Questions — OCR MEI C4 (332 questions)

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OCR MEI C4 2010 June Q2
2 Fig. 2 shows the curve \(y = \sqrt { 1 + x ^ { 2 } }\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5c149cb5-7392-4219-b285-486f4694aa6f-2_574_944_612_598} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure}
  1. The following table gives some values of \(x\) and \(y\).
    \(x\)00.250.50.751
    \(y\)11.03081.251.4142
    Find the missing value of \(y\), giving your answer correct to 4 decimal places.
    Hence show that, using the trapezium rule with four strips, the shaded area is approximately 1.151 square units.
  2. Jenny uses a trapezium rule with 8 strips, and obtains a value of 1.158 square units. Explain why she must have made a mistake.
  3. The shaded area is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Find the exact volume of the solid of revolution formed.
OCR MEI C4 2010 June Q3
3 The parametric equations of a curve are $$x = \cos 2 \theta , \quad y = \sin \theta \cos \theta \quad \text { for } 0 \leqslant \theta < \pi$$ Show that the cartesian equation of the curve is \(x ^ { 2 } + 4 y ^ { 2 } = 1\).
Sketch the curve.
OCR MEI C4 2010 June Q4
4 Find the first three terms in the binomial expansion of \(\sqrt { 4 + x }\) in ascending powers of \(x\).
State the set of values of \(x\) for which the expansion is valid.
show that \(\frac { y - 2 } { y + 1 } = A \mathrm { e } ^ { x ^ { 3 } }\), where \(A\) is a constant.
(ii) Hence, given that \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = x ^ { 2 } ( y - 2 ) ( y + 1 ) ,$$ 6 Solve the equation \(\tan \left( \theta + 45 ^ { \circ } \right) = 1 - 2 \tan \theta\), for \(0 ^ { \circ } \leqslant \theta \leqslant 90 ^ { \circ }\). Section B (36 marks)
7 A straight pipeline AB passes through a mountain. With respect to axes \(\mathrm { O } x y z\), with \(\mathrm { O } x\) due East, \(\mathrm { O } y\) due North and \(\mathrm { O } z\) vertically upwards, A has coordinates \(( - 200,100,0 )\) and B has coordinates \(( 100,200,100 )\), where units are metres.
OCR MEI C4 2010 June Q6
6 Solve the equation \(\tan \left( \theta + 45 ^ { \circ } \right) = 1 - 2 \tan \theta\), for \(0 ^ { \circ } \leqslant \theta \leqslant 90 ^ { \circ }\). Section B (36 marks)
OCR MEI C4 2010 June Q7
7 A straight pipeline AB passes through a mountain. With respect to axes \(\mathrm { O } x y z\), with \(\mathrm { O } x\) due East, \(\mathrm { O } y\) due North and \(\mathrm { O } z\) vertically upwards, A has coordinates \(( - 200,100,0 )\) and B has coordinates \(( 100,200,100 )\), where units are metres.
  1. Verify that \(\overrightarrow { \mathrm { AB } } = \left( \begin{array} { l } 300
    100
    100 \end{array} \right)\) and find the length of the pipeline.
  2. Write down a vector equation of the line AB , and calculate the angle it makes with the vertical. A thin flat layer of hard rock runs through the mountain. The equation of the plane containing this layer is \(x + 2 y + 3 z = 320\).
  3. Find the coordinates of the point where the pipeline meets the layer of rock.
  4. By calculating the angle between the line AB and the normal to the plane of the layer, find the angle at which the pipeline cuts through the layer.
OCR MEI C4 2010 June Q8
8 Part of the track of a roller-coaster is modelled by a curve with the parametric equations $$x = 2 \theta - \sin \theta , \quad y = 4 \cos \theta \quad \text { for } 0 \leqslant \theta \leqslant 2 \pi$$ This is shown in Fig. 8. B is a minimum point, and BC is vertical. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5c149cb5-7392-4219-b285-486f4694aa6f-4_602_1447_488_351} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure}
  1. Find the values of the parameter at A and B . Hence show that the ratio of the lengths OA and AC is \(( \pi - 1 ) : ( \pi + 1 )\).
