Questions C2 (1410 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 Q7
7 The gradient of a curve \(y = \mathrm { f } ( x )\) is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } - 10 x + 6\). The curve passes through the point \(( 2,3 )\) Find the equation of the curve.
OCR MEI C2 Q8
8 In the triangle ABC shown, \(\mathrm { AB } = 8 \mathrm {~cm}\). \(\mathrm { AC } = 12 \mathrm {~cm}\) and angle \(\mathrm { ABC } = 82 ^ { \circ }\). Find \(\theta\) correct to 3 significant figures.
\includegraphics[max width=\textwidth, alt={}, center]{1c52d6b5-84b4-455a-9620-c377ae457069-3_382_540_1492_718}
OCR MEI C2 Q9
9 Fig. 9 shows
\(P \quad\) The line \(y = x\)
\(Q\) The curve \(y = \sqrt { \frac { 1 } { 2 } \left( x + x ^ { 2 } \right) }\)
\(R \quad\) The curve \(\quad y = \sqrt { x }\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1c52d6b5-84b4-455a-9620-c377ae457069-4_471_1103_762_374} \captionsetup{labelformat=empty} \caption{Fig. 9}
\end{figure}
  1. Write down the area of the triangle formed by the line \(y = x\), the line \(x = 1\) and the \(x\)-axis.
  2. Show that the area of the region formed by the curve \(y = \sqrt { x }\), the line \(x = 1\) and the \(x\)-axis is \(\frac { 2 } { 3 }\). An estimate is required of the Area, \(A\), of the region formed by the curve \(y = \sqrt { \frac { 1 } { 2 } \left( x + x ^ { 2 } \right) }\), the line \(x = 1\) and the \(x\)-axis.
  3. Use results to parts (i) and (ii) to complete the statement $$\ldots \ldots \ldots \ldots . . < A < \ldots \ldots \ldots \ldots \ldots . .$$
  4. Use the Trapezium Rule with 4 strips to find an estimate for \(A\).
  5. Draw a sketch of Fig. 9. Use it to illustrate the area found as the trapezium rule estimate for \(A\).
    Explain how your diagram shows that the trapezium rule estimate must be:
    consistent with the answer to part (iv);
    an under-estimate for A .
OCR MEI C2 Q10
10 A culture of bacteria is observed during an experiment. The number of bacteria is denoted by \(N\) and the time in hours after the start of the experiment by \(t\).
The table gives observations of \(t\) and \(N\).
Time \(( t\) hours \()\)12345
Number of bacteria \(( N )\)120170250370530
  1. Plot the points \(( t , N )\) on graph paper and join them with a smooth curve.
  2. Explain why the curve suggests why the relationship connecting \(t\) and \(N\) may be of the form \(N = a b ^ { t }\).
  3. Explain how, by using logarithms, the curve given by plotting \(N\) against \(t\) can be transformed into a straight line.
    State the gradient of this straight line and its intercept with the vertical axis in terms of \(a\) and \(b\).
  4. Complete a table of values for \(\log _ { 10 } N\) and plot the points \(\left( t , \log _ { 10 } N \right)\) on graph paper. Draw the best fit line through the points and use it to estimate the values of \(a\) and \(b\).
OCR MEI C2 Q11
11 The sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots \ldots\) where \(a _ { 1 }\) is a given real number is defined by \(a _ { n + 1 } = 1 - \frac { 1 } { a _ { n } }\).
  1. For the case when \(a _ { 1 } = 2\), find \(a _ { 2 } , a _ { 3 }\) and \(a _ { 4 }\). Describe the behaviour of this sequence
  2. For the case when \(a _ { 1 } = k\), where \(k\) is an integer greater than 1 , find \(a _ { 2 }\) in terms of \(k\) as a single fraction.
    Find also \(a _ { 3 }\) in its simplest form and hence deduce that \(a _ { 4 } = k\).
  3. Show that \(a _ { 2 } a _ { 3 } a _ { 4 } = - 1\) for any integer \(k\).
  4. When \(a _ { 1 } = 2\) evaluate \(\sum _ { i = 1 } ^ { 99 } a _ { i }\).
OCR MEI C2 Q1
1
  1. Find \(\int \left( x ^ { 3 } - 2 x \right) \mathrm { d } x\). The graph below shows part of the curve \(y = x ^ { 3 } - 2 x\) for \(0 \leq x \leq 2\).
