Questions — OCR MEI (4333 questions)

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OCR MEI C4 2011 June Q8
18 marks Standard +0.8
8 Water is leaking from a container. After \(t\) seconds, the depth of water in the container is \(x \mathrm {~cm}\), and the volume of water is \(V \mathrm {~cm} ^ { 3 }\), where \(V = \frac { 1 } { 3 } x ^ { 3 }\). The rate at which water is lost is proportional to \(x\), so that \(\frac { \mathrm { d } V } { \mathrm {~d} t } = - k x\), where \(k\) is a constant.
  1. Show that \(x \frac { \mathrm {~d} x } { \mathrm {~d} t } = - k\). Initially, the depth of water in the container is 10 cm .
  2. Show by integration that \(x = \sqrt { 100 - 2 k t }\).
  3. Given that the container empties after 50 seconds, find \(k\). Once the container is empty, water is poured into it at a constant rate of \(1 \mathrm {~cm} ^ { 3 }\) per second. The container continues to lose water as before.
  4. Show that, \(t\) seconds after starting to pour the water in, \(\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { 1 - x } { x ^ { 2 } }\).
  5. Show that \(\frac { 1 } { 1 - x } - x - 1 = \frac { x ^ { 2 } } { 1 - x }\). Hence solve the differential equation in part (iv) to show that $$t = \ln \left( \frac { 1 } { 1 - x } \right) - \frac { 1 } { 2 } x ^ { 2 } - x .$$
  6. Show that the depth cannot reach 1 cm .
OCR MEI C4 2012 June Q1
5 marks Moderate -0.8
1 Solve the equation \(\frac { 4 x } { x + 1 } - \frac { 3 } { 2 x + 1 } = 1\).
OCR MEI C4 2012 June Q2
5 marks Moderate -0.8
2 Find the first four terms in the binomial expansion of \(\sqrt { 1 + 2 x }\). State the set of values of \(x\) for which the expansion is valid.
OCR MEI C4 2012 June Q3
8 marks Moderate -0.3
3 The total value of the sales made by a new company in the first \(t\) years of its existence is denoted by \(\pounds V\). A model is proposed in which the rate of increase of \(V\) is proportional to the square root of \(V\). The constant of proportionality is \(k\).
  1. Express the model as a differential equation. Verify by differentiation that \(V = \left( \frac { 1 } { 2 } k t + c \right) ^ { 2 }\), where \(c\) is an arbitrary constant, satisfies this differential equation.
  2. The value of the company’s sales in its first year is \(\pounds 10000\), and the total value of the sales in the first two years is \(\pounds 40000\). Find \(V\) in terms of \(t\).
OCR MEI C4 2012 June Q5
6 marks Moderate -0.3
5 Given the equation \(\sin \left( x + 45 ^ { \circ } \right) = 2 \cos x\), show that \(\sin x + \cos x = 2 \sqrt { 2 } \cos x\).
Hence solve, correct to 2 decimal places, the equation for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
OCR MEI C4 2012 June Q6
8 marks Moderate -0.3
6 Solve the differential equation \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y } { x ( x + 1 ) }\), given that when \(x = 1 , y = 1\). Your answer should express \(y\) explicitly in terms of \(x\).
OCR MEI C4 2012 June Q7
19 marks Standard +0.3
7 Fig. 7a shows the curve with the parametric equations $$x = 2 \cos \theta , \quad y = \sin 2 \theta , \quad - \frac { \pi } { 2 } \leqslant \theta \leqslant \frac { \pi } { 2 } .$$ The curve meets the \(x\)-axis at O and P . Q and R are turning points on the curve. The scales on the axes are the same. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9001b0d0-8d06-43f4-8831-23c0d6aef59d-3_513_661_632_685} \captionsetup{labelformat=empty} \caption{Fig. 7a}
\end{figure}
  1. State, with their coordinates, the points on the curve for which \(\theta = - \frac { \pi } { 2 } , \theta = 0\) and \(\theta = \frac { \pi } { 2 }\).
  2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\theta\). Hence find the gradient of the curve when \(\theta = \frac { \pi } { 2 }\), and verify that the two tangents to the curve at the origin meet at right angles.
  3. Find the exact coordinates of the turning point Q . When the curve is rotated about the \(x\)-axis, it forms a paperweight shape, as shown in Fig. 7b. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{9001b0d0-8d06-43f4-8831-23c0d6aef59d-3_321_385_1758_831} \captionsetup{labelformat=empty} \caption{Fig. 7b}
    \end{figure}
  4. Express \(\sin ^ { 2 } \theta\) in terms of \(x\). Hence show that the cartesian equation of the curve is \(y ^ { 2 } = x ^ { 2 } \left( 1 - \frac { 1 } { 4 } x ^ { 2 } \right)\).
