Questions C2 (1410 questions)

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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 C2 2012 January Q3
3 marks Moderate -0.8
3 Find the set of values of \(x\) for which \(x ^ { 2 } - 7 x\) is a decreasing function.
OCR MEI C2 2012 January Q4
3 marks Easy -1.8
4 Given that \(a > 0\), state the values of
  1. \(\log _ { a } 1\),
  2. \(\log _ { a } \left( a ^ { 3 } \right) ^ { 6 }\),
  3. \(\log _ { a } \sqrt { a }\).
OCR MEI C2 2012 January Q5
3 marks Moderate -0.8
5 Figs. 5.1 and 5.2 show the graph of \(y = \sin x\) for values of \(x\) from \(0 ^ { \circ }\) to \(360 ^ { \circ }\) and two transformations of this graph. State the equation of each graph after it has been transformed.
  1. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{ba9f2fd1-7a36-4749-86ec-40c93d23a84b-2_506_926_1324_571} \captionsetup{labelformat=empty} \caption{Fig. 5.1}
    \end{figure}
  2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{ba9f2fd1-7a36-4749-86ec-40c93d23a84b-2_513_936_2003_561} \captionsetup{labelformat=empty} \caption{Fig. 5.2}
    \end{figure}
OCR MEI C2 2012 January Q6
3 marks Easy -1.2
6 Use logarithms to solve the equation \(235 \times 5 ^ { x } = 987\), giving your answer correct to 3 decimal places.
OCR MEI C2 2012 January Q7
3 marks Moderate -0.8
7 Given that \(y = a + x ^ { b }\), find \(\log _ { 10 } x\) in terms of \(y\), \(a\) and \(b\).
OCR MEI C2 2012 January Q8
5 marks Moderate -0.8
8 Show that the equation \(4 \cos ^ { 2 } \theta = 1 + \sin \theta\) can be expressed as $$4 \sin ^ { 2 } \theta + \sin \theta - 3 = 0 .$$ Hence solve the equation for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
OCR MEI C2 2012 January Q9
5 marks Moderate -0.8
9 A geometric progression has a positive common ratio. Its first three terms are 32, \(b\) and 12.5.
Find the value of \(b\) and find also the sum of the first 15 terms of the progression.
OCR MEI C2 2012 January Q10
5 marks Moderate -0.8
10 In an arithmetic progression, the second term is 11 and the sum of the first 40 terms is 3030 . Find the first term and the common difference.
OCR MEI C2 2012 January Q11
12 marks Standard +0.3
11 The point A has \(x\)-coordinate 5 and lies on the curve \(y = x ^ { 2 } - 4 x + 3\).
  1. Sketch the curve.
  2. Use calculus to find the equation of the tangent to the curve at A .
  3. Show that the equation of the normal to the curve at A is \(x + 6 y = 53\). Find also, using an algebraic method, the \(x\)-coordinate of the point at which this normal crosses the curve again.
OCR MEI C2 2012 January Q12
12 marks Moderate -0.3
12 The equation of a curve is \(y = 9 x ^ { 2 } - x ^ { 4 }\).
  1. Show that the curve meets the \(x\)-axis at the origin and at \(x = \pm a\), stating the value of \(a\).
  2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\). Hence show that the origin is a minimum point on the curve. Find the \(x\)-coordinates of the maximum points.
  3. Use calculus to find the area of the region bounded by the curve and the \(x\)-axis between \(x = 0\) and \(x = a\), using the value you found for \(a\) in part (i).
OCR MEI C2 2012 January Q13
12 marks Moderate -0.3
13 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ba9f2fd1-7a36-4749-86ec-40c93d23a84b-4_709_709_262_303} \captionsetup{labelformat=empty} \caption{Fig. 13.1}
\end{figure}
\includegraphics[max width=\textwidth, alt={}]{ba9f2fd1-7a36-4749-86ec-40c93d23a84b-4_392_544_415_1197}
In a concert hall, seats are arranged along arcs of concentric circles, as shown in Fig. 13.1. As shown in Fig. 13.2, the stage is part of a sector ABO of radius 11 m . Fig. 13.2 also gives the dimensions of the stage.
  1. Show that angle \(\mathrm { COD } = 1.55\) radians, correct to 2 decimal places. Hence find the area of the stage.
  2. There are four rows of seats, with their backs along arcs, with centre O, of radii \(7.4 \mathrm {~m} , 8.6 \mathrm {~m} , 9.8 \mathrm {~m}\) and 11 m . Each seat takes up 80 cm of the arc.
    (A) Calculate how many seats can fit in the front row.
    (B) Calculate how many more seats can fit in the back row than the front row.
OCR MEI C2 2013 January Q1
3 marks Easy -1.8
1 Find \(\int 30 x ^ { \frac { 3 } { 2 } } \mathrm {~d} x\).
OCR MEI C2 2013 January Q2
3 marks Easy -1.2
2 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 2013 January Q3
4 marks Easy -1.2
3
  1. The point \(\mathrm { P } ( 4 , - 2 )\) lies on the curve \(y = \mathrm { f } ( x )\). Find the coordinates of the image of P when the curve is transformed to \(y = \mathrm { f } ( 5 x )\).
