Questions — OCR MEI (4301 questions)

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OCR MEI C4 2016 June Q3
4 marks Moderate -0.3
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
5 marks Moderate -0.3
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
7 marks Standard +0.8
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
7 marks Standard +0.8
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
5 marks Moderate -0.5
1 Find the coefficient of the term in \(x ^ { 3 }\) in the expansion of \(\frac { 1 } { ( 2 + 3 x ) ^ { 2 } }\).
OCR MEI C4 Q2
4 marks Moderate -0.8
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
5 marks Standard +0.3
3 Solve the equation \(\sec ^ { 2 } \theta = 2 \tan \theta + 4\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).
OCR MEI C4 Q4
6 marks Moderate -0.8
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
8 marks Standard +0.3
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
8 marks Moderate -0.8
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
18 marks Standard +0.3
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
18 marks Standard +0.3
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
4 marks Moderate -0.3
1 Solve the equation \(2 \sin 2 \theta = \cos \theta\) for \(0 ^ { \circ } \leq \theta \leq 360 ^ { \circ }\).
OCR MEI C4 Q2
4 marks Easy -1.2
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
5 marks Moderate -0.5
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
8 marks Moderate -0.8
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
7 marks Moderate -0.8
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?
OCR MEI C4 Q7
4 marks Moderate -0.5
7 When a stone is dropped into still water, ripples move outwards forming a circle of rippled water. At time \(t\) seconds after the stone hits the water the radius of the circle of ripples is increasing at a rate that is inversely proportional to the radius When the radius is 200 cm the rate of increase of the radius is 5 cm per second. Write down the differential equation that represents this situation.
OCR MEI C4 Q8
18 marks Standard +0.3
8
  1. Evaluate \(A _ { 0 } = \int _ { 0 } ^ { 2 } \left( 2 + 2 x - x ^ { 2 } \right) \mathrm { d } x\). Fig 8.1 illustrates the cross-section of a proposed tunnel. Lengths are in metres. The equation of the curved section is \(y = 2 + \sqrt { 2 x - x ^ { 2 } }\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{23771896-942c-4a1d-ab95-6b6d3cc5643c-3_419_515_1155_836} \captionsetup{labelformat=empty} \caption{Fig. 8.1}
    \end{figure} The designers need to know the area of the cross-section, \(A \mathrm {~m} ^ { 2 }\), so that they can work out the volume of the soil that will need to be removed when the tunnel is built.
  2. An initial estimate, \(A _ { 1 }\), is given by the area of the 8 rectangles shown in Fig 8.2. Calculate \(A _ { 1 }\), and state whether it is an overestimate or underestimate for \(A\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{23771896-942c-4a1d-ab95-6b6d3cc5643c-3_520_645_2053_644} \captionsetup{labelformat=empty} \caption{Fig. 8.2}
    \end{figure}
  3. On graph paper, draw the graphs of $$y = 2 + 2 x - x ^ { 2 } \text { and } y = 2 + \sqrt { 2 x - x ^ { 2 } } \text { for } 0 \leq x \leq 2 .$$ Make it clear which equation applies to which curve.
  4. State whether \(A _ { 0 }\), your answer to part (i), is an underestimate for \(A\) or an overestimate. Give a reason for your answer.
  5. The designers use the trapezium rule to estimate \(A\). What values does this give when they take
    (A) 2 strips,
    (B) 4 strips,
    (C) 8 strips? What can you conclude about the value of \(A\) ?
  6. The best estimate from the trapezium rule is denoted by \(A _ { 2 }\). State, with a reason, whether the true value of \(A\) is nearer \(A _ { 1 }\) or \(A _ { 2 }\).
OCR MEI C4 Q9
18 marks Standard +0.3
9 A laser beam is aimed from a point ( \(12,10,10\) ) in the direction \(- 2 \mathbf { i } - 2 \mathbf { j } - 3 \mathbf { k }\) towards a plane surface.
  1. Give the equation of the path of the laser beam in vector form. The points \(\mathrm { A } ( 1,1,1 ) , \mathrm { B } ( 1,4,2 )\) and \(\mathrm { C } ( 6,1,0 )\) lie on the plane.
  2. Show that the vector \(3 \mathbf { i } - 5 \mathbf { j } + 15 \mathbf { k }\) is perpendicular to the plane and hence find the cartesian equation of the plane.
  3. Find the coordinate of the point where the laser beam hits the surface of the plane.
  4. Find the angle between the laser beam and the plane. \section*{Insert for question 6.} The graph of \(y = \tan x\) is given below.
    On this graph sketch the graph of \(y = \cot x\).
    Show clearly where your graph crosses the graph of \(y = \tan x\) and indicate the asymptotes. [4]
    \includegraphics[max width=\textwidth, alt={}, center]{23771896-942c-4a1d-ab95-6b6d3cc5643c-5_853_1555_703_262}
OCR MEI C4 Q1
3 marks Moderate -0.8
1 Solve the equation. $$\frac { 8 } { x } - \frac { 9 } { x + 1 } = 1$$
OCR MEI C4 Q2
4 marks Standard +0.3
2 Solve the equation \(3 \operatorname { cosec } ^ { 2 } x = 2 \cot ^ { 2 } x + 3\) for values of \(x\) in the range \(0 ^ { \circ } < x < 360 ^ { \circ }\).
OCR MEI C4 Q3
4 marks Moderate -0.3
3 The curve \(y ^ { 2 } = x - 1\) for \(1 \leq x \leq 3\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Find the volume of the solid formed.
OCR MEI C4 Q4
5 marks Moderate -0.5
4 A curve is given by the parametric equations \(x = t ^ { 2 } , y = 3 t\) for all values of \(t\). Find the equation of the tangent to the curve at the point where \(t = - 2\).
OCR MEI C4 Q5
8 marks Moderate -0.3
5
  1. Express \(\frac { 1 + x } { ( 1 - x ) ( 1 - 2 x ) }\) in partial fractions.
  2. Hence find \(\int _ { 2 } ^ { 3 } \frac { 1 + x } { ( 1 - x ) ( 1 - 2 x ) } \mathrm { d } x\).