Questions — OCR (4628 questions)

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OCR C4 2009 June Q8
10 marks Standard +0.3
8
  1. Given that \(14 x ^ { 2 } - 7 x y + y ^ { 2 } = 2\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 28 x - 7 y } { 7 x - 2 y }\).
  2. The points \(L\) and \(M\) on the curve \(14 x ^ { 2 } - 7 x y + y ^ { 2 } = 2\) each have \(x\)-coordinate 1 . The tangents to the curve at \(L\) and \(M\) meet at \(N\). Find the coordinates of \(N\).
OCR C4 2009 June Q9
10 marks Moderate -0.3
9 A tank contains water which is heated by an electric water heater working under the action of a thermostat. The temperature of the water, \(\theta ^ { \circ } \mathrm { C }\), may be modelled as follows. When the water heater is first switched on, \(\theta = 40\). The heater causes the temperature to increase at a rate \(k _ { 1 } { } ^ { \circ } \mathrm { C }\) per second, where \(k _ { 1 }\) is a constant, until \(\theta = 60\). The heater then switches off.
  1. Write down, in terms of \(k _ { 1 }\), how long it takes for the temperature to increase from \(40 ^ { \circ } \mathrm { C }\) to \(60 ^ { \circ } \mathrm { C }\). The temperature of the water then immediately starts to decrease at a variable rate \(k _ { 2 } ( \theta - 20 ) ^ { \circ } \mathrm { C }\) per second, where \(k _ { 2 }\) is a constant, until \(\theta = 40\).
  2. Write down a differential equation to represent the situation as the temperature is decreasing.
  3. Find the total length of time for the temperature to increase from \(40 ^ { \circ } \mathrm { C }\) to \(60 ^ { \circ } \mathrm { C }\) and then decrease to \(40 ^ { \circ } \mathrm { C }\). Give your answer in terms of \(k _ { 1 }\) and \(k _ { 2 }\). 4
OCR C4 2010 June Q1
5 marks Moderate -0.8
1 Expand \(( 1 + 3 x ) ^ { - \frac { 5 } { 3 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
OCR C4 2010 June Q2
4 marks Moderate -0.3
2 Given that \(y = \frac { \cos x } { 1 - \sin x }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), simplifying your answer.
OCR C4 2010 June Q3
5 marks Moderate -0.3
3 Express \(\frac { x ^ { 2 } } { ( x - 1 ) ^ { 2 } ( x - 2 ) }\) in partial fractions.
OCR C4 2010 June Q4
7 marks Standard +0.3
4 Use the substitution \(u = \sqrt { x + 2 }\) to find the exact value of $$\int _ { - 1 } ^ { 7 } \frac { x ^ { 2 } } { \sqrt { x + 2 } } \mathrm {~d} x$$
OCR C4 2010 June Q5
7 marks Standard +0.8
5 Find the coordinates of the two stationary points on the curve with equation $$x ^ { 2 } + 4 x y + 2 y ^ { 2 } + 18 = 0$$
OCR C4 2010 June Q6
10 marks Standard +0.3
6 Lines \(l _ { 1 }\) and \(l _ { 2 }\) have vector equations $$\mathbf { r } = \mathbf { j } + \mathbf { k } + t ( 2 \mathbf { i } + a \mathbf { j } + \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = 3 \mathbf { i } - \mathbf { k } + s ( 2 \mathbf { i } + 2 \mathbf { j } - 6 \mathbf { k } )$$ respectively, where \(t\) and \(s\) are parameters and \(a\) is a constant.
  1. Given that \(l _ { 1 }\) and \(l _ { 2 }\) are perpendicular, find the value of \(a\).
  2. Given instead that \(l _ { 1 }\) and \(l _ { 2 }\) intersect, find
    (a) the value of \(a\),
    (b) the angle between the lines.
OCR C4 2010 June Q7
11 marks Moderate -0.3
7 The parametric equations of a curve are \(x = \frac { t + 2 } { t + 1 } , y = \frac { 2 } { t + 3 }\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } > 0\).
  2. Find the cartesian equation of the curve, giving your answer in a form not involving fractions.
OCR C4 2010 June Q8
10 marks Standard +0.3
8
  1. Find the quotient and the remainder when \(x ^ { 2 } - 5 x + 6\) is divided by \(x - 1\).
  2. (a) Find the general solution of the differential equation $$\left( \frac { x - 1 } { x ^ { 2 } - 5 x + 6 } \right) \frac { \mathrm { d } y } { \mathrm {~d} x } = y - 5 .$$ (b) Given that \(y = 7\) when \(x = 8\), find \(y\) when \(x = 6\).
OCR C4 2010 June Q9
13 marks Standard +0.3
9
  1. Find \(\int ( x + \cos 2 x ) ^ { 2 } \mathrm {~d} x\).
  2. \includegraphics[max width=\textwidth, alt={}, center]{80f94db1-39be-46f5-896e-277c93cbe4b8-3_538_935_383_646} The diagram shows the part of the curve \(y = x + \cos 2 x\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\). The shaded region bounded by the curve, the axes and the line \(x = \frac { 1 } { 2 } \pi\) is rotated completely about the \(x\)-axis to form a solid of revolution of volume \(V\). Find \(V\), giving your answer in an exact form.
OCR C4 2011 June Q1
4 marks Moderate -0.3
1 Simplify \(\frac { x ^ { 4 } - 10 x ^ { 2 } + 9 } { \left( x ^ { 2 } - 2 x - 3 \right) \left( x ^ { 2 } + 8 x + 15 \right) }\).
