Questions (33218 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 PURE Q4
6 marks Standard +0.3
4
  1. Find and simplify the first three terms in the expansion, in ascending powers of \(x\), of \(\left( 2 + \frac { 1 } { 3 } k x \right) ^ { 6 }\), where \(k\) is a constant.
  2. In the expansion of \(( 3 - 4 x ) \left( 2 + \frac { 1 } { 3 } k x \right) ^ { 6 }\), the constant term is equal to the coefficient of \(x ^ { 2 }\). Determine the exact value of \(k\), given that \(k\) is positive.
OCR PURE Q5
5 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{8c0b68bd-2257-4994-b444-def0b3f64334-4_591_547_262_242} The diagram shows the graphs of \(y = 2 ^ { 3 x }\) and \(y = 2 ^ { 3 x + 2 }\). The graph of \(y = 2 ^ { 3 x }\) can be transformed to the graph of \(y = 2 ^ { 3 x + 2 }\) by means of a stretch.
  1. Give details of the stretch. The point \(A\) lies on \(y = 2 ^ { 3 x }\) and the point \(B\) lies on \(y = 2 ^ { 3 x + 2 }\). The line segment \(A B\) is parallel to the \(y\)-axis and the difference between the \(y\)-coordinates of \(A\) and \(B\) is 36 .
  2. Determine the \(x\)-coordinate of \(A\). Give your answer in the form \(m \log _ { 2 } n\) where \(m\) and \(n\) are constants to be determined.
OCR PURE Q6
10 marks Moderate -0.3
6 The vertices of triangle \(A B C\) are \(A ( - 3,1 ) , B ( 5,0 )\) and \(C ( 9,7 )\).
  1. Show that \(A B = B C\).
  2. Show that angle \(A B C\) is not a right angle.
  3. Find the coordinates of the midpoint of \(A C\).
  4. Determine the equation of the line of symmetry of the triangle, giving your answer in the form \(p x + q y = r\), where \(p , q\) and \(r\) are integers to be determined.
  5. Write down an equation of the circle with centre \(A\) which passes through \(B\). This circle intersects the line of symmetry of the triangle at \(B\) and at a second point.
  6. Find the coordinates of this second point.
OCR PURE Q7
7 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{8c0b68bd-2257-4994-b444-def0b3f64334-5_944_938_260_244} The diagram shows the curve \(C\) with equation \(y = 4 x ^ { 2 } - 10 x + 7\) and two straight lines, \(l _ { 1 }\) and \(l _ { 2 }\). The line \(l _ { 1 }\) is the normal to \(C\) at the point \(\left( \frac { 1 } { 2 } , 3 \right)\). The line \(l _ { 2 }\) is the normal to \(C\) at the minimum point of \(C\).
  1. Determine the equation of \(l _ { 1 }\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers to be determined. The shaded region shown in the diagram is bounded by \(C , l _ { 1 }\) and \(l _ { 2 }\).
  2. Determine the inequalities that define the shaded region, including its boundaries.
OCR PURE Q9
4 marks Moderate -0.8
9 A cyclist travels along a straight horizontal road between house \(A\) and house \(B\). The cyclist starts from rest at \(A\) and moves with constant acceleration for 20 seconds, reaching a velocity of \(15 \mathrm {~ms} ^ { - 1 }\). The cyclist then moves at this constant velocity before decelerating at \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), coming to rest at \(B\).
  1. Find the time, in seconds, for which the cyclist is decelerating.
  2. Sketch a velocity-time graph for the motion of the cyclist between \(A\) and \(B\). [Your sketch need not be drawn to scale; numerical values need not be shown.] The total distance between \(A\) and \(B\) is 1950 m .
  3. Find the time, in seconds, for which the cyclist is moving at constant velocity.
OCR PURE Q10
8 marks Standard +0.3
10 A particle \(P\) is moving in a straight line. At time \(t\) seconds, where \(t \geqslant 0 , P\) has velocity \(v \mathrm {~ms} ^ { - 1 }\) and acceleration \(a \mathrm {~ms} ^ { - 2 }\) where \(a = 4 t - 9\). It is given that \(v = 2\) when \(t = 1\).
  1. Find an expression for \(v\) in terms of \(t\). The particle \(P\) is instantaneously at rest when \(t = t _ { 1 }\) and \(t = t _ { 2 }\), where \(t _ { 1 } < t _ { 2 }\).
