Questions — OCR MEI (4301 questions)

Browse by module
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
Browse by board
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 C2 2009 June Q8
5 marks Moderate -0.8
8 The gradient of a curve is \(3 \sqrt { x } - 5\). The curve passes through the point ( 4,6 ). Find the equation of the curve.
OCR MEI C2 2009 June Q9
3 marks Easy -1.2
9 Simplify
  1. \(10 - 3 \log _ { a } a\),
  2. \(\frac { \log _ { 10 } a ^ { 5 } + \log _ { 10 } \sqrt { a } } { \log _ { 10 } a }\). Section B (36 marks)
OCR MEI C2 2009 June Q11
12 marks Moderate -0.8
11
  1. In a 'Make Ten' quiz game, contestants get \(\pounds 10\) for answering the first question correctly, then a further \(\pounds 20\) for the second question, then a further \(\pounds 30\) for the third, and so on, until they get a question wrong and are out of the game.
    (A) Haroon answers six questions correctly. Show that he receives a total of \(\pounds 210\).
    (B) State, in a simple form, a formula for the total amount received by a contestant who answers \(n\) questions correctly. Hence find the value of \(n\) for a contestant who receives \(\pounds 10350\) from this game.
  2. In a 'Double Your Money' quiz game, contestants get \(\pounds 5\) for answering the first question correctly, then a further \(\pounds 10\) for the second question, then a further \(\pounds 20\) for the third, and so on doubling the amount for each question until they get a question wrong and are out of the game.
    (A) Gary received \(\pounds 75\) from the game. How many questions did he get right?
    (B) Bethan answered 9 questions correctly. How much did she receive from the game?
    (C) State a formula for the total amount received by a contestant who answers \(n\) questions correctly. Hence find the value of \(n\) for a contestant in this game who receives \(\pounds 2621435\).
OCR MEI C2 2009 June Q12
12 marks Moderate -0.8
12
  1. Calculate the gradient of the chord joining the points on the curve \(y = x ^ { 2 } - 7\) for which \(x = 3\) and \(x = 3.1\).
  2. Given that \(\mathrm { f } ( x ) = x ^ { 2 } - 7\), find and simplify \(\frac { \mathrm { f } ( 3 + h ) - \mathrm { f } ( 3 ) } { h }\).
  3. Use your result in part (ii) to find the gradient of \(y = x ^ { 2 } - 7\) at the point where \(x = 3\), showing your reasoning.
  4. Find the equation of the tangent to the curve \(y = x ^ { 2 } - 7\) at the point where \(x = 3\).
  5. This tangent crosses the \(x\)-axis at the point P . The curve crosses the positive \(x\)-axis at the point Q . Find the distance PQ , giving your answer correct to 3 decimal places.
OCR MEI C2 Q1
4 marks Easy -1.2
1 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 - 5 x\).
Find the equation of the curve given that it passes through the point \(( 0,1 )\).
OCR MEI C2 Q2
5 marks Easy -1.2
2
  1. Write \(\log _ { 2 } 5 + \log _ { 2 } 1.6\) as an integer.
  2. Solve the equation \(2 ^ { x } = 3\), giving your answer correct to 4 decimal places.
OCR MEI C2 Q3
6 marks Easy -1.2
3 On his \(1 ^ { \text {st } }\) birthday, John was given \(\pounds 5\) by his Uncle Fred. On each succeeding birthday, Uncle Fred gave a sum of money that was \(\pounds 3\) more than the amount he gave on the last birthday.
  1. How much did Uncle Fred give John on his \(8 { } ^ { \text {th } }\) birthday?
  2. On what birthday did the gift from Uncle Fred result in the total sum given on all birthdays exceeding £200?
OCR MEI C2 Q4
4 marks Moderate -0.8
4 Find the equation of the tangent to the curve \(y = x ^ { 3 } + 2 x - 7\) at the point where it cuts the \(y\) axis.
OCR MEI C2 Q5
5 marks Moderate -0.3
5
  1. Express \(2 \sin ^ { 2 } \theta + 3 \cos \theta\) as a quadratic function of \(\cos \theta\).
  2. Hence solve the equation \(2 \sin ^ { 2 } \theta + 3 \cos \theta = 3\), giving all values of \(\theta\) correct to the nearest degree in the range \(0 ^ { \circ } \leq \theta \leq 360 ^ { \circ }\).
OCR MEI C2 Q6
4 marks Easy -1.2
6 The angle of a sector of a circle is 2 radians and the length of the arc of the sector is 45 cm .
Find
  1. the radius of the circle,
  2. the area of the sector.
OCR MEI C2 Q7
3 marks Moderate -0.8
7 The first two terms of a geometric series are 5 and 4.
Find
  1. the sum of the first 10 terms,
  2. the sum to infinity.
OCR MEI C2 Q8
5 marks Easy -1.2
8 In the triangle \(\mathrm { ABC } , \mathrm { AB } = 5 \mathrm {~cm} , \mathrm { AC } = 6 \mathrm {~cm}\) and angle \(\mathrm { BAC } = 110 ^ { \circ }\).
Find the length of the side BC .
OCR MEI C2 Q9
13 marks Moderate -0.3
9 The equation of a curve is given by \(y = ( x - 1 ) ^ { 2 } ( x + 2 )\).
  1. Write \(( x - 1 ) ^ { 2 } ( x + 2 )\) in the form \(x ^ { 3 } + p x ^ { 2 } + q x + r\) where \(p , q\) and \(r\) are to be determined.