    \includegraphics[max width=\textwidth, alt={}, center]{c55a5f04-3573-4f36-a12c-3755bdd4a45b-2_528_1019_520_321}
  2. Show that the area of the shaded region \(P\) is the same as the area of the shaded region \(Q\).
OCR MEI C2 Q2
2 The growth in population \(P\) of a certain town after time \(t\) years can be modelled by the equation \(P = 11000 \times 10 ^ { k t }\) where \(k\) is a constant.
  1. State the initial population of the town.
  2. After three years the population of the town is 24000 . Use this information to find the value of \(k\) correct to two decimal places.
OCR MEI C2 Q3
3
  1. Write \(\log _ { 10 } ( x + 4 ) - 2 \log _ { 10 } x + \log _ { 10 } ( x + 16 )\) as a single logarithm.
  2. Without using your calculator, verify that \(x = 4\) is a root of the equation $$\log _ { 10 } ( x + 4 ) - 2 \log _ { 10 } x + \log _ { 10 } ( x + 16 ) = 1$$
OCR MEI C2 Q4
4 Find the values of \(\theta\) such that \(0 ^ { \circ } \leq \theta \leq 360 ^ { \circ }\) which satisfy the equation $$\cos \theta \tan \theta = \frac { \sqrt { 3 } } { 2 }$$
OCR MEI C2 Q5
5 The diagram shows the curve \(y = \mathrm { f } ( x )\) where \(a\) is a positive constant.
\includegraphics[max width=\textwidth, alt={}, center]{c55a5f04-3573-4f36-a12c-3755bdd4a45b-3_551_962_255_476} Sketch the following curves on separate diagrams, in each case stating the coordinates of points where they meet the \(x\) - and \(y\)-axes.
  1. \(\quad y = - \mathrm { f } ( x )\)
  2. \(\quad y = \mathrm { f } ( - x )\)
OCR MEI C2 Q6
6 A and B are points on the same side of a straight river. A and B are 180 metres apart. The angles made with a jetty J on the opposite side of the river \(78 ^ { \circ }\) and \(56 ^ { \circ }\) respectively as shown.
\includegraphics[max width=\textwidth, alt={}, center]{c55a5f04-3573-4f36-a12c-3755bdd4a45b-3_332_681_1451_565} Not to scale Calculate the width of the river correct to the nearest metre.
OCR MEI C2 Q7
7 For each of the following sequences, write down sufficient terms of the sequence in order to be able to describe its behaviour as divergent, periodic or convergent. For any convergent sequence, state its limit.
  1. \(a _ { 1 } = - 1 ; \quad a _ { k + 1 } = \frac { 4 } { a _ { k } }\)
  2. \(\quad a _ { 1 } = 1 ; \quad a _ { k } = 2 - 2 \times \left( \frac { 1 } { 2 } \right) ^ { k }\)
  3. \(\quad a _ { 1 } = 0 \quad a _ { k + 1 } = \left( 1 + a _ { k } \right) ^ { 2 }\).
OCR MEI C2 Q8
8 Fig. 8 shows a sector of a circle with centre O and radius 6 cm and a chord AB which subtends an angle of 1.8 radians at O . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c55a5f04-3573-4f36-a12c-3755bdd4a45b-4_341_485_310_771} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure}
  1. Calculate the area of the sector OAXB .
  2. Calculate the area of the triangle OAB and hence find the area of the shaded segment AXB.
OCR MEI C2 Q9
9 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } - 12 x + 9\). The curve passes through the point \(( 2 , - 2 )\).
  1. Find the equation of the curve.
  2. Show that the curve touches the \(x\)-axis at one point (A) and cuts it at another (B). State the coordinates of A and B.
  3. The curve cuts the \(y\)-axis at C . Show that the tangent at C is perpendicular to the normal at B.
OCR MEI C2 Q10
10 Fig. 10 shows the curve with equation \(y = x ^ { 2 } + \frac { 16 } { x }\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c55a5f04-3573-4f36-a12c-3755bdd4a45b-5_522_1019_403_394} \captionsetup{labelformat=empty} \caption{Fig. 10}
\end{figure}
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Hence calculate the coordinates of the stationary point on the curve.
  3. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) and explain why this confirms that he stationary point is a minimum.
  4. Using the trapezium rule with 4 intervals, estimate the area between the curve and the \(x\) axis between \(x = 2\) and \(x = 4\).