  2. Describe fully a single transformation which maps the curve \(y = \sin x ^ { \circ }\) onto the curve \(y = \sin ( x - 90 ) ^ { \circ }\).
OCR MEI C2 2013 January Q4
4 marks Moderate -0.8
4 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{19552108-0808-4946-a937-9074d58519b2-2_506_758_1292_657} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure} Fig. 4 shows sector OAB with sector angle 1.2 radians and arc length 4.2 cm . It also shows chord AB .
  1. Find the radius of this sector.
  2. Calculate the perpendicular distance of the chord AB from O .
    \(5 \quad \mathrm {~A}\) and B are points on the curve \(y = 4 \sqrt { x }\). Point A has coordinates \(( 9,12 )\) and point B has \(x\)-coordinate 9.5. Find the gradient of the chord AB . The gradient of AB is an approximation to the gradient of the curve at A . State the \(x\)-coordinate of a point C on the curve such that the gradient of AC is a closer approximation.
OCR MEI C2 2013 January Q6
4 marks Moderate -0.8
6 Differentiate \(2 x ^ { 3 } + 9 x ^ { 2 } - 24 x\). Hence find the set of values of \(x\) for which the function \(\mathrm { f } ( x ) = 2 x ^ { 3 } + 9 x ^ { 2 } - 24 x\) is increasing.
OCR MEI C2 2013 January Q7
5 marks Moderate -0.8
7 Fig. 7 shows a sketch of a village green ABC which is bounded by three straight roads. \(\mathrm { AB } = 92 \mathrm {~m}\), \(\mathrm { BC } = 75 \mathrm {~m}\) and \(\mathrm { AC } = 105 \mathrm {~m}\). Fig. 7 Calculate the area of the village green.
OCR MEI C2 2013 January Q8
5 marks Moderate -0.8
8
  1. Sketch the graph of \(y = 3 ^ { x }\).
  2. Solve the equation \(3 ^ { 5 x - 1 } = 500000\).
OCR MEI C2 2013 January Q9
5 marks Moderate -0.3
9
  1. Show that the equation \(\frac { \tan \theta } { \cos \theta } = 1\) may be rewritten as \(\sin \theta = 1 - \sin ^ { 2 } \theta\).
  2. Hence solve the equation \(\frac { \tan \theta } { \cos \theta } = 1\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
OCR MEI C2 2013 January Q10
11 marks Standard +0.3
10 Fig. 10 shows a sketch of the curve \(y = x ^ { 2 } - 4 x + 3\). The point A on the curve has \(x\)-coordinate 4 . At point B the curve crosses the \(x\)-axis. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{19552108-0808-4946-a937-9074d58519b2-4_768_734_500_667} \captionsetup{labelformat=empty} \caption{Fig. 10}
\end{figure}
  1. Use calculus to find the equation of the normal to the curve at A and show that this normal intersects the \(x\)-axis at \(\mathrm { C } ( 16,0 )\).
  2. Find the area of the region ABC bounded by the curve, the normal at A and the \(x\)-axis.
OCR MEI C2 2013 January Q11
12 marks Standard +0.3
11
  1. An arithmetic progression has first term \(A\) and common difference \(D\). The sum of its first two terms is 25 and the sum of its first four terms is 250 .
    (A) Find the values of \(A\) and \(D\).
    (B) Find the sum of the 21st to 50th terms inclusive of this sequence.
  2. A geometric progression has first term \(a\) and common ratio \(r\), with \(r \neq \pm 1\). The sum of its first two terms is 25 and the sum of its first four terms is 250 . Use the formula for the sum of a geometric progression to show that \(\frac { r ^ { 4 } - 1 } { r ^ { 2 } - 1 } = 10\) and hence or otherwise find algebraically the possible values of \(r\) and the corresponding values of \(a\).
OCR MEI C2 2013 January Q12
13 marks Moderate -0.3
12 The table shows population data for a country.
Year19691979198919992009
Population in
millions \(( p )\)
58.8180.35105.27134.79169.71
The data may be represented by an exponential model of growth. Using \(t\) as the number of years after 1960, a suitable model is \(p = a \times 10 ^ { k t }\).
  1. Derive an equation for \(\log _ { 10 } p\) in terms of \(a , k\) and \(t\).
  2. Complete the table and draw the graph of \(\log _ { 10 } p\) against \(t\), drawing a line of best fit by eye.
  3. Use your line of best fit to express \(\log _ { 10 } p\) in terms of \(t\) and hence find \(p\) in terms of \(t\).
  4. According to the model, what was the population in 1960 ?
  5. According to the model, when will the population reach 200 million?
OCR MEI C2 2011 June Q1
3 marks Easy -1.2
1 Find \(\int _ { 2 } ^ { 5 } \left( 2 x ^ { 3 } + 3 \right) \mathrm { d } x\).
OCR MEI C2 2011 June Q2
3 marks Moderate -0.8
2 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 2011 June Q3
5 marks Moderate -0.8
3 The equation of a curve is \(y = \sqrt { 1 + 2 x }\).
  1. Calculate the gradient of the chord joining the points on the curve where \(x = 4\) and \(x = 4.1\). Give your answer correct to 4 decimal places.
  2. Showing the points you use, calculate the gradient of another chord of the curve which is a closer approximation to the gradient of the curve when \(x = 4\).