Questions C4 (1219 questions)

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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.
OCR C4 2015 June Q1
5 marks Moderate -0.8
1
  1. Express \(\frac { 2 } { 3 - x } + \frac { 3 } { 1 + x }\) as a single fraction in its simplest form.
  2. Hence express \(\left( \frac { 2 } { 3 - x } + \frac { 3 } { 1 + x } \right) \times \frac { x ^ { 2 } + 8 x - 33 } { 121 - x ^ { 2 } }\) as a single fraction in its lowest terms.
OCR C4 2015 June Q2
6 marks Standard +0.3
2 A triangle has vertices at \(A ( 1,1,3 ) , B ( 5,9 , - 5 )\) and \(C ( 6,5 , - 4 ) . P\) is the point on \(A B\) such that \(A P : P B = 3 : 1\).
  1. Show that \(\overrightarrow { C P }\) is perpendicular to \(\overrightarrow { A B }\).
  2. Find the area of the triangle \(A B C\).
OCR C4 2015 June Q3
6 marks Standard +0.3
3 The equation of a curve is \(y = \mathrm { e } ^ { 2 x } \cos x\). Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence find the coordinates of any stationary points for which \(- \pi \leqslant x \leqslant \pi\). Give your answers correct to 3 significant figures.
OCR C4 2015 June Q4
5 marks Moderate -0.3
4
  1. Find the first three terms in the binomial expansion of \(( 8 - 9 x ) ^ { \frac { 2 } { 3 } }\) in ascending powers of \(x\).
  2. State the set of values of \(x\) for which this expansion is valid.
OCR C4 2015 June Q5
6 marks Standard +0.3
5 By first using the substitution \(t = \sqrt { x + 1 }\), find \(\int \mathrm { e } ^ { 2 \sqrt { x + 1 } } \mathrm {~d} x\).
OCR C4 2015 June Q6
8 marks Standard +0.8
6
  1. Use the quotient rule to show that the derivative of \(\frac { \cos x } { \sin x }\) is \(\frac { - 1 } { \sin ^ { 2 } x }\).
  2. Show that \(\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 4 } \pi } \frac { \sqrt { 1 + \cos 2 x } } { \sin x \sin 2 x } \mathrm {~d} x = \frac { 1 } { 2 } ( \sqrt { 6 } - \sqrt { 2 } )\).
OCR C4 2015 June Q7
7 marks Standard +0.3
7 A curve has equation \(( x + y ) ^ { 2 } = x y ^ { 2 }\). Find the gradient of the curve at the point where \(x = 1\).
OCR C4 2015 June Q8
8 marks Standard +0.3
8 In the year 2000 the population density, \(P\), of a village was 100 people per \(\mathrm { km } ^ { 2 }\), and was increasing at the rate of 1 person per \(\mathrm { km } ^ { 2 }\) per year. The rate of increase of the population density is thought to be inversely proportional to the size of the population density. The time in years after the year 2000 is denoted by \(t\).
  1. Write down a differential equation to model this situation, and solve it to express \(P\) in terms of \(t\).
  2. In 2008 the population density of the village was 108 people per \(\mathrm { km } ^ { 2 }\) and in 2013 it was 128 people per \(\mathrm { km } ^ { 2 }\). Determine how well the model fits these figures.
OCR C4 2015 June Q9
7 marks Standard +0.3
9 Two lines have equations $$\mathbf { r } = 3 \mathbf { i } + 5 \mathbf { j } - \mathbf { k } + \lambda ( 2 \mathbf { i } + \mathbf { j } + \mathbf { k } ) \text { and } \mathbf { r } = 4 \mathbf { i } + 10 \mathbf { j } + 19 \mathbf { k } + \mu ( \mathbf { i } - \mathbf { j } + \alpha \mathbf { k } ) ,$$ where \(\alpha\) is a constant.
Find the value of \(\alpha\) in each of the following cases.
  1. The lines intersect at the point (7,7,1).
  2. The angle between their directions is \(60 ^ { \circ }\).
OCR C4 2015 June Q10
14 marks Standard +0.3
10
  1. Express \(\frac { x + 8 } { x ( x + 2 ) }\) in partial fractions.
  2. By first using division, express \(\frac { 7 x ^ { 2 } + 16 x + 16 } { x ( x + 2 ) }\) in the form \(P + \frac { Q } { x } + \frac { R } { x + 2 }\). A curve has parametric equations \(x = \frac { 2 t } { 1 - t } , y = 3 t + \frac { 4 } { t }\).
  3. Show that the cartesian equation of the curve is \(y = \frac { 7 x ^ { 2 } + 16 x + 16 } { x ( x + 2 ) }\).
  4. Find the area of the region bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 2\). Give your answer in the form \(L + M \ln 2 + N \ln 3\).
OCR C4 2016 June Q1
3 marks Moderate -0.5
1 Find the quotient and the remainder when \(4 x ^ { 3 } + 8 x ^ { 2 } - 5 x + 12\) is divided by \(2 x ^ { 2 } + 1\).
OCR C4 2016 June Q2
5 marks Standard +0.3
2 Use integration to find the exact value of \(\int _ { \frac { 1 } { 16 } \pi } ^ { \frac { 1 } { 8 } \pi } \left( 9 - 6 \cos ^ { 2 } 4 x \right) \mathrm { d } x\).