  2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\theta\). Find the gradient of the track at A .
  3. Show that, when the gradient of the track is \(1 , \theta\) satisfies the equation $$\cos \theta - 4 \sin \theta = 2 .$$
  4. Express \(\cos \theta - 4 \sin \theta\) in the form \(R \cos ( \theta + \alpha )\). Hence solve the equation \(\cos \theta - 4 \sin \theta = 2\) for \(0 \leqslant \theta \leqslant 2 \pi\). {www.ocr.org.uk}) after the live examination series.
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    MATHEMATICS (MEI)} 4754B
    Applications of Advanced Mathematics (C4) Paper B: Comprehension \section*{Candidates answer on the Question Paper} OCR Supplied Materials:
    • Insert (inserted)
    • MEI Examination Formulae and Tables (MF2)
    \section*{Other Materials Required:}
    • Rough paper
    • Scientific or graphical calculator
    Wednesday 9 June 2010 Afternoon
    \includegraphics[max width=\textwidth, alt={}, center]{5c149cb5-7392-4219-b285-486f4694aa6f-5_264_456_881_1361} 1 The train journey from Swansea to London is 307 km and that by road is 300 km . Carry out the calculations performed on the First Great Western website to estimate how much lower the carbon dioxide emissions are when travelling by rail rather than road.
    2 The equation of the curve in Fig. 3 is $$y = \frac { 1 } { 10 ^ { 4 } } \left( x ^ { 3 } - 100 x ^ { 2 } - 10000 x + 2100100 \right)$$ Calculate the speed at which the car has its lowest carbon dioxide emissions and the value of its emissions at that speed.
    [0pt] [An answer obtained from the graph will be given no marks.]
    3
  5. In line 109 the carbon dioxide emissions for a particular train journey from Exeter to London are estimated to be 3.7 tonnes. Obtain this figure.
  6. The text then goes on to state that the emissions per extra passenger on this journey are less than \(\frac { 1 } { 2 } \mathrm {~kg}\). Justify this figure.
  7. \(\_\_\_\_\)
  8. \(\_\_\_\_\)
    4 The daily number of trains, \(n\), on a line in another country may be modelled by the function defined below, where \(P\) is the annual number of passengers. $$\begin{aligned} & n = 10 \text { for } 0 \leqslant P < 10 ^ { 6 }
    & n = 11 \text { for } 10 ^ { 6 } \leqslant P < 1.5 \times 10 ^ { 6 }
    & n = 12 \text { for } 1.5 \times 10 ^ { 6 } \leqslant P < 2 \times 10 ^ { 6 }
    & n = 13 \text { for } 2 \times 10 ^ { 6 } \leqslant P < 2.5 \times 10 ^ { 6 }
    & n = 14 \text { for } 2.5 \times 10 ^ { 6 } \leqslant P < 3 \times 10 ^ { 6 }
    & \ldots \text { and so on } \ldots \end{aligned}$$
  9. Sketch the graph of \(n\) against \(P\).
  10. Describe, in words, the relationship between the daily number of trains and the annual number of passengers.

  11. \includegraphics[max width=\textwidth, alt={}, center]{5c149cb5-7392-4219-b285-486f4694aa6f-7_716_1249_1011_440}
  12. \(\_\_\_\_\)
    5 The FGW website gives the conversion factor for miles to kilometres to 7 significant figures.
    "We got the distance between the two stations by road from \href{http://theaa.com}{theaa.com}. We then converted this distance to kilometres by multiplying it by \(1.609344 . "\) Suppose this conversion factor is applied to a distance of exactly 100 miles.
    State which one of the following best expresses the level of accuracy for the distance in metric units, justifying your answer. A : to the nearest millimetre
    B : to the nearest 10 centimetres
    C : to the nearest metre
OCR MEI C4 2016 June Q1
1 Express \(\cos \theta - 3 \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\).
Hence show that the equation \(\cos \theta - 3 \sin \theta = 4\) has no solution.
