Questions — OCR MEI C2 (480 questions)

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OCR MEI C2 Q4
4 marks Moderate -0.8
4 Fig. 2 shows the coordinates at certain points on a curve. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f56da008-e7f5-45b9-8db8-e2ba09ab0161-3_646_1149_285_530} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} Use the trapezium rule with 6 strips to calculate an estimate of the area of the region bounded by this curve and the axes.
OCR MEI C2 Q5
13 marks Standard +0.3
5 Fig. 10 shows a sketch of the graph of \(y = 7 x - x ^ { 2 } - 6\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f56da008-e7f5-45b9-8db8-e2ba09ab0161-4_608_908_290_663} \captionsetup{labelformat=empty} \caption{Fig. 10}
\end{figure}
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence find the equation of the tangent to the curve at the point on the curve where \(x = 2\). Show that this tangent crosses the \(x\)-axis where \(x = \frac { 2 } { 3 }\).
  2. Show that the curve crosses the \(x\)-axis where \(x = 1\) and find the \(x\)-coordinate of the other point of intersection of the curve with the \(x\)-axis.
  3. Find \(\int _ { 1 } ^ { 2 } \left( 7 x - x ^ { 2 } - 6 \right) \mathrm { d } x\). Hence find the area of the region bounded by the curve, the tangent and the \(x\)-axis, shown shaded on Fig. 10.
OCR MEI C2 Q1
12 marks Moderate -0.8
1
  1. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{1a6d059d-8ab8-41e0-8bf3-54e248f820e4-1_650_759_252_762} \captionsetup{labelformat=empty} \caption{Fig. 12}
    \end{figure} Fig. 12 shows part of the curve \(y = x ^ { 4 }\) and the line \(y = 8 x\), which intersect at the origin and the point P .
    (A) Find the coordinates of P , and show that the area of triangle OPQ is 16 square units.
    (B) Find the area of the region bounded by the line and the curve.
  2. You are given that \(\mathrm { f } ( x ) = x ^ { 4 }\).
    (A) Complete this identity for \(\mathrm { f } ( x + h )\). $$\mathrm { f } ( x + h ) = ( x + h ) ^ { 4 } = x ^ { 4 } + 4 x ^ { 3 } h + \ldots$$ (B) Simplify \(\frac { \mathrm { f } ( x + h ) - \mathrm { f } ( x ) } { h }\).
    (C) Find \(\lim _ { h \rightarrow 0 } \frac { \mathrm { f } ( x + h ) - \mathrm { f } ( x ) } { h }\).
    (D) State what this limit represents.
OCR MEI C2 Q2
5 marks Moderate -0.8
2 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1a6d059d-8ab8-41e0-8bf3-54e248f820e4-2_622_979_232_553} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure} Fig. 4 shows a curve which passes through the points shown in the following table.
\(x\)11.522.533.54
\(y\)8.26.45.55.04.74.44.2
Use the trapezium rule with 6 strips to estimate the area of the region bounded by the curve, the lines \(x = 1\) and \(x = 4\), and the \(x\)-axis. State, with a reason, whether the trapezium rule gives an overestimate or an underestimate of the area of this region.
[0pt] [5]
OCR MEI C2 Q3
12 marks Moderate -0.3
3
  1. A tunnel is 100 m long. Its cross-section, shown in Fig. 9.1, is modelled by the curve $$y = \frac { 1 } { 4 } \left( 10 x - x ^ { 2 } \right) ,$$ where \(x\) and \(y\) are horizontal and vertical distances in metres. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{1a6d059d-8ab8-41e0-8bf3-54e248f820e4-3_512_819_493_700} \captionsetup{labelformat=empty} \caption{Figure 9.1}
    \end{figure} Using this model,
    (A) find the greatest height of the tunnel,
    (B) explain why \(100 \int _ { 0 } ^ { 10 } y \mathrm {~d} x\) gives the volume, in cubic metres, of earth removed to make the tunnel. Calculate this volume.
    [0pt] [5]
  2. The roof of the tunnel is re-shaped to allow for larger vehicles. Fig. 9.2 shows the new crosssection. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{1a6d059d-8ab8-41e0-8bf3-54e248f820e4-3_506_942_1703_629} \captionsetup{labelformat=empty} \caption{Not to scale}
    \end{figure} Fig. 9.2 Use the trapezium rule with 5 strips to estimate the new cross-sectional area.