  5. State, giving a reason, whether this estimate of the area under-estimates or over-estimates the true area beneath the curve.
OCR MEI C2 Q11
11 When Fred joined a computer firm his salary was \(\pounds 28000\) per annum. In each subsequent year he received an annual increase of \(12 \%\) of his previous year's salary.
  1. State Fred's salary for each of his first 3 years with the company. State also the common ratio of the geometric sequence formed by his salaries.
  2. How much did Fred earn in the tenth year?
  3. Show that the total amount Fred earned over the ten years was between \(\pounds 400000\) and £500000.
  4. When Fred joined the computer firm, his brother Archie set up a plumbing business. He earned \(\pounds 35000\) in his first year and each year earned \(\pounds d\) more than in the previous year. At the end of ten years, he had earned exactly the same total amount as Fred. Calculate the value of \(d\).
OCR C2 Q2
2. \(f ( x ) = x ^ { 3 } + k x - 20\). Given that \(\mathrm { f } ( x )\) is exactly divisible by ( \(x + 1\) ),
  1. find the value of the constant \(k\),
  2. solve the equation \(\mathrm { f } ( x ) = 0\).
OCR C2 Q3
3. Given that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 \sqrt { x } - x ^ { 2 }$$ and that \(y = \frac { 2 } { 3 }\) when \(x = 1\), find the value of \(y\) when \(x = 4\).
OCR C2 Q4
4. A geometric progression has third term 36 and fourth term 27. Find
  1. the common ratio,
  2. the fifth term,
  3. the sum to infinity.
OCR C2 Q5
5. (i) Solve the equation $$\log _ { 2 } ( 6 - x ) = 3 - \log _ { 2 } x$$ (ii) Find the smallest integer \(n\) such that $$3 ^ { n - 2 } > 8 ^ { 250 }$$
OCR C2 Q6
  1. \(f ( x ) = \cos 2 x , 0 \leq x \leq \pi\).
    1. Sketch the curve \(y = \mathrm { f } ( x )\).
    2. Write down the coordinates of any points where the curve \(y = \mathrm { f } ( x )\) meets the coordinate axes.
    3. Solve the equation \(\mathrm { f } ( x ) = 0.5\), giving your answers in terms of \(\pi\).
    4. (i) Find
    $$\int \left( x + 5 + \frac { 3 } { \sqrt { x } } \right) \mathrm { d } x$$
  2. Evaluate $$\int _ { - 2 } ^ { 0 } ( 3 x - 1 ) ^ { 2 } d x$$
OCR C2 Q8
  1. (a) An arithmetic series has a common difference of 7 .
Given that the sum of the first 20 terms of the series is 530 , find
  1. the first term of the series,
  2. the smallest positive term of the series.
    (b) The terms of a sequence are given by $$u _ { n } = ( n + k ) ^ { 2 } , \quad n \geq 1$$ where \(k\) is a positive constant.
    Given that \(u _ { 2 } = 2 u _ { 1 }\),
  3. find the value of \(k\),
  4. show that \(u _ { 3 } = 11 + 6 \sqrt { 2 }\).
OCR C2 Q9
9.
\includegraphics[max width=\textwidth, alt={}, center]{61af807c-1f2c-417a-85cf-86f2cf566cb9-3_670_1022_1263_374} The diagram shows the curve \(y = 2 x ^ { 2 } + 6 x + 7\) and the straight line \(y = 2 x + 13\).
  1. Find the coordinates of the points where the curve and line intersect.
  2. Show that the area of the shaded region bounded by the curve and line is given by $$\int _ { - 3 } ^ { 1 } \left( 6 - 4 x - 2 x ^ { 2 } \right) d x$$
  3. Hence find the area of the shaded region.
OCR C2 Q1
  1. A sequence is defined by
$$u _ { n + 1 } = \frac { u _ { n } + 1 } { 3 } , \quad n = 1,2,3 , \ldots$$ Given that \(u _ { 3 } = 5\),
  1. find the value of \(u _ { 4 }\),
  2. find the value of \(u _ { 1 }\).
OCR C2 Q2
2.
\includegraphics[max width=\textwidth, alt={}, center]{5025c118-e763-424b-b2c1-5452953a43a9-1_550_901_817_468} The diagram shows the curve with equation \(y = \sqrt { x } + \frac { 8 } { x ^ { 2 } } , x > 0\).
Show that the area of the shaded region bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 9\) is \(24 \frac { 4 } { 9 }\).