Questions C2 (1410 questions)

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OCR MEI C2 Q6
2 marks Easy -1.2
6 You are given that $$\begin{aligned} u _ { 1 } & = 1 \\ u _ { n + 1 } & = \frac { u _ { n } } { 1 + u _ { n } } \end{aligned}$$ Find the values of \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\). Give your answers as fractions.
OCR MEI C2 Q7
4 marks Easy -1.3
7
  1. Evaluate \(\sum _ { r = 2 } ^ { 5 } \frac { 1 } { r - 1 }\).
  2. Express the series \(2 \times 3 + 3 \times 4 + 4 \times 5 + 5 \times 6 + 6 \times 7\) in the form \(\sum _ { r = 2 } ^ { a } \mathrm { f } ( r )\) where \(\mathrm { f } ( r )\) and \(a\) are to be determined.
OCR MEI C2 Q8
3 marks Moderate -0.8
8
  1. Find \(\sum _ { k = 3 } ^ { 8 } \left( k ^ { 2 } - 1 \right)\).
  2. State whether the sequence with \(k\) th term \(k ^ { 2 } - 1\) is convergent or divergent, giving a reason for your answer.
OCR MEI C2 Q9
4 marks Easy -1.2
9
  1. Find the second and third terms of the sequence defined by the following. $$\begin{aligned} t _ { n + 1 } & = 2 t _ { n } + 5 \\ t _ { 1 } & = 3 \end{aligned}$$
  2. Find \(\sum _ { k = 1 } ^ { 3 } k ( k + 1 )\).
OCR MEI C2 Q10
3 marks Easy -1.2
10 For each of the following sequences, state with a reason whether it is convergent, periodic or neither. Each sequence continues in the pattern established by the given terms.
  1. \(3 , \frac { 3 } { 2 } , \frac { 3 } { 4 } , \frac { 3 } { 8 } , \ldots\)
  2. \(3,7,11,15 , \ldots\)
  3. \(3,5 , - 3 , - 5,3,5 , - 3 , - 5 , \ldots\)
OCR MEI C2 Q11
2 marks Easy -1.2
11 Find \(\sum _ { r = 3 } ^ { 6 } r ( r + 2 )\).
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
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 \(\tan \theta\).