Questions — OCR MEI C2 (480 questions)

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OCR MEI C2 Q4
5 marks Standard +0.3
4 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 Q5
3 marks Moderate -0.8
5 You are given that \(\sin \theta = \frac { \sqrt { 2 } } { 3 }\) and that \(\theta\) is an acute angle. Find the exact value of \(\tan \theta\).
OCR MEI C2 Q6
3 marks Moderate -0.8
6 Solve the equation \(\sin 2 x = - 0.5\) for \(0 ^ { \circ } < x < 180 ^ { \circ }\).
OCR MEI C2 Q7
3 marks Moderate -0.8
7 You are given that \(\tan \theta = \frac { 1 } { 2 }\) and the angle \(\theta\) is acute. Show, without using a calculator, that \(\cos ^ { 2 } \theta = \frac { 4 } { 5 }\).
OCR MEI C2 Q8
3 marks Moderate -0.8
8 Given that \(\cos \theta = \frac { 1 } { 3 }\) and \(\theta\) is acute, find the exact value of \(\tan \theta\).
OCR MEI C2 Q9
3 marks Easy -1.8
9 Fig. 3 Beginning with the triangle shown in Fig. 3, prove that \(\sin 60 ^ { \circ } = \frac { \sqrt { 3 } } { 2 }\).
OCR MEI C2 Q10
5 marks Moderate -0.8
10
  1. Sketch the graph of \(y = \cos x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
    On the same axes, sketch the graph of \(y = \cos 2 x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\). Label each graph clearly.
  2. Solve the equation \(\cos 2 x = 0.5\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
OCR MEI C2 Q12
5 marks Moderate -0.8
12
  1. Sketch the graph of \(y = \tan x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
  2. Solve the equation \(4 \sin x = 3 \cos x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
OCR MEI C2 2009 January Q1
4 marks Easy -1.3
1 Find \(\int \left( 20 x ^ { 4 } + 6 x ^ { - \frac { 3 } { 2 } } \right) \mathrm { d } x\).
[0pt] [4]
OCR MEI C2 2009 January Q2
4 marks Easy -1.2
2 Fig. 2 shows the coordinates at certain points on a curve. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{838d6b15-69a9-4e67-bc36-5bf60254a767-2_645_1146_589_497} \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 2009 January Q3
2 marks Easy -1.8
3 Find \(\sum _ { k = 1 } ^ { 5 } \frac { 1 } { 1 + k }\).
OCR MEI C2 2009 January Q4
3 marks Moderate -0.8
4 Solve the equation \(\sin 2 x = - 0.5\) for \(0 ^ { \circ } < x < 180 ^ { \circ }\).
OCR MEI C2 2009 January Q6
5 marks Moderate -0.8
6 An arithmetic progression has first term 7 and third term 12.
  1. Find the 20th term of this progression.
  2. Find the sum of the 21st to the 50th terms inclusive of this progression.
OCR MEI C2 2009 January Q7
5 marks Moderate -0.8
7 Differentiate \(4 x ^ { 2 } + \frac { 1 } { x }\) and hence find the \(x\)-coordinate of the stationary point of the curve \(y = 4 x ^ { 2 } + \frac { 1 } { x }\).
OCR MEI C2 2009 January Q8
5 marks Moderate -0.8
8 The terms of a sequence are given by $$\begin{aligned} u _ { 1 } & = 192 \\ u _ { n + 1 } & = - \frac { 1 } { 2 } u _ { n } \end{aligned}$$
  1. Find the third term of this sequence and state what type of sequence it is.
  2. Show that the series \(u _ { 1 } + u _ { 2 } + u _ { 3 } + \ldots\) converges and find its sum to infinity.
OCR MEI C2 2009 January Q9
4 marks Easy -1.8
9
  1. State the value of \(\log _ { a } a\).
  2. Express each of the following in terms of \(\log _ { a } x\).
    (A) \(\log _ { a } x ^ { 3 } + \log _ { a } \sqrt { x }\) (B) \(\log _ { a } \frac { 1 } { x }\) Section B (36 marks)
OCR MEI C2 2009 January Q10
13 marks Standard +0.3
10 Fig. 10 shows a sketch of the graph of \(y = 7 x - x ^ { 2 } - 6\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{838d6b15-69a9-4e67-bc36-5bf60254a767-4_609_908_1338_621} \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 2009 January Q11
11 marks Standard +0.3
11
  1. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{838d6b15-69a9-4e67-bc36-5bf60254a767-5_469_878_274_671} \captionsetup{labelformat=empty} \caption{Fig. 11.1}
    \end{figure} Fig. 11.1 shows the surface ABCD of a TV presenter's desk. AB and CD are arcs of circles with centre O and sector angle 2.5 radians. \(\mathrm { OC } = 60 \mathrm {~cm}\) and \(\mathrm { OB } = 140 \mathrm {~cm}\).
    (A) Calculate the length of the arc CD.
    (B) Calculate the area of the surface ABCD of the desk.
  2. The TV presenter is at point P , shown in Fig. 11.2. A TV camera can move along the track EF , which is of length 3.5 m . \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{838d6b15-69a9-4e67-bc36-5bf60254a767-5_378_877_1334_675} \captionsetup{labelformat=empty} \caption{Fig. 11.2}
    \end{figure} When the camera is at E , the TV presenter is 1.6 m away. When the camera is at F , the TV presenter is 2.8 m away.
    (A) Calculate, in degrees, the size of angle EFP.
    (B) Calculate the shortest possible distance between the camera and the TV presenter.
OCR MEI C2 2011 January Q1
2 marks Easy -1.2
1 Calculate \(\sum _ { r = 3 } ^ { 6 } \frac { 12 } { r }\).
OCR MEI C2 2011 January Q2
4 marks Easy -1.3
2 Find \(\int \left( 3 x ^ { 5 } + 2 x ^ { - \frac { 1 } { 2 } } \right) \mathrm { d } x\).
OCR MEI C2 2011 January Q3
3 marks Easy -1.2
3 At a place where a river is 7.5 m wide, its depth is measured every 1.5 m across the river. The table shows the results.
Distance across river \(( \mathrm { m } )\)01.534.567.5
Depth of river \(( \mathrm { m } )\)0.62.33.12.81.80.7
Use the trapezium rule with 5 strips to estimate the area of cross-section of the river.
OCR MEI C2 2011 January Q4
4 marks Easy -1.2
4 The curve \(y = \mathrm { f } ( x )\) has a minimum point at \(( 3,5 )\).
State the coordinates of the corresponding minimum point on the graph of
  1. \(y = 3 \mathrm { f } ( x )\),
  2. \(y = \mathrm { f } ( 2 x )\).
OCR MEI C2 2011 January Q5
5 marks Moderate -0.3
5 The second term of a geometric sequence is 6 and the fifth term is - 48 .
Find the tenth term of the sequence.
Find also, in simplified form, an expression for the sum of the first \(n\) terms of this sequence.
OCR MEI C2 2011 January Q6
5 marks Moderate -0.8
6 The third term of an arithmetic progression is 24 . The tenth term is 3 .
Find the first term and the common difference. Find also the sum of the 21st to 50th terms inclusive.
OCR MEI C2 2011 January Q7
4 marks Easy -1.3
7 Simplify
  1. \(\log _ { 10 } x ^ { 5 } + 3 \log _ { 10 } x ^ { 4 }\),
  2. \(\log _ { a } 1 - \log _ { a } a ^ { b }\).