  2. Find the values of \(t _ { 1 }\) and \(t _ { 2 }\).
  3. Determine the total distance travelled by \(P\) between times \(t = 0\) and \(t = t _ { 2 }\).
OCR PURE Q11
13 marks Challenging +1.2
11 Two balls \(P\) and \(Q\) have masses 0.6 kg and 0.4 kg respectively. The balls are attached to the ends of a string. The string passes over a pulley which is fixed at the edge of a rough horizontal surface. Ball \(P\) is held at rest on the surface 2 m from the pulley. Ball \(Q\) hangs vertically below the pulley. Ball \(Q\) is attached to a third ball \(R\) of mass \(m \mathrm {~kg}\) by another string and \(R\) hangs vertically below \(Q\) (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{8c0b68bd-2257-4994-b444-def0b3f64334-7_419_945_493_246} The system is released from rest with the strings taut. Ball \(P\) moves towards the pulley with acceleration \(3.5 \mathrm {~ms} ^ { - 2 }\) and a constant frictional force of magnitude 4.5 N opposes the motion of \(P\). The balls are modelled as particles, the pulley is modelled as being small and smooth, and the strings are modelled as being light and inextensible.
  1. By considering the motion of \(P\), find the tension in the string connecting \(P\) and \(Q\).
  2. Hence determine the value of \(m\). Give your answer correct to \(\mathbf { 3 }\) significant figures. When the balls have been in motion for 0.4 seconds the string connecting \(Q\) and \(R\) breaks.
  3. Show that, according to the model, \(P\) does not reach the pulley. It is given that in fact ball \(P\) does reach the pulley.
  4. Identify one factor in the modelling that could account for this difference.
OCR MEI AS Paper 1 2018 June Q1
2 marks Easy -1.2
1 Write \(\frac { 8 } { 3 - \sqrt { 5 } }\) in the form \(a + b \sqrt { 5 }\), where \(a\) and \(b\) are integers to be found.
OCR MEI AS Paper 1 2018 June Q2
4 marks Easy -1.8
2 Find the binomial expansion of \(( 3 - 2 x ) ^ { 3 }\).
OCR MEI AS Paper 1 2018 June Q3
3 marks Moderate -0.8
3 A particle is in equilibrium under the action of three forces in newtons given by $$\mathbf { F } _ { 1 } = \binom { 8 } { 0 } , \quad \mathbf { F } _ { 2 } = \binom { 2 a } { - 3 a } \quad \text { and } \quad \mathbf { F } _ { 3 } = \binom { 0 } { b } .$$ Find the values of the constants \(a\) and \(b\).
OCR MEI AS Paper 1 2018 June Q4
4 marks Moderate -0.3
4 Fig. 4 shows a block of mass \(4 m \mathrm {~kg}\) and a particle of mass \(m \mathrm {~kg}\) connected by a light inextensible string passing over a smooth pulley. The block is on a horizontal table, and the particle hangs freely. The part of the string between the pulley and the block is horizontal. The block slides towards the pulley and the particle descends. In this motion, the friction force between the table and the block is \(\frac { 1 } { 2 } m g \mathrm {~N}\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1513048a-d53b-4b85-82f4-c86e0d81f8f8-3_204_741_1151_662} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure} Find expressions for
  • the acceleration of the system,
  • the tension in the string.
OCR MEI AS Paper 1 2018 June Q5
7 marks Standard +0.8
5
  1. Sketch the graphs of \(y = 4 \cos x\) and \(y = 2 \sin x\) for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\) on the same axes.
  2. Find the exact coordinates of the point of intersection of these graphs, giving your answer in the form (arctan \(a , k \sqrt { b }\) ), where \(a\) and \(b\) are integers and \(k\) is rational.
  3. A student argues that without the condition \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\) all the points of intersection of the graphs would occur at intervals of \(360 ^ { \circ }\) because both \(\sin x\) and \(\cos x\) are periodic functions with this period. Comment on the validity of the student's argument.
OCR MEI AS Paper 1 2018 June Q6
5 marks Moderate -0.8
6 In this question you must show detailed reasoning.
You are given that \(\mathrm { f } ( x ) = 4 x ^ { 3 } - 3 x + 1\).
  1. Use the factor theorem to show that \(( x + 1 )\) is a factor of \(\mathrm { f } ( x )\).
  2. Solve the equation \(\mathrm { f } ( x ) = 0\).
OCR MEI AS Paper 1 2018 June Q7
6 marks Standard +0.3
7 A toy boat of mass 1.5 kg is pushed across a pond, starting from rest, for 2.5 seconds. During this time, the boat has an acceleration of \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Subsequently, when the only horizontal force acting on the boat is a constant resistance to motion, the boat travels 10 m before coming to rest. Calculate the magnitude of the resistance to motion.
OCR MEI AS Paper 1 2018 June Q9
9 marks Standard +0.3
9 The curve \(y = ( x - 1 ) ^ { 2 }\) maps onto the curve \(\mathrm { C } _ { 1 }\) following a stretch scale factor \(\frac { 1 } { 2 }\) in the \(x\)-direction.