OCR MEI C4 2016 June Q2
2 Given that \(\left( 1 + \frac { x } { p } \right) ^ { q } = 1 - x + \frac { 3 } { 4 } x ^ { 2 } + \ldots\), find \(p\) and \(q\), and state the set of values of \(x\) for which the expansion is valid.
OCR MEI C4 2016 June Q3
3 Fig. 3 shows the curve \(y = x ^ { 4 }\) and the line \(y = 4\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8b807b2e-777b-4c9a-b3dd-890d21d33174-2_509_510_778_774} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} The finite region enclosed by the curve and the line is rotated through \(180 ^ { \circ }\) about the \(y\)-axis. Find the exact volume of revolution generated.
OCR MEI C4 2016 June Q4
4 Solve the equation \(2 \sin 2 \theta = 1 + \cos 2 \theta\) for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
OCR MEI C4 2016 June Q5
5 In Fig. 5, triangles \(\mathrm { ABC } , \mathrm { ACD }\) and ADE are all right-angled, and angles \(\mathrm { BAC } , \mathrm { CAD }\) and DAE are all \(\theta\). \(\mathrm { AB } = x\) and \(\mathrm { AE } = 2 x\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8b807b2e-777b-4c9a-b3dd-890d21d33174-2_567_465_1905_799} \captionsetup{labelformat=empty} \caption{Fig. 5}
\end{figure}
  1. Show that \(\sec ^ { 3 } \theta = 2\).
  2. Hence show the ratio of lengths ED to CB is \(2 ^ { \frac { 2 } { 3 } } : 1\).
OCR MEI C4 2016 June Q6
6 P is a general point on the curve with parametric equations \(x = 2 t , y = \frac { 2 } { t }\). This is shown in Fig. 6. The tangent at P intersects the \(x\) - and \(y\)-axes at the points Q and R respectively. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8b807b2e-777b-4c9a-b3dd-890d21d33174-3_487_684_388_685} \captionsetup{labelformat=empty} \caption{Fig. 6}
\end{figure} Show that the area of the triangle OQR , where O is the origin, is independent of \(t\).
OCR MEI C4 Q1
1 Find the coefficient of the term in \(x ^ { 3 }\) in the expansion of \(\frac { 1 } { ( 2 + 3 x ) ^ { 2 } }\).
OCR MEI C4 Q2
2 The graph shows part of the curve \(y = x ^ { 2 } + 1\).
\includegraphics[max width=\textwidth, alt={}, center]{62dbc58e-f498-483f-a9aa-05cb5aa44881-2_380_876_715_575} Find the volume when the area between this curve, the axes and the line \(x = 2\) is rotated through \(360 ^ { 0 }\) about the \(x\)-axis.
OCR MEI C4 Q3
3 Solve the equation \(\sec ^ { 2 } \theta = 2 \tan \theta + 4\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).
OCR MEI C4 Q4
4 You are given that \(\mathbf { a } = \left( \begin{array} { c } 1
2
- 1 \end{array} \right)\) and \(\mathbf { b } = \left( \begin{array} { c } 3
- 1
k \end{array} \right)\).
  1. Find the angle between \(\mathbf { a }\) and \(\mathbf { b }\) when \(k = 2\).
  2. Find the value of \(k\) such that \(\mathbf { a }\) and \(\mathbf { b }\) are perpendicular.
OCR MEI C4 Q5
5 A curve is given by the parametric equations \(x = a t ^ { 2 } , y = 2 a\) (where \(a\) is a constant). A point P on the curve has coordinates ( \(a p ^ { 2 }\), 2ap).
  1. Find the coordinates of the point, T , where the tangent to the curve at P meets the \(x\)-axis and the coordinates of the point N where the normal to the curve at P meets the \(x\)-axis.
