Questions — OCR (4628 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 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 Stats 1 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 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 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 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 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 January Q2
7 marks Moderate -0.3
2
  1. Find, in the form \(\mathbf { r } = \mathbf { a } + t \mathbf { b }\), an equation of the line \(l\) through the points ( \(4,2,7\) ) and ( \(5 , - 4 , - 1\) ).
  2. Find the acute angle between the line \(l\) and a line in the direction of the vector \(\left( \begin{array} { l } 1 \\ 2 \\ 3 \end{array} \right)\).
OCR C4 2012 January Q3
8 marks Standard +0.3
3 The equation of a curve \(C\) is \(( x + 3 ) ( y + 4 ) = x ^ { 2 } + y ^ { 2 }\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
  2. The line \(2 y = x + 3\) meets \(C\) at two points. What can be said about the tangents to \(C\) at these points? Justify your answer.
  3. Find the equation of the tangent at the point ( 6,0 ), giving your answer in the form \(a x + b y = c\), where \(a , b\) and \(c\) are integers.
OCR C4 2012 January Q4
9 marks Standard +0.8
4
  1. Expand \(( 1 - 4 x ) ^ { \frac { 1 } { 4 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
  2. The term of lowest degree in the expansion of $$( 1 + a x ) \left( 1 + b x ^ { 2 } \right) ^ { 7 } - ( 1 - 4 x ) ^ { \frac { 1 } { 4 } }$$ in ascending powers of \(x\) is the term in \(x ^ { 3 }\). Find the values of the constants \(a\) and \(b\).
OCR C4 2012 January Q5
6 marks Standard +0.3
5 Use the substitution \(u = \cos x\) to find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \sin ^ { 3 } x \cos ^ { 2 } x d x$$
OCR C4 2012 January Q6
7 marks Standard +0.8
6 \includegraphics[max width=\textwidth, alt={}, center]{cf154c94-6248-4dda-91e8-61349cc10482-3_606_846_251_614} The diagram shows the curves \(y = \cos x\) and \(y = \sin x\), for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\). The region \(R\) is bounded by the curves and the \(x\)-axis. Find the volume of the solid of revolution formed when \(R\) is rotated completely about the \(x\)-axis, giving your answer in terms of \(\pi\).
OCR C4 2012 January Q7
6 marks Standard +0.3
7 The equation of a straight line \(l\) is $$\mathbf { r } = \left( \begin{array} { l } 1 \\ 0 \\ 2 \end{array} \right) + t \left( \begin{array} { r } 1 \\ - 1 \\ 0 \end{array} \right) .$$ \(O\) is the origin.
  1. Find the position vector of the point \(P\) on \(l\) such that \(O P\) is perpendicular to \(l\).
  2. A point \(Q\) on \(l\) is such that the length of \(O Q\) is 3 units. Find the two possible position vectors of \(Q\). [3]
OCR C4 2012 January Q8
10 marks Standard +0.3
8 A curve is defined by the parametric equations $$x = \sin ^ { 2 } \theta , \quad y = 4 \sin \theta - \sin ^ { 3 } \theta ,$$ where \(- \frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 4 - 3 \sin ^ { 2 } \theta } { 2 \sin \theta }\).
  2. Find the coordinates of the point on the curve at which the gradient is 2 .
  3. Show that the curve has no stationary points.
  4. Find a cartesian equation of the curve, giving your answer in the form \(y ^ { 2 } = \mathrm { f } ( x )\).
OCR C4 2012 January Q9
7 marks Standard +0.3
9 Find the exact value of \(\int _ { 0 } ^ { 1 } \left( x ^ { 2 } + 1 \right) \mathrm { e } ^ { 2 x } \mathrm {~d} x\).
OCR C4 2012 January Q10
9 marks Standard +0.3
10
  1. Write down the derivative of \(\sqrt { y ^ { 2 } + 1 }\) with respect to \(y\).
  2. Given that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { ( x - 1 ) \sqrt { y ^ { 2 } + 1 } } { x y }\) and that \(y = \sqrt { \mathrm { e } ^ { 2 } - 2 \mathrm { e } }\) when \(x = \mathrm { e }\),
    find a relationship between \(x\) and \(y\).
OCR C4 2013 January Q1
4 marks Moderate -0.3
1 Find \(\int x \cos 3 x \mathrm {~d} x\).
OCR C4 2013 January Q2
5 marks Moderate -0.3
2 Find the first three terms in the expansion of \(( 9 - 16 x ) ^ { \frac { 3 } { 2 } }\) in ascending powers of \(x\), and state the set of values for which this expansion is valid.
OCR C4 2013 January Q3
7 marks Standard +0.3
3 The equation of a curve is \(x y ^ { 2 } = x ^ { 2 } + 1\). Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\), and hence find the coordinates of the stationary points on the curve.
OCR C4 2013 January Q4
8 marks Standard +0.3
4 The equations of two lines are $$\mathbf { r } = \mathbf { i } + 2 \mathbf { j } + \lambda ( 2 \mathbf { i } + \mathbf { j } + 3 \mathbf { k } ) \text { and } \mathbf { r } = 6 \mathbf { i } + 8 \mathbf { j } + \mathbf { k } + \mu ( \mathbf { i } + 4 \mathbf { j } - 5 \mathbf { k } ) .$$
  1. Show that these lines meet, and find the coordinates of the point of intersection.
  2. Find the acute angle between these lines.
OCR C4 2013 January Q5
7 marks Moderate -0.3
5 The parametric equations of a curve are $$x = 2 + 3 \sin \theta \text { and } y = 1 - 2 \cos \theta \text { for } 0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi$$
  1. Find the coordinates of the point on the curve where the gradient is \(\frac { 1 } { 2 }\).
  2. Find the cartesian equation of the curve.
OCR C4 2013 January Q6
7 marks Moderate -0.3
6 Use the substitution \(u = 2 x + 1\) to evaluate \(\int _ { 0 } ^ { \frac { 1 } { 2 } } \frac { 4 x - 1 } { ( 2 x + 1 ) ^ { 5 } } \mathrm {~d} x\).
