Questions — OCR MEI C2 (480 questions)

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OCR MEI C2 Q12
5 marks Moderate -0.8
12 Calculate \(\sum _ { r = 3 } ^ { 6 } \frac { 12 } { r }\). 12 A sequence begins $$\begin{array} { l l l l l l l l l l l } 1 & 3 & 5 & 3 & 1 & 3 & 5 & 3 & 1 & 3 & \ldots \end{array}$$ and continues in this pattern.
  1. Find the 55th term of this sequence, showing your method.
  2. Find the sum of the first 55 terms of the sequence.
OCR MEI C2 Q1
2 marks Easy -1.2
1 Find \(\sum _ { k = 1 } ^ { 5 } \frac { 1 } { 1 + k }\).
OCR MEI C2 Q2
5 marks Moderate -0.8
2 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 Q3
3 marks Easy -1.8
3 A sequence begins $$\begin{array} { l l l l l l l l l l l l } 1 & 2 & 3 & 4 & 5 & 1 & 2 & 3 & 4 & 5 & 1 & \ldots \end{array}$$ and continues in this pattern.
  1. Find the 48th term of this sequence.
  2. Find the sum of the first 48 terms of this sequence.
OCR MEI C2 Q4
3 marks Moderate -0.8
4 Sequences A, B and C are shown below. They each continue in the pattern established by the given terms.
A:1,2,4,32,\(\ldots\)
B:20,- 10,5,- 2.5,1.25,- 0.625,\(\ldots\)
C:20,5,1,20,5,\(\ldots\)
  1. Which of these sequences is periodic?
  2. Which of these sequences is convergent?
  3. Find, in terms of \(n\), the \(n\)th term of sequence A .
OCR MEI C2 Q5
2 marks Easy -1.8
5 Find the numerical value of \(\sum _ { k = 2 } ^ { 5 } k ^ { 3 }\).
OCR MEI C2 Q6
5 marks Easy -1.3
6
  1. Find \(\sum _ { k = 2 } ^ { 5 } 2 ^ { k }\).
  2. Find the value of \(n\) for which \(2 ^ { n } = \frac { 1 } { 64 }\).
  3. Sketch the curve with equation \(y = 2 ^ { x }\).
OCR MEI C2 Q2
14 marks Standard +0.3
2 Fig. 10.1 shows Jean's back garden. This is a quadrilateral ABCD with dimensions as shown. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cd507fa5-97b2-4edd-ae37-a58aea1de5ed-2_711_1018_292_549} \captionsetup{labelformat=empty} \caption{Fig. 10.1}
\end{figure}
  1. (A) Calculate AC and angle ACB . Hence calculate AD .
    (B) Calculate the area of the garden.
  2. The shape of the fence panels used in the garden is shown in Fig. 10.2. EH is the arc of a sector of a circle with centre at the midpoint, M , of side FG , and sector angle 1.1 radians, as shown. \(\mathrm { FG } = 1.8 \mathrm {~m}\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{cd507fa5-97b2-4edd-ae37-a58aea1de5ed-2_579_981_1512_567} \captionsetup{labelformat=empty} \caption{Fig. 10.2}
    \end{figure} Calculate the area of one of these fence panels.
OCR MEI C2 Q3
5 marks Moderate -0.3
3 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cd507fa5-97b2-4edd-ae37-a58aea1de5ed-3_596_689_244_534} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} \section*{Not to scale} In Fig. 3, BCD is a straight line. \(\mathrm { AC } = 9.8 \mathrm {~cm} , \mathrm { BC } = 7.3 \mathrm {~cm}\) and \(\mathrm { CD } = 6.4 \mathrm {~cm}\); angle \(\mathrm { ACD } = 53.4 ^ { \circ }\).
  1. Calculate the length AD .
  2. Calculate the area of triangle ABC .
OCR MEI C2 Q4
11 marks Standard +0.3
4
  1. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{cd507fa5-97b2-4edd-ae37-a58aea1de5ed-4_492_1018_256_567} \captionsetup{labelformat=empty} \caption{Fig. 10.1}
    \end{figure} At a certain time, ship S is 5.2 km from lighthouse L on a bearing of \(048 ^ { \circ }\). At the same time, ship T is 6.3 km from L on a bearing of \(105 ^ { \circ }\), as shown in Fig. 10.1. For these positions, calculate
    (A) the distance between ships S and T ,
    (B) the bearing of S from T .
  2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{cd507fa5-97b2-4edd-ae37-a58aea1de5ed-4_430_698_1350_573} \captionsetup{labelformat=empty} \caption{Fig. 10.2}
    \end{figure} Not to scale Ship S then travels at \(24 \mathrm {~km} \mathrm {~h} { } ^ { 1 }\) anticlockwise along the arc of a circle, keeping 5.2 km from the lighthouse L, as shown in Fig. 10.2. Find, in radians, the angle \(\theta\) that the line LS has turned through in 26 minutes.
