Questions — OCR MEI C2 (454 questions)

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OCR MEI C2 Q10
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
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
1 Find \(\int \left( 20 x ^ { 4 } + 6 x ^ { - \frac { 3 } { 2 } } \right) \mathrm { d } x\).
[0pt] [4]
OCR MEI C2 2009 January Q2
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
3 Find \(\sum _ { k = 1 } ^ { 5 } \frac { 1 } { 1 + k }\).
OCR MEI C2 2009 January Q4
4 Solve the equation \(\sin 2 x = - 0.5\) for \(0 ^ { \circ } < x < 180 ^ { \circ }\).
OCR MEI C2 2009 January Q6
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
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
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
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
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
  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 2010 January Q1
1 Find \(\int \left( x - \frac { 3 } { x ^ { 2 } } \right) \mathrm { d } x\).
OCR MEI C2 2010 January Q2
2 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 2010 January Q3
3 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 2010 January Q4
4 A sector of a circle has area \(8.45 \mathrm {~cm} ^ { 2 }\) and sector angle 0.4 radians. Calculate the radius of the sector.
OCR MEI C2 2010 January Q5
5 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{053009a4-e88f-4711-ad97-cebb1740744b-2_547_991_1340_577} \captionsetup{labelformat=empty} \caption{Fig. 5}
\end{figure} Fig. 5 shows a sketch of the graph of \(y = \mathrm { f } ( x )\). On separate diagrams, sketch the graphs of the following, showing clearly the coordinates of the points corresponding to \(\mathrm { P } , \mathrm { Q }\) and R .
  1. \(y = \mathrm { f } ( 2 x )\)
  2. \(y = \frac { 1 } { 4 } \mathrm { f } ( x )\)
OCR MEI C2 2010 January Q6
6
  1. Find the 51 st term of the sequence given by $$\begin{aligned} u _ { 1 } & = 5
    u _ { n + 1 } & = u _ { n } + 4 \end{aligned}$$
  2. Find the sum to infinity of the geometric progression which begins $$5 \quad 2 \quad 0.8 \quad \ldots .$$ \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{053009a4-e88f-4711-ad97-cebb1740744b-3_531_969_744_589} \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
  3. angle BAD ,
  4. the length BC .
OCR MEI C2 2010 January Q8
8 Find the equation of the tangent to the curve \(y = 6 \sqrt { x }\) at the point where \(x = 16\).
OCR MEI C2 2010 January Q9
9
  1. Sketch the graph of \(y = 3 ^ { x }\).
  2. Use logarithms to solve \(3 ^ { 2 x + 1 } = 10\), giving your answer correct to 2 decimal places.
OCR MEI C2 2010 January Q10
10
  1. Differentiate \(x ^ { 3 } - 3 x ^ { 2 } - 9 x\). Hence find the \(x\)-coordinates of the stationary points on the curve \(y = x ^ { 3 } - 3 x ^ { 2 } - 9 x\), showing which is the maximum and which the minimum.
  2. Find, in exact form, the coordinates of the points at which the curve crosses the \(x\)-axis.
  3. Sketch the curve.
OCR MEI C2 2010 January Q11
11 Fig. 11 shows the cross-section of a school hall, with measurements of the height in metres taken at 1.5 m intervals from O . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{053009a4-e88f-4711-ad97-cebb1740744b-4_579_1381_861_383} \captionsetup{labelformat=empty} \caption{Fig. 11}
\end{figure}
  1. Use the trapezium rule with 8 strips to calculate an estimate of the area of the cross-section.
  2. Use 8 rectangles to calculate a lower bound for the area of the cross-section. The curve of the roof may be modelled by \(y = - 0.013 x ^ { 3 } + 0.16 x ^ { 2 } - 0.082 x + 2.4\), where \(x\) metres is the horizontal distance from O across the hall, and \(y\) metres is the height.
  3. Use integration to find the area of the cross-section according to this model.
  4. Comment on the accuracy of this model for the height of the hall when \(x = 7.5\).
OCR MEI C2 2011 January Q1
1 Calculate \(\sum _ { r = 3 } ^ { 6 } \frac { 12 } { r }\).
OCR MEI C2 2011 January Q2
2 Find \(\int \left( 3 x ^ { 5 } + 2 x ^ { - \frac { 1 } { 2 } } \right) \mathrm { d } x\).
OCR MEI C2 2011 January Q3
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.