Questions — OCR MEI C2 (480 questions)

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OCR MEI C2 2011 January Q8
5 marks Moderate -0.3
8 Showing your method clearly, solve the equation $$5 \sin ^ { 2 } \theta = 5 + \cos \theta \quad \text { for } 0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ } .$$
OCR MEI C2 2011 January Q9
4 marks Standard +0.3
9 Charles has a slice of cake; its cross-section is a sector of a circle, as shown in Fig. 9. The radius is \(r \mathrm {~cm}\) and the sector angle is \(\frac { \pi } { 6 }\) radians. He wants to give half of the slice to Jan. He makes a cut across the sector as shown. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{aa8688bb-3608-45ab-85b7-bf84889dd189-3_420_657_497_744} \captionsetup{labelformat=empty} \caption{Fig. 9}
\end{figure} Show that when they each have half the slice, \(a = r \sqrt { \frac { \pi } { 6 } }\). Section B (36 marks)
OCR MEI C2 2011 January Q10
12 marks Standard +0.3
10 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{aa8688bb-3608-45ab-85b7-bf84889dd189-3_645_793_1377_676} \captionsetup{labelformat=empty} \caption{Fig. 10}
\end{figure} A is the point with coordinates \(( 1,4 )\) on the curve \(y = 4 x ^ { 2 }\). B is the point with coordinates \(( 0,1 )\), as shown in Fig. 10.
  1. The line through A and B intersects the curve again at the point C . Show that the coordinates of C are \(\left( - \frac { 1 } { 4 } , \frac { 1 } { 4 } \right)\).
  2. Use calculus to find the equation of the tangent to the curve at A and verify that the equation of the tangent at C is \(y = - 2 x - \frac { 1 } { 4 }\).
  3. The two tangents intersect at the point D . Find the \(y\)-coordinate of D . \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{aa8688bb-3608-45ab-85b7-bf84889dd189-4_773_1027_255_557} \captionsetup{labelformat=empty} \caption{Fig. 11}
    \end{figure} Fig. 11 shows the curve \(y = x ^ { 3 } - 3 x ^ { 2 } - x + 3\).
OCR MEI C2 2011 January Q12
13 marks Moderate -0.3
12 The table shows the size of a population of house sparrows from 1980 to 2005.
Year198019851990199520002005
Population250002200018750162501350012000
The 'red alert' category for birds is used when a population has decreased by at least \(50 \%\) in the previous 25 years.
  1. Show that the information for this population is consistent with the house sparrow being on red alert in 2005. The size of the population may be modelled by a function of the form \(P = a \times 10 ^ { - k t }\), where \(P\) is the population, \(t\) is the number of years after 1980, and \(a\) and \(k\) are constants.
  2. Write the equation \(P = a \times 10 ^ { - k t }\) in logarithmic form using base 10, giving your answer as simply as possible.
  3. Complete the table and draw the graph of \(\log _ { 10 } P\) against \(t\), drawing a line of best fit by eye.
  4. Use your graph to find the values of \(a\) and \(k\) and hence the equation for \(P\) in terms of \(t\).
  5. Find the size of the population in 2015 as predicted by this model. Would the house sparrow still be on red alert? Give a reason for your answer.
OCR MEI C2 2012 January Q1
2 marks Easy -1.2
1 Find \(\sum _ { r = 3 } ^ { 6 } r ( r + 2 )\).
OCR MEI C2 2012 January Q2
4 marks Easy -1.3
2 Find \(\int \left( x ^ { 5 } + 10 x ^ { \frac { 3 } { 2 } } \right) \mathrm { d } x\).
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 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 \(x = 4\).
OCR MEI C2 2011 June Q4
3 marks Moderate -0.8
4 The graph of \(y = a b ^ { x }\) passes through the points \(( 1,6 )\) and \(( 2,3.6 )\). Find the values of \(a\) and \(b\).
OCR MEI C2 2011 June Q5
5 marks Moderate -0.8
5 Find the equation of the normal to the curve \(y = 8 x ^ { 4 } + 4\) at the point where \(x = \frac { 1 } { 2 }\).
OCR MEI C2 2011 June Q6
5 marks Moderate -0.8
6 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 \sqrt { x } - 2\). Given also that the curve passes through the point \(( 9,4 )\), find the equation of the curve.
OCR MEI C2 2011 June Q7
4 marks Moderate -0.3
7 Solve the equation \(\tan \theta = 2 \sin \theta\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
OCR MEI C2 2011 June Q8
3 marks Easy -1.2
8 Using logarithms, rearrange \(p = s t ^ { n }\) to make \(n\) the subject.