    Hence estimate the volume of earth removed when the tunnel is re-shaped.
OCR MEI C2 Q1
12 marks Moderate -0.8
1
  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 Q2
3 marks Easy -1.2
2 The first term of a geometric series is 5.4 and the common ratio is 0.1.
  1. Find the fourth term of the series.
  2. Find the sum to infinity of the series.
OCR MEI C2 Q3
5 marks Moderate -0.3
3 The 11th term of an arithmetic progression is 1. The sum of the first 10 terms is 120. Find the 4th term.
OCR MEI C2 Q4
12 marks Moderate -0.3
4
  1. André is playing a game where he makes piles of counters. He puts 3 counters in the first pile. Each successive pile he makes has 2 more counters in it than the previous one.
    1. How many counters are there in his sixth pile?
    2. André makes ten piles of counters. How many counters has he used altogether?
  2. In another game, played with an ordinary fair die and counters, Betty needs to throw a six to start. The probability \(\mathrm { P } _ { n }\) of Betty starting on her \(n\)th throw is given by $$P _ { n } = \frac { 1 } { 6 } \times \left( \frac { 5 } { 6 } \right) ^ { n - 1 }$$
    1. Calculate \(\mathrm { P } _ { 4 }\). Give your answer as a fraction.
    2. The values \(\mathrm { P } _ { 1 } , \mathrm { P } _ { 2 } , \mathrm { P } _ { 3 } , \ldots\) form an infinite geometric progression. State the first term and the common ratio of this progression. Hence show that \(\mathrm { P } _ { 1 } + \mathrm { P } _ { 2 } + \mathrm { P } _ { 3 } + \ldots = 1\).
    3. Given that \(\mathrm { P } _ { n } < 0.001\), show that \(n\) satisfies the inequality $$n > \frac { \log _ { 10 } 0.006 } { \log _ { 10 } \left( \frac { 5 } { 6 } \right) } + 1$$ Hence find the least value of \(n\) for which \(\mathrm { P } _ { n } < 0.001\).
OCR MEI C2 Q5
3 marks Moderate -0.8
5 The first term of a geometric series is 8. The sum to infinity of the series is 10 .
Find the common ratio.
OCR MEI C2 Q1
5 marks Standard +0.3
1 The 7th term of an arithmetic progression is 6. The sum of the first 10 terms of the progression is 30. Find the 5th term of the progression.
OCR MEI C2 Q2
5 marks Easy -1.2
2 The first three terms of a geometric progression are 4, 2, 1.
Find the twentieth term, expressing your answer as a power of 2 .
Find also the sum to infinity of this progression.
OCR MEI C2 Q3
5 marks Easy -1.2
3 A sequence is given by $$\begin{gathered} a _ { 1 } = 4 \\ a _ { r + 1 } = a _ { r } + 3 \end{gathered}$$ Write down the first 4 terms of this sequence.
Find the sum of the first 100 terms of the sequence.
OCR MEI C2 Q4
11 marks Standard +0.3
4 There is a flowerhead at the end of each stem of an oleander plant. The next year, each flowerhead is replaced by three stems and flowerheads, as shown in Fig. 11. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f291e6e3-975e-4d1e-aab6-67308f305da2-2_517_1116_356_455} \captionsetup{labelformat=empty} \caption{Fig. 11}
\end{figure}
  1. How many flowerheads are there in year 5 ?
  2. How many flowerheads are there in year \(n\) ?
  3. As shown in Fig. 11, the total number of stems in year 2 is 4, (that is, 1 old one and 3 new ones). Similarly, the total number of stems in year 3 is 13 , (that is, \(1 + 3 + 9\) ). Show that the total number of stems in year \(n\) is given by \(\frac { 3 ^ { n } - 1 } { 2 }\).
  4. Kitty's oleander has a total of 364 stems. Find
    (A) its age,
    (B) how many flowerheads it has.
  5. Abdul's oleander has over 900 flowerheads. Show that its age, \(y\) years, satisfies the inequality \(y > \frac { \log _ { 10 } 900 } { \log _ { 10 } 3 } + 1\).
    Find the smallest integer value of \(y\) for which this is true.
OCR MEI C2 Q1
3 marks Moderate -0.8
1 A sequence is defined by \(u _ { 1 } = 2\) and \(u _ { k + 1 } = \frac { 10 } { u _ { k } ^ { 2 } }\).
Calculate \(\sum _ { k = 1 } ^ { 4 } u _ { k }\).
