Questions — OCR (4628 questions)

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OCR C4 2012 June Q6
7 marks Standard +0.3
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 marks Standard +0.3
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
10 marks Moderate -0.3
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 marks Standard +0.3
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 marks Standard +0.3
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
5 marks Standard +0.3
1 Express \(\frac { ( x - 7 ) ( x - 2 ) } { ( x + 2 ) ( x - 1 ) ^ { 2 } }\) in partial fractions.
OCR C4 2013 June Q2
5 marks Standard +0.3
2 Find \(\int x ^ { 8 } \ln ( 3 x ) \mathrm { d } x\).
OCR C4 2013 June Q3
6 marks Moderate -0.3
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
6 marks Standard +0.3
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
7 marks Standard +0.3
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 marks Standard +0.3
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
10 marks Standard +0.3
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
9 marks Standard +0.3
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 marks Standard +0.3
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
9 marks Standard +0.3
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
3 marks Easy -1.2
1 Express \(x + \frac { 1 } { 1 - x } + \frac { 2 } { 1 + x }\) as a single fraction, simplifying your answer.
OCR C4 2014 June Q2
5 marks Standard +0.3
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
5 marks Standard +0.3
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 \(\frac { x + 3 } { \sqrt { 1 - 2 x } }\).
OCR C4 2014 June Q4
5 marks Standard +0.8
4 Show that \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \frac { 1 - 2 \sin ^ { 2 } x } { 1 + 2 \sin x \cos x } \mathrm {~d} x = \frac { 1 } { 2 } \ln 2\).
OCR C4 2014 June Q5
6 marks Standard +0.3
5 The equations of three lines are as follows. $$\begin{array} { l l } \text { Line } A : & \mathbf { r } = \mathbf { i } + 4 \mathbf { j } + \mathbf { k } + s ( - \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k } ) \\ \text { Line } B : & \mathbf { r } = 2 \mathbf { i } + 8 \mathbf { j } + 2 \mathbf { k } + t ( \mathbf { i } + 3 \mathbf { j } + 5 \mathbf { k } ) \\ \text { Line } C : & \mathbf { r } = - \mathbf { i } + 19 \mathbf { j } + 15 \mathbf { k } + u ( 2 \mathbf { i } - 4 \mathbf { j } - 4 \mathbf { k } ) \end{array}$$
  1. Show that lines \(A\) and \(B\) are skew.
  2. Determine, giving reasons, the geometrical relationship between lines \(A\) and \(C\).
OCR C4 2014 June Q6
8 marks Standard +0.8
6 \includegraphics[max width=\textwidth, alt={}, center]{02e31b5d-10dd-42b1-885a-6db610d788c3-2_570_1191_1509_420} The diagram shows the curve with equation \(x ^ { 2 } + y ^ { 3 } - 8 x - 12 y = 4\). At each of the points \(P\) and \(Q\) the tangent to the curve is parallel to the \(y\)-axis. Find the coordinates of \(P\) and \(Q\).
OCR C4 2014 June Q7
11 marks Standard +0.3
7 A curve has parametric equations $$x = 2 \sin t , \quad y = \cos 2 t + 2 \sin t$$ for \(- \frac { 1 } { 2 } \pi \leqslant t \leqslant \frac { 1 } { 2 } \pi\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1 - 2 \sin t\) and hence find the coordinates of the stationary point.
  2. Find the cartesian equation of the curve.
  3. State the set of values that \(x\) can take and hence sketch the curve.
OCR C4 2014 June Q8
9 marks Standard +0.8
8
  1. Use division to show that \(\frac { t ^ { 3 } } { t + 2 } \equiv t ^ { 2 } - 2 t + 4 - \frac { 8 } { t + 2 }\).
  2. Find \(\int _ { 1 } ^ { 2 } 6 t ^ { 2 } \ln ( t + 2 ) \mathrm { d } t\). Give your answer in the form \(A + B \ln 3 + C \ln 4\).
OCR C4 2014 June Q9
9 marks Standard +0.3
9 Express \(\frac { 2 + x ^ { 2 } } { ( 1 + 2 x ) ( 1 - x ) ^ { 2 } }\) in partial fractions and hence show that \(\int _ { 0 } ^ { \frac { 1 } { 4 } } \frac { 2 + x ^ { 2 } } { ( 1 + 2 x ) ( 1 - x ) ^ { 2 } } \mathrm {~d} x = \frac { 1 } { 2 } \ln \frac { 3 } { 2 } + \frac { 1 } { 3 }\).
OCR C4 2014 June Q10
11 marks Standard +0.3
10 A container in the shape of an inverted cone of radius 3 metres and vertical height 4.5 metres is initially filled with liquid fertiliser. This fertiliser is released through a hole in the bottom of the container at a rate of \(0.01 \mathrm {~m} ^ { 3 }\) per second. At time \(t\) seconds the fertiliser remaining in the container forms an inverted cone of height \(h\) metres.
[0pt] [The volume of a cone is \(V = \frac { 1 } { 3 } \pi r ^ { 2 } h\).]
  1. Show that \(h ^ { 2 } \frac { \mathrm {~d} h } { \mathrm {~d} t } = - \frac { 9 } { 400 \pi }\).
  2. Express \(h\) in terms of \(t\).
  3. Find the time it takes to empty the container, giving your answer to the nearest minute.