Questions — OCR MEI C2 (454 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 Q2
2 Use an isosceles right-angled triangle to show that \(\cos 45 ^ { \circ } = \frac { 1 } { \sqrt { 2 } }\).
OCR MEI C2 Q3
3
  1. On the same axes, sketch the graphs of \(y = \cos x\) and \(y = \cos 2 x\) for values of \(x\) from 0 to \(2 \pi\).
  2. Describe the transformation which maps the graph of \(y = \cos x\) onto the graph of \(y = 3 \cos x\).
    \(4 \theta\) is an acute angle and \(\sin \theta = \frac { 1 } { 4 }\). Find the exact value of \(\tan \theta\).
OCR MEI C2 Q5
5
  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 Q7
7 Sketch the curve \(y = \sin x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
Solve the equation \(\sin x = - 0.68\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
OCR MEI C2 Q8
8
  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 Q9
9 Sketch the graph of \(y = \sin x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
Solve the equation \(\sin x = - 0.2\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
OCR MEI C2 Q1
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
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
3 Simplify \(\frac { \sqrt { 1 - \cos ^ { 2 } \theta } } { \tan \theta }\), where \(\theta\) is an acute angle.
[0pt] [3]
OCR MEI C2 Q4
3 marks
4 Solve the equation \(\tan 2 \theta = 3\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).
[0pt] [3]
OCR MEI C2 Q5
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
6 Solve the equation \(\tan \theta = 2 \sin \theta\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
OCR MEI C2 Q7
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
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
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
4 marks
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
1 Given that \(\sin \theta = \frac { \sqrt { 3 } } { 4 }\), find in surd form the possible values of \(\cos \theta\).
OCR MEI C2 Q2
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
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 \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
OCR MEI C2 Q4
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
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
6 Solve the equation \(\sin 2 x = - 0.5\) for \(0 ^ { \circ } < x < 180 ^ { \circ }\).
OCR MEI C2 Q7
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
8 Given that \(\cos \theta = \frac { 1 } { 3 }\) and \(\theta\) is acute, find the exact value of \(\tan \theta\).
OCR MEI C2 Q9
9 Fig. 3 Beginning with the triangle shown in Fig. 3, prove that \(\sin 60 ^ { \circ } = \frac { \sqrt { 3 } } { 2 }\).