  2. Hence show that the area of the triangle PTN is \(2 a ^ { 2 } p \left( p ^ { 2 } + 1 \right)\) square units.
OCR MEI C4 Q6
6 The graph shows part of the curve \(y = \frac { 1 } { 1 + x ^ { 2 } }\).
\includegraphics[max width=\textwidth, alt={}, center]{62dbc58e-f498-483f-a9aa-05cb5aa44881-3_474_961_406_479} Use the trapezium rule to estimate the area between the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 2\) using
  1. 2 strips,
  2. 4 strips. What can you conclude about the true value of the area?
OCR MEI C4 Q7
7 A quantity of oil is dropped into water. When the oil hits the water it spreads out as a circle. The radius of the circle is \(r \mathrm {~cm}\) after \(t\) seconds and when \(t = 3\) the radius of the circle is increasing at the rate of 0.5 centimetres per second.
One observer believes that the radius increases at a rate which is proportional to \(\frac { 1 } { ( t + 1 ) }\).
  1. Write down a differential equation for this situation, using \(k\) as a constant of proportionality.
  2. Show that \(k = 2\).
  3. Calculate the radius of the circle after 10 seconds according to this model. Another observer believes that the rate of increase of the radius of the circle is proportional to \(\frac { 1 } { ( t + 1 ) ( t + 2 ) }\).
  4. Write down a new differential equation for this new situation. Using the same initial conditions as before, find the value of the new constant of proportionality.
  5. Hence solve the differential equation.
  6. Calculate the radius of the circle after 10 seconds according to this model.
OCR MEI C4 Q8
8 The height of tide at the entrance to a harbour on a particular day may be modelled by the function \(h = 3 + 2 \sin 30 t + 1.5 \cos 30 t\) where \(h\) is measured in metres, \(t\) in hours after midnight and \(30 t\) is in degrees.
[0pt] [The values 2 and 1.5 represent the relative effects of the moon and sun respectively.]
  1. Show that \(2 \sin 30 t + 1.5 \cos 30 t\) can be written in the form \(2.5 \sin ( 30 t + \alpha )\), where \(\alpha\) is to be determined.
  2. Find the height of tide at high water and the first time that this occurs after midnight.
  3. Find the range of tide during the day.
  4. Sketch the graph of \(h\) against \(t\) for \(0 \leq t \leq 12\), indicating the maximum and minimum points.
  5. A sailing boat may enter the harbour only if there is at least 2 metres of water. Find the times during this morning when it may enter the harbour.
  6. From your graph estimate the time at which the water falling fastest and the rate at which it is falling.
OCR MEI C4 Q1
1 Solve the equation \(2 \sin 2 \theta = \cos \theta\) for \(0 ^ { \circ } \leq \theta \leq 360 ^ { \circ }\).
OCR MEI C4 Q2
2 Show that the curve, given by the parametric equations given below, represents a circle. $$x = 2 \cos \theta + 3 , y = 2 \sin \theta - 3$$ State the radius and centre of this circle.
OCR MEI C4 Q3
3 Find the first three terms of the binomial expansion of \(\frac { 1 } { 2 - 3 x }\).
Give the range of values of \(x\) for which the expansion is valid.
OCR MEI C4 Q4
4 The points \(\mathrm { A } , \mathrm { B }\) and C are given by the position vectors \(\mathbf { a } = \binom { - 2 } { 1 } , \mathbf { b } = \binom { 0 } { 5 }\) and \(\mathbf { c } = \binom { 4 } { 3 }\). M is the midpoint of AC .
  1. Find the position vector of M .
  2. Find the vector \(\overrightarrow { B C }\).
  3. Find the position vector of the point D such that \(\overrightarrow { \mathrm { BC } } = \overrightarrow { \mathrm { AD } }\).
  4. Show that D lies on BM .
OCR MEI C4 Q5
5 A ball is thrown towards a hedge. Its position relative to the point from which it was thrown is given by the parametric equations $$x = 8 t , y = 10 t - 5 t ^ { 2 }$$
  1. Find the cartesian equation of the trajectory of the ball.
  2. The ball just clears the hedge. What can you say about the height of the hedge?