  5. Find the volume of the paperweight shape.
OCR MEI C4 2012 June Q8
17 marks Standard +0.3
8 With respect to cartesian coordinates Oxyz, a laser beam ABC is fired from the point A(1, 2, 4), and is reflected at point B off the plane with equation \(x + 2 y - 3 z = 0\), as shown in Fig. 8. \(\mathrm { A } ^ { \prime }\) is the point (2, 4, 1), and M is the midpoint of \(\mathrm { AA } ^ { \prime }\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9001b0d0-8d06-43f4-8831-23c0d6aef59d-4_563_716_413_635} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure}
  1. Show that \(\mathrm { AA } ^ { \prime }\) is perpendicular to the plane \(x + 2 y - 3 z = 0\), and that M lies in the plane. The vector equation of the line AB is \(\mathbf { r } = \left( \begin{array} { l } 1 \\ 2 \\ 4 \end{array} \right) + \lambda \left( \begin{array} { r } 1 \\ - 1 \\ 2 \end{array} \right)\).
  2. Find the coordinates of B , and a vector equation of the line \(\mathrm { A } ^ { \prime } \mathrm { B }\).
  3. Given that \(\mathrm { A } ^ { \prime } \mathrm { BC }\) is a straight line, find the angle \(\theta\).
  4. Find the coordinates of the point where BC crosses the Oxz plane (the plane containing the \(x\) - and \(z\)-axes).
OCR MEI C4 2013 June Q1
8 marks Standard +0.3
1
  1. Express \(\frac { x } { ( 1 + x ) ( 1 - 2 x ) }\) in partial fractions.
  2. Hence use binomial expansions to show that \(\frac { x } { ( 1 + x ) ( 1 - 2 x ) } = a x + b x ^ { 2 } + \ldots\), where \(a\) and \(b\) are
    constants to be determined. State the set of values of \(x\) for which the expansion is valid.
OCR MEI C4 2013 June Q3
7 marks Moderate -0.8
3 Using appropriate right-angled triangles, show that \(\tan 45 ^ { \circ } = 1\) and \(\tan 30 ^ { \circ } = \frac { 1 } { \sqrt { 3 } }\). Hence show that \(\tan 75 ^ { \circ } = 2 + \sqrt { 3 }\).
OCR MEI C4 2013 June Q4
8 marks Moderate -0.3
4
  1. Find a vector equation of the line \(l\) joining the points \(( 0,1,3 )\) and \(( - 2,2,5 )\).
  2. Find the point of intersection of the line \(l\) with the plane \(x + 3 y + 2 z = 4\).
  3. Find the acute angle between the line \(l\) and the normal to the plane.
OCR MEI C4 2013 June Q5
6 marks Moderate -0.3
5 The points \(\mathrm { A } , \mathrm { B }\) and C have coordinates \(\mathrm { A } ( 3,2 , - 1 ) , \mathrm { B } ( - 1,1,2 )\) and \(\mathrm { C } ( 10,5 , - 5 )\), relative to the origin O . Show that \(\overrightarrow { \mathrm { OC } }\) can be written in the form \(\lambda \overrightarrow { \mathrm { OA } } + \mu \overrightarrow { \mathrm { OB } }\), where \(\lambda\) and \(\mu\) are to be determined. What can you deduce about the points \(\mathrm { O } , \mathrm { A } , \mathrm { B }\) and C from the fact that \(\overrightarrow { \mathrm { OC } }\) can be expressed as a combination of \(\overrightarrow { \mathrm { OA } }\) and \(\overrightarrow { \mathrm { OB } }\) ?
OCR MEI C4 2013 June Q6
18 marks Standard +0.3
6 The motion of a particle is modelled by the differential equation $$v \frac { \mathrm {~d} v } { \mathrm {~d} x } + 4 x = 0 ,$$ where \(x\) is its displacement from a fixed point, and \(v\) is its velocity.
Initially \(x = 1\) and \(v = 4\).
  1. Solve the differential equation to show that \(v ^ { 2 } = 20 - 4 x ^ { 2 }\). Now consider motion for which \(x = \cos 2 t + 2 \sin 2 t\), where \(x\) is the displacement from a fixed point at time \(t\).
  2. Verify that, when \(t = 0 , x = 1\). Use the fact that \(v = \frac { \mathrm { d } x } { \mathrm {~d} t }\) to verify that when \(t = 0 , v = 4\).
