Questions — OCR MEI (4301 questions)

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All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics Stats 1 Unit 1 Unit 2 Unit 3 Unit 4
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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
OCR MEI S1 Q3
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
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 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
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 2006 January Q2
2 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{7a4a7ff2-d196-4645-96f1-9c994caab0a2-2_526_524_541_767} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} Fig. 2 shows graphs \(A\) and \(B\).
  1. State the transformation which maps graph \(A\) onto graph \(B\).
  2. The equation of graph \(A\) is \(y = \mathrm { f } ( x )\). Which one of the following is the equation of graph \(B\) ? $$\begin{array} { l l l l } y = \mathrm { f } ( x ) + 2 & y = \mathrm { f } ( x ) - 2 & y = \mathrm { f } ( x + 2 ) & y = \mathrm { f } ( x - 2 )
    y = 2 \mathrm { f } ( x ) & y = \mathrm { f } ( x + 3 ) & y = \mathrm { f } ( x - 3 ) & y = 3 \mathrm { f } ( x ) \end{array}$$
OCR MEI C1 2006 January Q3
3 Find the binomial expansion of \(( 2 + x ) ^ { 4 }\), writing each term as simply as possible.
OCR MEI C1 2006 January Q4
4 Solve the inequality \(\frac { 3 ( 2 x + 1 ) } { 4 } > - 6\).
OCR MEI C1 2006 January Q5
5 Make \(C\) the subject of the formula \(P = \frac { C } { C + 4 }\).
OCR MEI C1 2006 January Q6
6 When \(x ^ { 3 } + 3 x + k\) is divided by \(x - 1\), the remainder is 6 . Find the value of \(k\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{7a4a7ff2-d196-4645-96f1-9c994caab0a2-3_577_1013_351_662} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure} The line AB has equation \(y = 4 x - 5\) and passes through the point \(\mathrm { B } ( 2,3 )\), as shown in Fig. 7. The line BC is perpendicular to AB and cuts the \(x\)-axis at C . Find the equation of the line BC and the \(x\)-coordinate of C .
OCR MEI C1 2006 January Q8
8
  1. Simplify \(5 \sqrt { 8 } + 4 \sqrt { 50 }\). Express your answer in the form \(a \sqrt { b }\), where \(a\) and \(b\) are integers and \(b\) is as small as possible.
  2. Express \(\frac { \sqrt { 3 } } { 6 - \sqrt { 3 } }\) in the form \(p + q \sqrt { 3 }\), where \(p\) and \(q\) are rational.
OCR MEI C1 2006 January Q9
9
  1. Find the range of values of \(k\) for which the equation \(x ^ { 2 } + 5 x + k = 0\) has one or more real roots.
  2. Solve the equation \(4 x ^ { 2 } + 20 x + 25 = 0\).
OCR MEI C1 2006 January Q10
10 A circle has equation \(x ^ { 2 } + y ^ { 2 } = 45\).
  1. State the centre and radius of this circle.
  2. The circle intersects the line with equation \(x + y = 3\) at two points, A and B. Find algebraically the coordinates of A and B . Show that the distance AB is \(\sqrt { 162 }\).
OCR MEI C1 2006 January Q11
11
  1. Write \(x ^ { 2 } - 7 x + 6\) in the form \(( x - a ) ^ { 2 } + b\).
  2. State the coordinates of the minimum point on the graph of \(y = x ^ { 2 } - 7 x + 6\).
  3. Find the coordinates of the points where the graph of \(y = x ^ { 2 } - 7 x + 6\) crosses the axes and sketch the graph.
  4. Show that the graphs of \(y = x ^ { 2 } - 7 x + 6\) and \(y = x ^ { 2 } - 3 x + 4\) intersect only once. Find the \(x\)-coordinate of the point of intersection.
OCR MEI C1 2006 January Q12
12
  1. Sketch the graph of \(y = x ( x - 3 ) ^ { 2 }\).
  2. Show that the equation \(x ( x - 3 ) ^ { 2 } = 2\) can be expressed as \(x ^ { 3 } - 6 x ^ { 2 } + 9 x - 2 = 0\).
  3. Show that \(x = 2\) is one root of this equation and find the other two roots, expressing your answers in surd form. Show the location of these roots on your sketch graph in part (i).
OCR MEI C1 2008 January Q1
1 Make \(v\) the subject of the formula \(E = \frac { 1 } { 2 } m v ^ { 2 }\).
OCR MEI C1 2008 January Q2
2 Factorise and hence simplify \(\frac { 3 x ^ { 2 } - 7 x + 4 } { x ^ { 2 } - 1 }\).
OCR MEI C1 2008 January Q3
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 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
  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
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
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
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
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
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
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.