Questions — OCR C4 (310 questions)

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OCR C4 2011 June Q8
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
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
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
2 Use integration by parts to find \(\int \ln ( x + 2 ) \mathrm { d } x\).
OCR C4 2012 June Q3
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
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
\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. \(| \mathbf { a } - \mathbf { b } |\).
OCR C4 2012 June Q6
6 Use the substitution \(u = 1 + \sqrt { x }\) to show that $$\int _ { 4 } ^ { 9 } \frac { 1 } { 1 + \sqrt { x } } \mathrm {~d} x = 2 + 2 \ln \frac { 3 } { 4 }$$
OCR C4 2012 June Q7
7 Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } ( 1 - \sin 3 x ) ^ { 2 } \mathrm {~d} x\).
OCR C4 2012 June Q8
8
  1. Find the gradient of the curve \(x ^ { 2 } + x y + y ^ { 2 } = 3\) at the point \(( - 1 , - 1 )\).
  2. A curve \(C\) has parametric equations $$x = 2 t ^ { 2 } - 1 , y = t ^ { 3 } + t$$
    1. Find the coordinates of the point on \(C\) at which the tangent is parallel to the \(y\)-axis.
    2. Find the values of \(t\) for which \(x\) and \(y\) have the same rate of change with respect to \(t\).
OCR C4 2012 June Q9
9
  1. Express \(\frac { x ^ { 2 } - x - 11 } { ( x + 1 ) ( x - 2 ) ^ { 2 } }\) in partial fractions.
  2. Find the exact value of \(\int _ { 3 } ^ { 4 } \frac { x ^ { 2 } - x - 11 } { ( x + 1 ) ( x - 2 ) ^ { 2 } } \mathrm {~d} x\), giving your answer in the form \(a + \ln b\), where \(a\) and \(b\) are rational numbers.
OCR C4 2012 June Q10
10 Lines \(l _ { 1 }\) and \(l _ { 2 }\) have vector equations $$\mathbf { r } = - \mathbf { i } + 2 \mathbf { j } + 7 \mathbf { k } + t ( 2 \mathbf { i } + 2 \mathbf { j } + \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = 2 \mathbf { i } + 9 \mathbf { j } - 4 \mathbf { k } + s ( \mathbf { i } + 3 \mathbf { j } - 2 \mathbf { k } )$$ respectively. The point \(A\) has coordinates ( \(- 3,0,6\) ) relative to the origin \(O\).
  1. Show that \(A\) lies on \(l _ { 1 }\) and that \(O A\) is perpendicular to \(l _ { 1 }\).
  2. Show that the line through \(O\) and \(A\) intersects \(l _ { 2 }\).
  3. Given that the point of intersection in part (ii) is \(B\), find the ratio \(| \overrightarrow { O A } | : | \overrightarrow { B A } |\). \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE.}
OCR C4 2013 June Q1
1 Express \(\frac { ( x - 7 ) ( x - 2 ) } { ( x + 2 ) ( x - 1 ) ^ { 2 } }\) in partial fractions.
OCR C4 2013 June Q2
2 Find \(\int x ^ { 8 } \ln ( 3 x ) \mathrm { d } x\).
OCR C4 2013 June Q3
3 Determine whether the lines whose equations are $$\mathbf { r } = ( 1 + 2 \lambda ) \mathbf { i } - \lambda \mathbf { j } + ( 3 + 5 \lambda ) \mathbf { k } \text { and } \mathbf { r } = ( \mu - 1 ) \mathbf { i } + ( 5 - \mu ) \mathbf { j } + ( 2 - 5 \mu ) \mathbf { k }$$ are parallel, intersect or are skew.
OCR C4 2013 June Q4
4 The equation of a curve is \(y = \cos 2 x + 2 \sin x\). Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence find the coordinates of the stationary points on the curve for \(0 < x < \pi\).