  3. Express \(x\) in the form \(R \cos ( 2 t - \alpha )\), where \(R\) and \(\alpha\) are constants to be determined, and obtain the corresponding expression for \(v\). Hence or otherwise verify that, for this motion too, \(v ^ { 2 } = 20 - 4 x ^ { 2 }\).
  4. Use your answers to part (iii) to find the maximum value of \(x\), and the earliest time at which \(x\) reaches this maximum value.
OCR MEI C4 2013 June Q7
18 marks Standard +0.3
7 Fig. 7 shows the curve BC defined by the parametric equations $$x = 5 \ln u , y = u + \frac { 1 } { u } , \quad 1 \leqslant u \leqslant 10 .$$ The point A lies on the \(x\)-axis and AC is parallel to the \(y\)-axis. The tangent to the curve at C makes an angle \(\theta\) with AC, as shown. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4924020c-4df1-4bd5-aae9-95149a09f8c4-03_497_579_1612_744} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure}
  1. Find the lengths \(\mathrm { OA } , \mathrm { OB }\) and AC .
  2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(u\). Hence find the angle \(\theta\).
  3. Show that the cartesian equation of the curve is \(y = \mathrm { e } ^ { \frac { 1 } { 5 } x } + \mathrm { e } ^ { - \frac { 1 } { 5 } x }\). An object is formed by rotating the region OACB through \(360 ^ { \circ }\) about \(\mathrm { O } x\).
  4. Find the volume of the object. \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE.}
OCR MEI C4 2014 June Q1
5 marks Standard +0.3
1 Express \(\frac { 3 x } { ( 2 - x ) \left( 4 + x ^ { 2 } \right) }\) in partial fractions.
OCR MEI C4 2014 June Q3
5 marks Moderate -0.8
3 Fig. 3 shows the curve \(y = x ^ { 3 } + \sqrt { ( \sin x ) }\) for \(0 \leqslant x \leqslant \frac { \pi } { 4 }\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{413a0c52-b506-46d4-b1e4-fe13466abcc1-02_577_538_744_758} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure}
  1. Use the trapezium rule with 4 strips to estimate the area of the region bounded by the curve, the \(x\)-axis and the line \(x = \frac { \pi } { 4 }\), giving your answer to 3 decimal places.
  2. Suppose the number of strips in the trapezium rule is increased. Without doing further calculations, state, with a reason, whether the area estimate increases, decreases, or it is not possible to say.
OCR MEI C4 2014 June Q4
8 marks Standard +0.3
4
  1. Show that \(\cos ( \alpha + \beta ) = \frac { 1 - \tan \alpha \tan \beta } { \sec \alpha \sec \beta }\).
  2. Hence show that \(\cos 2 \alpha = \frac { 1 - \tan ^ { 2 } \alpha } { 1 + \tan ^ { 2 } \alpha }\).
  3. Hence or otherwise solve the equation \(\frac { 1 - \tan ^ { 2 } \theta } { 1 + \tan ^ { 2 } \theta } = \frac { 1 } { 2 }\) for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
OCR MEI C4 2014 June Q5
7 marks Standard +0.3
5 A curve has parametric equations \(x = \mathrm { e } ^ { 3 t } , y = t \mathrm { e } ^ { 2 t }\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\). Hence find the exact gradient of the curve at the point with parameter \(t = 1\).
  2. Find the cartesian equation of the curve in the form \(y = a x ^ { b } \ln x\), where \(a\) and \(b\) are constants to be determined.
OCR MEI C4 2014 June Q6
6 marks Standard +0.3
6 Fig. 6 shows the region enclosed by the curve \(y = \left( 1 + 2 x ^ { 2 } \right) ^ { \frac { 1 } { 3 } }\) and the line \(y = 2\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{413a0c52-b506-46d4-b1e4-fe13466abcc1-03_426_673_340_678} \captionsetup{labelformat=empty} \caption{Fig. 6}
\end{figure} This region is rotated about the \(y\)-axis. Find the volume of revolution formed, giving your answer as a multiple of \(\pi\). \section*{Question 7 begins on page 4.}
OCR MEI C4 2014 June Q7
18 marks Standard +0.3
7 Fig. 7 shows a tetrahedron ABCD . The coordinates of the vertices, with respect to axes \(\mathrm { O } x y z\), are \(\mathrm { A } ( - 3,0,0 ) , \mathrm { B } ( 2,0 , - 2 ) , \mathrm { C } ( 0,4,0 )\) and \(\mathrm { D } ( 0,4,5 )\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{413a0c52-b506-46d4-b1e4-fe13466abcc1-04_794_844_456_589} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure}
  1. Find the lengths of the edges AB and AC , and the size of the angle CAB . Hence calculate the area of triangle ABC .
  2. (A) Verify that \(4 \mathbf { i } - 3 \mathbf { j } + 10 \mathbf { k }\) is normal to the plane ABC .