  2. Show that the curve \(y = ( x - 1 ) ^ { 2 } ( x + 2 )\) has a maximum point when \(x = - 1\) and find the coordinates of the minimum point.
  3. Sketch the curve \(y = ( x - 1 ) ^ { 2 } ( x + 2 )\).
  4. For what values of \(k\) does \(( x - 1 ) ^ { 2 } ( x + 2 ) = k\) have exactly one root.
OCR MEI C2 Q10
12 marks Moderate -0.3
10 A function \(y = \mathrm { f } ( x )\) may be modelled by the equation \(y = a x ^ { b }\).
  1. Show why, if this is so, then plotting \(\log y\) against \(\log x\) will produce a straight line graph. Explain how \(a\) and \(b\) may be determined experimentally from the graph.
  2. Values of \(x\) and \(y\) are given below. By plotting a graph of logy against log \(x\), show that the model above is appropriate for this set of data and find values of \(a\) and \(b\) given that \(a\) is an integer and \(b\) can be written as a fraction with a denominator less than 10 .
    \(x\)23456
    \(y\)4.65.05.35.55.7
  3. Use your formula from part (ii) to estimate the value of \(y\) when \(x = 2.8\).
OCR MEI C2 Q11
11 marks Moderate -0.3
11 The cross-section of a brick wall built on horizontal ground is given, for \(0 \leq x \leq 6\), by the following function $$\begin{array} { l l } 0 \leq x \leq 2 & y = 1 \\ 2 \leq x \leq 4 & y = - \frac { 1 } { 2 } x ^ { 2 } + 3 x - 3 \\ 4 \leq x \leq 6 & y = 1 \end{array}$$
\includegraphics[max width=\textwidth, alt={}]{13bfa97b-ec49-4f41-b3dd-d9a31a2c30e8-4_523_1327_633_413}
Units are metres.
  1. Show that the highest point on the wall is 1.5 metres above the ground.
  2. Find the area of the cross-section of the wall.
OCR MEI C2 Q1
3 marks Easy -1.2
1 Find all the angles in the range \(0 ^ { 0 } \leq x \leq 360 ^ { 0 }\) satisfying the equation \(\sin x + \frac { 1 } { 2 } \sqrt { 3 } = 0\).
OCR MEI C2 Q2
3 marks Easy -1.2
2 Solve the equation \(3 ^ { x } = 15\), giving your answer correct to 4 decimal places.
OCR MEI C2 Q3
3 marks Moderate -0.8
3 The sum to infinity of a geometric series is 5 and the first term is 2 .
Find the common ratio of the series.
OCR MEI C2 Q4
5 marks Moderate -0.3
4 The first 3 terms of an arithmetical progression are 7, 5.9 and 4.8.
Find
  1. the common difference,
  2. the smallest value of \(n\) for which the sum to \(n\) terms is negative.
OCR MEI C2 Q5
4 marks Easy -1.2
5 The gradient of a curve is given by the function \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 - x\).
The curve passes through the point \(( 1,2 )\).
Find the equation of the curve.
OCR MEI C2 Q6
5 marks Easy -1.2
6 Evaluate \(\int _ { 1 } ^ { 2 } \left( x ^ { 2 } + \frac { 1 } { x ^ { 2 } } \right) \mathrm { d } x\).
OCR MEI C2 Q7
5 marks Easy -1.2
7
  1. Using the triangle, show that \(\sin ^ { 2 } x + \cos ^ { 2 } x = 1\).
  2. Hence prove that
    \includegraphics[max width=\textwidth, alt={}]{73d1c02b-1b7b-426d-a171-c762597cfed4-2_255_501_1779_1022} \(1 + \tan ^ { 2 } x = \frac { 1 } { \cos ^ { 2 } x }\).
OCR MEI C2 Q8
4 marks Easy -1.8
8 Draw two sketches of the graph of \(y = \sin x\) in the range \(0 ^ { \circ } \leq x \leq 360 ^ { \circ }\).
  1. On the first sketch, draw also a sketch of \(y = \sin ( 2 x )\).
  2. On the second sketch, draw also a sketch of \(y = 2 \sin x\).
OCR MEI C2 Q9
4 marks Moderate -0.8
9 A sector of a circle has an angle of 0.8 radians. The arc length is 5 cm . Calculate the radius of the circle and the area of the sector.
OCR MEI C2 Q10
12 marks Standard +0.3
10 At 1200 the captain of a ship observes that the bearing of a lighthouse is \(340 ^ { \circ }\). His position is at A.
At 1230 he takes another bearing of the lighthouse and finds it to be \(030 ^ { \circ }\). During this time the ship moves on a constant course of \(280 ^ { \circ }\) to the point B . His plot on the chart is as shown in Fig. 11 below. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{73d1c02b-1b7b-426d-a171-c762597cfed4-3_501_1156_661_387} \captionsetup{labelformat=empty} \caption{Fig. 11}
\end{figure}
  1. Write down the size of the angles LAB and LBA .
  2. The captain believes that at A he is 5 km from L . Assuming that LA is exactly 5 km , show that LB is 4.61 km , correct to 2 decimal places, and find AB . Hence calculate the speed of the ship.
  3. The speed of the ship is actually 10 kilometres per hour. Given that the bearings of \(340 ^ { \circ }\) and \(030 ^ { \circ }\) and the ship's course of \(280 ^ { \circ }\) are all accurate, calculate the true value of the distance LA.