    Hence find, in degrees, the bearing of ship S from the lighthouse at this time.
OCR MEI C2 Q5
5 marks Moderate -0.8
5 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 Q6
5 marks Standard +0.3
6
Not to scale \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cd507fa5-97b2-4edd-ae37-a58aea1de5ed-5_484_968_1516_617} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure} Fig. 7 shows triangle ABC , with \(\mathrm { AB } = 8.4 \mathrm {~cm}\). D is a point on AC such that angle \(\mathrm { ADB } = 79 ^ { \circ }\), \(\mathrm { BD } = 5.6 \mathrm {~cm}\) and \(\mathrm { CD } = 7.8 \mathrm {~cm}\). Calculate
  1. angle BAD ,
  2. the length BC .
OCR MEI C2 Q1
5 marks Moderate -0.3
1 Show that the equation \(\sin ^ { 2 } x = 3 \cos x - 2\) can be expressed as a quadratic equation in \(\cos x\) and hence solve the equation for values of \(x\) between 0 and \(2 \pi\).
OCR MEI C2 Q2
11 marks Moderate -0.3
2 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4dcf71fc-2585-4247-a21d-8b14f11ce0d0-1_239_1478_439_335} \captionsetup{labelformat=empty} \caption{Fig. 9.1}
\end{figure}
  1. Jean is designing a model aeroplane. Fig. 9.1 shows her first sketch of the wing's cross-section. Calculate angle A and the area of the cross-section.
  2. Jean then modifies her design for the wing. Fig. 9.2 shows the new cross-section, with 1 unit for each of \(x\) and \(y\) representing one centimetre. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{4dcf71fc-2585-4247-a21d-8b14f11ce0d0-1_415_1662_1081_240} \captionsetup{labelformat=empty} \caption{Fig. 9.2}
    \end{figure} Here are some of the coordinates that Jean used to draw the new cross-section.
    Upper surfaceLower surface
    \(x\)\(y\)\(x\)\(y\)
    0000
    41.454- 0.85
    81.568- 0.76
    121.2712- 0.55
    161.0416- 0.30
    200200
    Use the trapezium rule with trapezia of width 4 cm to calculate an estimate of the area of this cross-section.
OCR MEI C2 Q3
3 marks Moderate -0.8
3 Simplify \(\frac { \sqrt { 1 - \cos ^ { 2 } \theta } } { \tan \theta }\), where \(\theta\) is an acute angle.
[0pt] [3]
OCR MEI C2 Q4
3 marks Moderate -0.8
4 Solve the equation \(\tan 2 \theta = 3\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).
[0pt] [3]
OCR MEI C2 Q5
5 marks Moderate -0.3
5 Solve the equation \(\sin 2 \theta = 0.7\) for values of \(\theta\) between 0 and \(2 \pi\), giving your answers in radians correct to 3 significant figures.
OCR MEI C2 Q6
4 marks Moderate -0.3
6 Solve the equation \(\tan \theta = 2 \sin \theta\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
OCR MEI C2 Q7
5 marks Moderate -0.3
7 Showing your method clearly, solve the equation \(4 \sin ^ { 2 } \theta = 3 + \cos ^ { 2 } \theta\), for values of \(\theta\) between \(0 ^ { \circ }\) and \(360 ^ { \circ }\).
OCR MEI C2 Q8
5 marks Moderate -0.3
8 Show that the equation \(4 \cos ^ { 2 } \theta = 4 - \sin \theta\) may be written in the form $$4 \sin ^ { 2 } \theta - \sin \theta = 0$$ Hence solve the equation \(4 \cos ^ { 2 } \theta = 4 - \sin \theta\) for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
OCR MEI C2 Q9
5 marks Standard +0.3
9 Showing your method, solve the equation \(2 \sin ^ { 2 } \theta = \cos \theta + 2\) for values of \(\theta\) between \(0 ^ { \circ }\) and \(360 ^ { \circ }\).
OCR MEI C2 Q10
5 marks Moderate -0.3
10
  1. Show that the equation \(2 \cos ^ { 2 } \theta + 7 \sin \theta = 5\) may be written in the form $$2 \sin ^ { 2 } \theta - 7 \sin \theta + 3 = 0$$
  2. By factorising this quadratic equation, solve the equation for values of \(\theta\) between \(0 ^ { \circ }\) and \(180 ^ { \circ }\). [4]
OCR MEI C2 Q1
3 marks Moderate -0.8
1 Given that \(\sin \theta = \frac { \sqrt { 3 } } { 4 }\), find in surd form the possible values of \(\cos \theta\).
OCR MEI C2 Q2
5 marks Moderate -0.3
2
  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 Q3
5 marks Moderate -0.3
3 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 \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).