OCR MEI C2 Q2
5 marks Easy -1.2
2 The \(n\)th term of a sequence, \(u _ { n }\), is given by $$u _ { n } = 12 - \frac { 1 } { 2 } n .$$
  1. Write down the values of \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\). State what type of sequence this is.
  2. Find \(\sum _ { n = 1 } ^ { 30 } u _ { n }\).
OCR MEI C2 Q3
3 marks Moderate -0.8
3 A sequence is defined by $$\begin{aligned} u _ { 1 } & = 10 \\ u _ { r + 1 } & = \frac { 5 } { u _ { r } ^ { 2 } } \end{aligned}$$ Calculate the values of \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\).
What happens to the terms of the sequence as \(r\) tends to infinity?
OCR MEI C2 Q4
2 marks Moderate -0.8
4 The \(n\)th term, \(t _ { n }\), of a sequence is given by $$t _ { n } = \sin ( \theta + 180 n ) ^ { \circ }$$ Express \(t _ { 1 }\) and \(t _ { 2 }\) in terms of \(\sin \theta ^ { \circ }\).
OCR MEI C2 Q5
13 marks Moderate -0.8
5 Jim and Mary are each planning monthly repayments for money they want to borrow.
  1. Jim's first payment is \(\pounds 500\), and he plans to pay \(\pounds 10\) less each month, so that his second payment is \(\pounds 490\), his third is \(\pounds 480\), and so on.
    (A) Calculate his 12th payment.
    (B) He plans to make 24 payments altogether. Show that he pays \(\pounds 9240\) in total.
  2. Mary's first payment is \(\pounds 460\) and she plans to pay \(2 \%\) less each month than the previous month, so that her second payment is \(\pounds 450.80\), her third is \(\pounds 441.784\), and so on.
    (A) Calculate her 12th payment.
    (B) Show that Jim's 20th payment is less than Mary's 20th payment but that his 19th is not less than her 19th.
    (C) Mary plans to make 24 payments altogether. Calculate how much she pays in total.
    (D) How much would Mary's first payment need to be if she wishes to pay \(2 \%\) less each month as before, but to pay the same in total as Jim, \(\pounds 9240\), over the 24 months?
OCR MEI C2 Q6
2 marks Easy -1.2
6 You are given that $$\begin{aligned} u _ { 1 } & = 1 \\ u _ { n + 1 } & = \frac { u _ { n } } { 1 + u _ { n } } \end{aligned}$$ Find the values of \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\). Give your answers as fractions.
OCR MEI C2 Q7
4 marks Easy -1.3
7
  1. Evaluate \(\sum _ { r = 2 } ^ { 5 } \frac { 1 } { r - 1 }\).
  2. Express the series \(2 \times 3 + 3 \times 4 + 4 \times 5 + 5 \times 6 + 6 \times 7\) in the form \(\sum _ { r = 2 } ^ { a } \mathrm { f } ( r )\) where \(\mathrm { f } ( r )\) and \(a\) are to be determined.
OCR MEI C2 Q8
3 marks Moderate -0.8
8
  1. Find \(\sum _ { k = 3 } ^ { 8 } \left( k ^ { 2 } - 1 \right)\).
  2. State whether the sequence with \(k\) th term \(k ^ { 2 } - 1\) is convergent or divergent, giving a reason for your answer.
OCR MEI C2 Q9
4 marks Easy -1.2
9
  1. Find the second and third terms of the sequence defined by the following. $$\begin{aligned} t _ { n + 1 } & = 2 t _ { n } + 5 \\ t _ { 1 } & = 3 \end{aligned}$$
  2. Find \(\sum _ { k = 1 } ^ { 3 } k ( k + 1 )\).
OCR MEI C2 Q10
3 marks Easy -1.2
10 For each of the following sequences, state with a reason whether it is convergent, periodic or neither. Each sequence continues in the pattern established by the given terms.
  1. \(3 , \frac { 3 } { 2 } , \frac { 3 } { 4 } , \frac { 3 } { 8 } , \ldots\)
  2. \(3,7,11,15 , \ldots\)
  3. \(3,5 , - 3 , - 5,3,5 , - 3 , - 5 , \ldots\)
OCR MEI C2 Q11
2 marks Easy -1.2
11 Find \(\sum _ { r = 3 } ^ { 6 } r ( r + 2 )\).