    (B) Hence find the equation of this plane.
  3. Write down a vector equation for the line through D perpendicular to the plane ABC . Hence find the point of intersection of this line with the plane ABC . The volume of a tetrahedron is \(\frac { 1 } { 3 } \times\) area of base × height.
  4. Find the volume of the tetrahedron ABCD .
OCR MEI C4 2014 June Q8
18 marks Standard +0.3
8 Fig. 8.1 shows an upright cylindrical barrel containing water. The water is leaking out of a hole in the side of the barrel. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{413a0c52-b506-46d4-b1e4-fe13466abcc1-05_254_442_347_794} \captionsetup{labelformat=empty} \caption{Fig. 8.1}
\end{figure} The height of the water surface above the hole \(t\) seconds after opening the hole is \(h\) metres, where $$\frac { \mathrm { d } h } { \mathrm {~d} t } = - A \sqrt { h }$$ and where \(A\) is a positive constant. Initially the water surface is 1 metre above the hole.
  1. Verify that the solution to this differential equation is $$h = \left( 1 - \frac { 1 } { 2 } A t \right) ^ { 2 } .$$ The water stops leaking when \(h = 0\). This occurs after 20 seconds.
  2. Find the value of \(A\), and the time when the height of the water surface above the hole is 0.5 m . Fig. 8.2 shows a similar situation with a different barrel; \(h\) is in metres. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{413a0c52-b506-46d4-b1e4-fe13466abcc1-05_232_447_1489_794} \captionsetup{labelformat=empty} \caption{Fig. 8.2}
    \end{figure} For this barrel, $$\frac { \mathrm { d } h } { \mathrm {~d} t } = - B \frac { \sqrt { h } } { ( 1 + h ) ^ { 2 } } ,$$ where \(B\) is a positive constant. When \(t = 0 , h = 1\).
  3. Solve this differential equation, and hence show that $$h ^ { \frac { 1 } { 2 } } \left( 30 + 20 h + 6 h ^ { 2 } \right) = 56 - 15 B t .$$
  4. Given that \(h = 0\) when \(t = 20\), find \(B\). Find also the time when the height of the water surface above the hole is 0.5 m . \section*{END OF QUESTION PAPER}
OCR MEI C4 2015 June Q2
7 marks Moderate -0.3
2 Express \(6 \cos 2 \theta + \sin \theta\) in terms of \(\sin \theta\).
Hence solve the equation \(6 \cos 2 \theta + \sin \theta = 0\), for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
OCR MEI C4 2015 June Q3
8 marks Standard +0.3
3
  1. Find the first three terms of the binomial expansion of \(\frac { 1 } { \sqrt [ 3 ] { 1 - 2 x } }\). State the set of values of \(x\) for which
    the expansion is valid. the expansion is valid.
  2. Hence find \(a\) and \(b\) such that \(\frac { 1 - 3 x } { \sqrt [ 3 ] { 1 - 2 x } } = 1 + a x + b x ^ { 2 } + \ldots\).
OCR MEI C4 2015 June Q4
8 marks Moderate -0.3
4 You are given that \(\mathrm { f } ( x ) = \cos x + \lambda \sin x\) where \(\lambda\) is a positive constant.
  1. Express \(\mathrm { f } ( x )\) in the form \(R \cos ( x - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\), giving \(R\) and \(\alpha\) in terms of \(\lambda\).
  2. Given that the maximum value (as \(x\) varies) of \(\mathrm { f } ( x )\) is 2 , find \(R , \lambda\) and \(\alpha\), giving your answers in exact form.
OCR MEI C4 2015 June Q5
8 marks Standard +0.3
5 A curve has parametric equations \(x = \sec \theta , y = 2 \tan \theta\).
  1. Given that the derivative of \(\sec \theta\) is \(\sec \theta \tan \theta\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 \operatorname { cosec } \theta\).
  2. Verify that the cartesian equation of the curve is \(y ^ { 2 } = 4 x ^ { 2 } - 4\). Fig. 5 shows the region enclosed by the curve and the line \(x = 2\). This region is rotated through \(180 ^ { \circ }\) about the \(x\)-axis. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{132ae754-bd4c-4819-80ef-4823ac2ead4f-02_545_853_1738_607} \captionsetup{labelformat=empty} \caption{Fig. 5}
    \end{figure}
  3. Find the volume of revolution produced, giving your answer in exact form.