Questions — OCR C4 (317 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 S1 S2 S3 S4 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 PURE S1 S2 S3 S4 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
Sort by: Default | Easiest first | Hardest first
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. \(| \mathbf { a } - \mathbf { b } |\).
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 \(Q\).