Questions — OCR MEI (4301 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure 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 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 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 Stats 1 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
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 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
12 marks Standard +0.3
1 Fig. 11.1 shows a village green which is bordered by 3 straight roads \(\mathrm { AB } , \mathrm { BC }\) and CA . The road AC runs due North and the measurements shown are in metres. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{139b1905-2035-4503-9ffb-3e6e81f78ef9-1_459_1141_424_629} \captionsetup{labelformat=empty} \caption{Fig. 11.1}
\end{figure}
  1. Calculate the bearing of B from C , giving your answer to the nearest \(0.1 ^ { \circ }\).
  2. Calculate the area of the village green. The road \(A B\) is replaced by a new road, as shown in Fig. 11.2. The village green is extended up to the new road. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{139b1905-2035-4503-9ffb-3e6e81f78ef9-1_432_574_1377_779} \captionsetup{labelformat=empty} \caption{Not to scale}
    \end{figure} Fig. 11.2 The new road is an arc of a circle with centre O and radius 130 m .
  3. (A) Show that angle AOB is 1.63 radians, correct to 3 significant figures.
    (B) Show that the area of land added to the village green is \(5300 \mathrm {~m} ^ { 2 }\) correct to 2 significant figures. Fig. 4 For triangle ABC shown in Fig. 4, calculate
  4. the length of BC , Not to scale
  5. the area of triangle ABC .
OCR MEI C2 Q3
13 marks Standard +0.3
3
  1. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{139b1905-2035-4503-9ffb-3e6e81f78ef9-3_769_766_174_770} \captionsetup{labelformat=empty} \caption{Fig. 11.1}
    \end{figure} A boat travels from P to Q and then to R . As shown in Fig. 11.1, Q is 10.6 km from P on a bearing of \(045 ^ { \circ }\). R is 9.2 km from P on a bearing of \(113 ^ { \circ }\), so that angle QPR is \(68 ^ { \circ }\). Calculate the distance and bearing of R from Q .
  2. Fig. 11.2 shows the cross-section, EBC, of the rudder of a boat. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{139b1905-2035-4503-9ffb-3e6e81f78ef9-3_517_1472_1433_414} \captionsetup{labelformat=empty} \caption{Fig. 11.2}
    \end{figure} BC is an arc of a circle with centre A and radius 80 cm . Angle \(\mathrm { CAB } = \frac { 2 \pi } { 3 }\) radians.
    EC is an arc of a circle with centre D and radius \(r \mathrm {~cm}\). Angle CDE is a right angle.
    1. Calculate the area of sector ABC .
    2. Show that \(r = 40 \sqrt { 3 }\) and calculate the area of triangle CDA.
    3. Hence calculate the area of cross-section of the rudder.
OCR MEI C2 Q4
12 marks Standard +0.3
4 Arrowline Enterprises is considering two possible logos: \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{139b1905-2035-4503-9ffb-3e6e81f78ef9-4_1139_1673_252_240} \captionsetup{labelformat=empty} \caption{Fig. 10.1}
\end{figure}
  1. Fig. 10.1 shows the first logo ABCD . It is symmetrical about AC . Find the length of AB and hence find the area of this logo.
  2. Fig. 10.2 shows a circle with centre O and radius 12.6 cm . ST and RT are tangents to the circle and angle SOR is 1.82 radians. The shaded region shows the second logo. Show that \(\mathrm { ST } = 16.2 \mathrm {~cm}\) to 3 significant figures.
    Find the area and perimeter of this logo.
OCR MEI C2 Q5
12 marks Moderate -0.3
5
  1. The course for a yacht race is a triangle, as shown in Fig. 11.1. The yachts start at A , then travel to B , then to C and finally back to A . \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{139b1905-2035-4503-9ffb-3e6e81f78ef9-5_659_867_348_716} \captionsetup{labelformat=empty} \caption{Fig. 11.1}
    \end{figure} (A) Calculate the total length of the course for this race.
    (B) Given that the bearing of the first stage, AB , is \(175 ^ { \circ }\), calculate the bearing of the second stage, BC .
  2. Fig. 11.2 shows the course of another yacht race. The course follows the arc of a circle from P to \(Q\), then a straight line back to \(P\). The circle has radius 120 m and centre \(O\); angle \(P O Q = 136 ^ { \circ }\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{139b1905-2035-4503-9ffb-3e6e81f78ef9-5_705_822_1624_739} \captionsetup{labelformat=empty} \caption{Fig. 11.2}
    \end{figure}
OCR MEI C2 Q1
5 marks Moderate -0.8
1
  1. Starting with an equilateral triangle, prove that \(\cos 30 ^ { \circ } = \frac { \sqrt { 3 } } { 2 }\).
  2. Solve the equation \(2 \sin \theta = - 1\) for \(0 \leqslant \theta \leqslant 2 \pi\), giving your answers in terms of \(\pi\).
OCR MEI C2 Q2
2 marks Easy -1.8
2 Use an isosceles right-angled triangle to show that \(\cos 45 ^ { \circ } = \frac { 1 } { \sqrt { 2 } }\).
OCR MEI C2 Q3
5 marks Moderate -0.8
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 marks Moderate -0.8
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
4 marks Moderate -0.8
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
5 marks Moderate -0.8
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
4 marks Moderate -0.8
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
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 \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).