \(v\) the subject of the formula \(E = \frac { 1 } { 2 } m v ^ { 2 }\).
OCR MEI C1 2008 January Q2
3 marks Moderate -0.8
2 Factorise and hence simplify \(\frac { 3 x ^ { 2 } - 7 x + 4 } { x ^ { 2 } - 1 }\).
OCR MEI C1 2008 January Q3
4 marks Easy -1.8
3
  1. Write down the value of \(\left( \frac { 1 } { 4 } \right) ^ { 0 }\).
  2. Find the value of \(16 ^ { - \frac { 3 } { 2 } }\).
OCR MEI C1 2008 January Q4
4 marks Easy -1.8
4 Find, algebraically, the coordinates of the point of intersection of the lines \(y = 2 x - 5\) and \(6 x + 2 y = 7\).
OCR MEI C1 2008 January Q5
5 marks Easy -1.2
5
  1. Find the gradient of the line \(4 x + 5 y = 24\).
  2. A line parallel to \(4 x + 5 y = 24\) passes through the point \(( 0,12 )\). Find the coordinates of its point of intersection with the \(x\)-axis.
OCR MEI C1 2008 January Q6
3 marks Moderate -0.8
6 When \(x ^ { 3 } + k x + 7\) is divided by \(( x - 2 )\), the remainder is 3 . Find the value of \(k\).
OCR MEI C1 2008 January Q7
4 marks Easy -1.3
7
  1. Find the value of \({ } ^ { 8 } \mathrm { C } _ { 3 }\).
  2. Find the coefficient of \(x ^ { 3 }\) in the binomial expansion of \(\left( 1 - \frac { 1 } { 2 } x \right) ^ { 8 }\).
OCR MEI C1 2008 January Q8
5 marks Easy -1.2
8
  1. Write \(\sqrt { 48 } + \sqrt { 3 }\) in the form \(a \sqrt { b }\), where \(a\) and \(b\) are integers and \(b\) is as small as possible.
  2. Simplify \(\frac { 1 } { 5 + \sqrt { 2 } } + \frac { 1 } { 5 - \sqrt { 2 } }\).
OCR MEI C1 2008 January Q9
5 marks Moderate -0.3
9
  1. Prove that 12 is a factor of \(3 n ^ { 2 } + 6 n\) for all even positive integers \(n\).
  2. Determine whether 12 is a factor of \(3 n ^ { 2 } + 6 n\) for all positive integers \(n\).
OCR MEI C1 2008 January Q10
11 marks Moderate -0.8
10
  1. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{450c1c3a-9290-4afa-a051-112b60cf19c0-3_753_775_360_726} \captionsetup{labelformat=empty} \caption{Fig. 10}
    \end{figure} Fig. 10 shows a sketch of the graph of \(y = \frac { 1 } { x }\).
    Sketch the graph of \(y = \frac { 1 } { x - 2 }\), showing clearly the coordinates of any points where it crosses the axes.
  2. Find the value of \(x\) for which \(\frac { 1 } { x - 2 } = 5\).
  3. Find the \(x\)-coordinates of the points of intersection of the graphs of \(y = x\) and \(y = \frac { 1 } { x - 2 }\). Give your answers in the form \(a \pm \sqrt { b }\). Show the position of these points on your graph in part (i).
OCR MEI C1 2008 January Q11
12 marks Moderate -0.8
11
  1. Write \(x ^ { 2 } - 5 x + 8\) in the form \(( x - a ) ^ { 2 } + b\) and hence show that \(x ^ { 2 } - 5 x + 8 > 0\) for all values of \(x\).
  2. Sketch the graph of \(y = x ^ { 2 } - 5 x + 8\), showing the coordinates of the turning point.
  3. Find the set of values of \(x\) for which \(x ^ { 2 } - 5 x + 8 > 14\).
  4. If \(\mathrm { f } ( x ) = x ^ { 2 } - 5 x + 8\), does the graph of \(y = \mathrm { f } ( x ) - 10\) cross the \(x\)-axis? Show how you decide.
OCR MEI C1 2008 January Q12
13 marks Moderate -0.3
12 A circle has equation \(x ^ { 2 } + y ^ { 2 } - 8 x - 4 y = 9\).
  1. Show that the centre of this circle is \(\mathrm { C } ( 4,2 )\) and find the radius of the circle.
  2. Show that the origin lies inside the circle.
  3. Show that AB is a diameter of the circle, where A has coordinates (2, 7) and B has coordinates \(( 6 , - 3 )\).
  4. Find the equation of the tangent to the circle at A . Give your answer in the form \(y = m x + c\).
OCR MEI C1 2009 January Q1
2 marks Easy -1.8
1 State the value of each of the following.
  1. \(2 ^ { - 3 }\)
  2. \(9 ^ { 0 }\)
OCR MEI C1 2009 January Q2
3 marks Easy -1.2
2 Find the equation of the line passing through \(( - 1 , - 9 )\) and \(( 3,11 )\). Give your answer in the form \(y = m x + c\).
OCR MEI C1 2009 January Q3
3 marks Easy -1.8
3 Solve the inequality \(7 - x < 5 x - 2\).
OCR MEI C1 2009 January Q4
3 marks Moderate -0.8
4 You are given that \(\mathrm { f } ( x ) = x ^ { 4 } + a x - 6\) and that \(x - 2\) is a factor of \(\mathrm { f } ( x )\).
Find the value of \(a\).
OCR MEI C1 2009 January Q5
5 marks Easy -1.2
5
  1. Find the coefficient of \(x ^ { 3 }\) in the expansion of \(\left( x ^ { 2 } - 3 \right) \left( x ^ { 3 } + 7 x + 1 \right)\).
  2. Find the coefficient of \(x ^ { 2 }\) in the binomial expansion of \(( 1 + 2 x ) ^ { 7 }\).
OCR MEI C1 2009 January Q6
3 marks Easy -1.8
6 Solve the equation \(\frac { 3 x + 1 } { 2 x } = 4\).
OCR MEI C1 2009 January Q7
4 marks Easy -1.3
7
  1. Express \(125 \sqrt { 5 }\) in the form \(5 ^ { k }\).
  2. Simplify \(\left( 4 a ^ { 3 } b ^ { 5 } \right) ^ { 2 }\).
OCR MEI C1 2009 January Q8
4 marks Moderate -0.5
8 Find the range of values of \(k\) for which the equation \(2 x ^ { 2 } + k x + 18 = 0\) does not have real roots.