OCR C4 2013 January Q7
7 marks Standard +0.3
7
  1. Given that \(y = \ln ( 1 + \sin x ) - \ln ( \cos x )\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \cos x }\).
  2. Using this result, evaluate \(\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \sec x \mathrm {~d} x\), giving your answer as a single logarithm.
OCR C4 2013 January Q8
7 marks Standard +0.3
8 The points \(A ( 3,2,1 ) , B ( 5,4 , - 3 ) , C ( 3,17 , - 4 )\) and \(D ( 1,6,3 )\) form a quadrilateral \(A B C D\).
  1. Show that \(A B = A D\).
  2. Find a vector equation of the line through \(A\) and the mid-point of \(B D\).
  3. Show that \(C\) lies on the line found in part (ii).
  4. What type of quadrilateral is \(A B C D\) ?
OCR C4 2013 January Q9
9 marks Standard +0.3
9 The temperature of a freezer is \(- 20 ^ { \circ } \mathrm { C }\). A container of a liquid is placed in the freezer. The rate at which the temperature, \(\theta ^ { \circ } \mathrm { C }\), of a liquid decreases is proportional to the difference in temperature between the liquid and its surroundings. The situation is modelled by the differential equation $$\frac { \mathrm { d } \theta } { \mathrm {~d} t } = - k ( \theta + 20 ) ,$$ where time \(t\) is in minutes and \(k\) is a positive constant.
  1. Express \(\theta\) in terms of \(t , k\) and an arbitrary constant. Initially the temperature of the liquid in the container is \(40 ^ { \circ } \mathrm { C }\) and, at this instant, the liquid is cooling at a rate of \(3 ^ { \circ } \mathrm { C }\) per minute. The liquid freezes at \(0 ^ { \circ } \mathrm { C }\).
  2. Find the value of \(k\) and find also the time it takes (to the nearest minute) for the liquid to freeze. The procedure is repeated on another occasion with a different liquid. The initial temperature of this liquid is \(90 ^ { \circ } \mathrm { C }\). After 19 minutes its temperature is \(0 ^ { \circ } \mathrm { C }\).
  3. Without any further calculation, explain what you can deduce about the value of \(k\) in this case.
OCR C4 2013 January Q10
11 marks Moderate -0.3
10
  1. Use algebraic division to express \(\frac { x ^ { 3 } - 2 x ^ { 2 } - 4 x + 13 } { x ^ { 2 } - x - 6 }\) in the form \(A x + B + \frac { C x + D } { x ^ { 2 } - x - 6 }\), where \(A , B , C\) and \(D\) are constants.
  2. Hence find \(\int _ { 4 } ^ { 6 } \frac { x ^ { 3 } - 2 x ^ { 2 } - 4 x + 13 } { x ^ { 2 } - x - 6 } \mathrm {~d} x\), giving your answer in the form \(a + \ln b\).
OCR C4 2009 June Q1
4 marks Moderate -0.3
1 Find the quotient and the remainder when \(3 x ^ { 4 } - x ^ { 3 } - 3 x ^ { 2 } - 14 x - 8\) is divided by \(x ^ { 2 } + x + 2\).
OCR C4 2009 June Q2
7 marks Standard +0.3
2 Use the substitution \(x = \tan \theta\) to find the exact value of $$\int _ { 1 } ^ { \sqrt { 3 } } \frac { 1 - x ^ { 2 } } { 1 + x ^ { 2 } } \mathrm {~d} x$$
OCR C4 2009 June Q3
7 marks Standard +0.3
3
  1. Expand \(( a + x ) ^ { - 2 }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 2 }\).
  2. When \(( 1 - x ) ( a + x ) ^ { - 2 }\) is expanded, the coefficient of \(x ^ { 2 }\) is 0 . Find the value of \(a\).
OCR C4 2009 June Q4
7 marks Standard +0.3
4
  1. Differentiate \(\mathrm { e } ^ { x } ( \sin 2 x - 2 \cos 2 x )\), simplifying your answer.
  2. Hence find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \mathrm { e } ^ { x } \sin 2 x \mathrm {~d} x\).
OCR C4 2009 June Q5
9 marks Standard +0.3
5 A curve has parametric equations $$x = 2 t + t ^ { 2 } , \quad y = 2 t ^ { 2 } + t ^ { 3 }$$
  1. Express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\) and find the gradient of the curve at the point \(( 3 , - 9 )\).
  2. By considering \(\frac { y } { x }\), find a cartesian equation of the curve, giving your answer in a form not involving fractions.
OCR C4 2009 June Q6
9 marks Moderate -0.3
6 The expression \(\frac { 4 x } { ( x - 5 ) ( x - 3 ) ^ { 2 } }\) is denoted by \(\mathrm { f } ( x )\).
  1. Express f \(( x )\) in the form \(\frac { A } { x - 5 } + \frac { B } { x - 3 } + \frac { C } { ( x - 3 ) ^ { 2 } }\), where \(A , B\) and \(C\) are constants.
  2. Hence find the exact value of \(\int _ { 1 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x\).