OCR C4 2013 June Q5
5
  1. Show that \(\frac { 1 } { 1 - \tan x } - \frac { 1 } { 1 + \tan x } \equiv \tan 2 x\).
  2. Hence evaluate \(\int _ { \frac { 1 } { 12 } \pi } ^ { \frac { 1 } { 6 } \pi } \left( \frac { 1 } { 1 - \tan x } - \frac { 1 } { 1 + \tan x } \right) \mathrm { d } x\), giving your answer in the form \(a \ln b\).
OCR C4 2013 June Q6
6 Use the substitution \(u = 1 + \ln x\) to find \(\int \frac { \ln x } { x ( 1 + \ln x ) ^ { 2 } } \mathrm {~d} x\).
OCR C4 2013 June Q7
7 Points \(A ( 2,2,5 ) , B ( 1 , - 1 , - 4 ) , C ( 3,3,10 )\) and \(D ( 8,6,3 )\) are the vertices of a pyramid with a triangular base.
  1. Calculate the lengths \(A B\) and \(A C\), and the angle \(B A C\).
  2. Show that \(\overrightarrow { A D }\) is perpendicular to both \(\overrightarrow { A B }\) and \(\overrightarrow { A C }\).
  3. Calculate the volume of the pyramid \(A B C D\).
    [0pt] [The volume of the pyramid is \(V = \frac { 1 } { 3 } \times\) base area × perpendicular height.]
OCR C4 2013 June Q8
8 At time \(t\) seconds, the radius of a spherical balloon is \(r \mathrm {~cm}\). The balloon is being inflated so that the rate of increase of its radius is inversely proportional to the square root of its radius. When \(t = 5 , r = 9\) and, at this instant, the radius is increasing at \(1.08 \mathrm {~cm} \mathrm {~s} ^ { - 1 }\).
  1. Write down a differential equation to model this situation, and solve it to express \(r\) in terms of \(t\).
  2. How much air is in the balloon initially?
    [0pt] [The volume of a sphere is \(V = \frac { 4 } { 3 } \pi r ^ { 3 }\).]
OCR C4 2013 June Q9
9 A curve has parametric equations \(x = \frac { 1 } { t } - 1\) and \(y = 2 t + \frac { 1 } { t ^ { 2 } }\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\), simplifying your answer.
  2. Find the coordinates of the stationary point and, by considering the gradient of the curve on either side of this point, determine its nature.
  3. Find a cartesian equation of the curve.
OCR C4 2013 June Q10
10
  1. Show that \(\frac { x } { ( 1 - x ) ^ { 3 } } \approx x + 3 x ^ { 2 } + 6 x ^ { 3 }\) for small values of \(x\).
  2. Use this result, together with a suitable value of \(x\), to obtain a decimal estimate of the value of \(\frac { 100 } { 729 }\).
  3. Show that \(\frac { x } { ( 1 - x ) ^ { 3 } } = - \frac { 1 } { x ^ { 2 } } \left( 1 - \frac { 1 } { x } \right) ^ { - 3 }\). Hence find the first three terms of the binomial expansion of \(\frac { x } { ( 1 - x ) ^ { 3 } }\) in powers of \(\frac { 1 } { x }\).
  4. Comment on the suitability of substituting the same value of \(x\) as used in part (ii) in the expansion in part (iii) to estimate the value of \(\frac { 100 } { 729 }\).
OCR C4 2014 June Q1
1 Express \(x + \frac { 1 } { 1 - x } + \frac { 2 } { 1 + x }\) as a single fraction, simplifying your answer.
OCR C4 2014 June Q2
2 The points \(O ( 0,0,0 ) , A ( 2,8,2 ) , B ( 5,5,8 )\) and \(C ( 3 , - 3,6 )\) form a parallelogram \(O A B C\). Use a scalar product to find the acute angle between the diagonals of this parallelogram.
OCR C4 2014 June Q3
3
  1. Find the first three terms in the expansion of \(( 1 - 2 x ) ^ { - \frac { 1 } { 2 } }\) in ascending powers of \(x\), where \(| x | < \frac { 1 } { 2 }\).
  2. Hence find the coefficient of \(x ^ { 2 }\) in the expansion of \(\frac { x + 3 } { \sqrt { 1 - 2 x } }\).