OCR C4 2011 June Q2
3 marks Easy -1.2
2 Find the unit vector in the direction of \(\left( \begin{array} { c } 2 \\ - 3 \\ \sqrt { 12 } \end{array} \right)\).
OCR C4 2011 June Q3
8 marks Standard +0.3
3
  1. Find the quotient when \(3 x ^ { 3 } - x ^ { 2 } + 10 x - 3\) is divided by \(x ^ { 2 } + 3\), and show that the remainder is \(x\).
  2. Hence find the exact value of $$\int _ { 0 } ^ { 1 } \frac { 3 x ^ { 3 } - x ^ { 2 } + 10 x - 3 } { x ^ { 2 } + 3 } \mathrm {~d} x$$
OCR C4 2011 June Q4
6 marks Standard +0.8
4 Use the substitution \(x = \frac { 1 } { 3 } \sin \theta\) to find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 6 } } \frac { 1 } { \left( 1 - 9 x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x$$
OCR C4 2011 June Q5
10 marks Standard +0.3
5 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations $$\mathbf { r } = \left( \begin{array} { l } 4
OCR C4 2011 June Q6
8 marks Standard +0.3
6
4 \end{array} \right) + s \left( \begin{array} { l } 3
2
1 \end{array} \right) \quad \text { and } \quad \mathbf { r } = \left( \begin{array} { l } 1
0
0 \end{array} \right) + t \left( \begin{array} { r } 0
1
- 1 \end{array} \right)$$ respectively.
  1. Show that \(l _ { 1 }\) and \(l _ { 2 }\) are skew.
  2. Find the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\).
  3. The point \(A\) lies on \(l _ { 1 }\) and \(O A\) is perpendicular to \(l _ { 1 }\), where \(O\) is the origin. Find the position vector of \(A\). 6 Find the coefficient of \(x ^ { 2 }\) in the expansion in ascending powers of \(x\) of $$\sqrt { \frac { 1 + a x } { 4 - x } } ,$$ giving your answer in terms of \(a\).
OCR C4 2011 June Q7
7 marks Moderate -0.3
7 The gradient of a curve at the point \(( x , y )\), where \(x > - 2\), is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 3 y ^ { 2 } ( x + 2 ) }$$ The points \(( 1,2 )\) and \(( q , 1.5 )\) lie on the curve. Find the value of \(q\), giving your answer correct to 3 significant figures.
OCR C4 2011 June Q8
13 marks Standard +0.3
8 A curve has parametric equations $$x = \frac { 1 } { t + 1 } , \quad y = t - 1 .$$ The line \(y = 3 x\) intersects the curve at two points.
  1. Show that the value of \(t\) at one of these points is - 2 and find the value of \(t\) at the other point.
  2. Find the equation of the normal to the curve at the point for which \(t = - 2\).
  3. Find the value of \(t\) at the point where this normal meets the curve again.
  4. Find a cartesian equation of the curve, giving your answer in the form \(y = \mathrm { f } ( x )\).
OCR C4 2011 June Q9
13 marks Standard +0.3
9
  1. Show that \(\frac { \mathrm { d } } { \mathrm { d } x } ( x \ln x - x ) = \ln x\).
  2. \includegraphics[max width=\textwidth, alt={}, center]{8492b214-aaac-4354-8649-e317bf7b3535-3_481_725_1064_751} In the diagram, \(C\) is the curve \(y = \ln x\). The region \(R\) is bounded by \(C\), the \(x\)-axis and the line \(x = \mathrm { e }\).
    (a) Find the exact volume of the solid of revolution formed by rotating \(R\) completely about the \(x\)-axis.
    (b) The region \(R\) is rotated completely about the \(y\)-axis. Explain why the volume of the solid of revolution formed is given by $$\pi \mathrm { e } ^ { 2 } - \pi \int _ { 0 } ^ { 1 } \mathrm { e } ^ { 2 y } \mathrm {~d} y ,$$ and find this volume.
OCR C4 2012 June Q1
6 marks Moderate -0.8
1 Simplify
  1. \(\frac { 1 - x } { x ^ { 2 } - 3 x + 2 }\),
  2. \(\frac { ( x + 1 ) } { ( x - 1 ) ( x - 3 ) } - \frac { ( x - 5 ) } { ( x - 3 ) ( x - 4 ) }\).
OCR C4 2012 June Q2
5 marks Standard +0.3
2 Use integration by parts to find \(\int \ln ( x + 2 ) \mathrm { d } x\).
OCR C4 2012 June Q3
7 marks Standard +0.3
3
  1. Expand \(\frac { 1 + x ^ { 2 } } { \sqrt { 1 + 4 x } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
  2. State the set of values of \(x\) for which this expansion is valid.
OCR C4 2012 June Q4
6 marks Moderate -0.3
4 Solve the differential equation $$\mathrm { e } ^ { 2 y } \frac { \mathrm {~d} y } { \mathrm {~d} x } + \tan x = 0 ,$$ given that \(x = 0\) when \(y = 0\). Give your answer in the form \(y = \mathrm { f } ( x )\).
OCR C4 2012 June Q5
5 marks Moderate -0.8
5 \includegraphics[max width=\textwidth, alt={}, center]{b5d85e48-0d5a-4edf-bf58-eba4f8d28d3d-2_425_680_1302_689} In the diagram the points \(A\) and \(B\) have position vectors \(\mathbf { a }\) and \(\mathbf { b }\) with respect to the origin \(O\). Given that \(| \mathbf { a } | = 3 , | \mathbf { b } | = 4\) and \(\mathbf { a . b } = 6\), find
  1. the angle \(A O B\),
  2. \(| \mathbf { a } - \mathbf { b } |\).