  1. Show that the equation of \(\mathrm { C } _ { 1 }\) can be written as \(y = 4 x ^ { 2 } - 4 x + 1\). The curve \(\mathrm { C } _ { 2 }\) is a translation of \(y = 4.25 x - x ^ { 2 }\) by \(\binom { 0 } { - 3 }\).
  2. Show that the normal to the curve \(\mathrm { C } _ { 1 }\) at the point \(( 0,1 )\) is a tangent to the curve \(\mathrm { C } _ { 2 }\).
OCR MEI AS Paper 1 2018 June Q10
9 marks Standard +0.3
10 Rory runs a distance of 45 m in 12.5 s . He starts from rest and accelerates to a speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). He runs the remaining distance at \(4 \mathrm {~ms} ^ { - 1 }\). Rory proposes a model in which the acceleration is constant until time \(T\) seconds.
  1. Sketch the velocity-time graph for Rory's run using this model.
  2. Calculate \(T\).
  3. Find an expression for Rory's displacement at time \(t \mathrm {~s}\) for \(0 \leqslant t \leqslant T\).
  4. Use this model to find the time taken for Rory to run the first 4 m . Rory proposes a refined model in which the velocity during the acceleration phase is a quadratic function of \(t\). The graph of Rory's quadratic goes through \(( 0,0 )\) and has its maximum point at \(( S , 4 )\). In this model the acceleration phase lasts until time \(S\) seconds, after which the velocity is constant.
  5. Sketch a velocity-time graph that represents Rory's run using this refined model.
  6. State with a reason whether \(S\) is greater than \(T\) or less than \(T\). (You are not required to calculate the value of \(S\).)
OCR MEI AS Paper 1 2018 June Q11
13 marks Moderate -0.8
11 The intensity of the sun's radiation, \(y\) watts per square metre, and the average distance from the sun, \(x\) astronomical units, are shown in Fig. 11 for the planets Mercury and Jupiter. \begin{table}[h]
\(x\)\(y\)
Mercury0.307514400
Jupiter4.95055.8
\captionsetup{labelformat=empty} \caption{Fig. 11}
\end{table} The intensity \(y\) is proportional to a power of the distance \(x\).
  1. Write down an equation for \(y\) in terms of \(x\) and two constants.
  2. Show that the equation can be written in the form \(\ln y = a + b \ln x\).
  3. In the Printed Answer Booklet, complete the table for \(\ln x\) and \(\ln y\) correct to 4 significant figures.
  4. Use the values from part (iii) to find \(a\) and \(b\).
  5. Hence rewrite your equation from part (i) for \(y\) in terms of \(x\), using suitable numerical values for the constants.
  6. Sketch a graph of the equation found in part (v).
  7. Earth is 1 astronomical unit from the sun. Find the intensity of the sun's radiation for Earth.
OCR MEI AS Paper 1 2019 June Q3
4 marks Moderate -0.8
3 Given that \(k\) is an integer, express \(\frac { 3 \sqrt { 2 } - k } { \sqrt { 8 } + 1 }\) in the form \(a + b \sqrt { 2 }\) where \(a\) and \(b\) are rational expressions in terms of \(k\).
OCR MEI AS Paper 1 2019 June Q4
5 marks Moderate -0.3
4 A triangle ABC has sides \(\mathrm { AB } = 5 \mathrm {~cm} , \mathrm { AC } = 9 \mathrm {~cm}\) and \(\mathrm { BC } = 10 \mathrm {~cm}\).
  1. Find the cosine of angle BAC, giving your answer as a fraction in its lowest terms.
  2. Find the exact area of the triangle.
OCR MEI AS Paper 1 2019 June Q5
3 marks Moderate -0.8
5 In this question, the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal and vertically upwards respectively. A particle has mass 2.5 kg .
  1. Write the weight of the particle as a vector. The particle moves under the action of its weight and two external forces ( \(3 \mathbf { i } - 2 \mathbf { j }\) ) N and \(( - \mathbf { i } + 18 \mathbf { j } ) N\).
  2. Find the acceleration of the particle, giving your answer in vector form.
OCR MEI AS Paper 1 2019 June Q6
7 marks Moderate -0.3
6 Fig. 6 shows a train consisting of an engine of mass 80 tonnes pulling two trucks each of mass 25 tonnes. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0b1c272a-f0f4-4931-be89-5d045804a7af-4_189_1262_938_246} \captionsetup{labelformat=empty} \caption{Fig. 6}
\end{figure} The engine exerts a driving force of \(D \mathrm {~N}\) and experiences a resistance to motion of 2000 N . Each truck experiences a resistance of 600 N . The train travels in a straight line on a level track with an acceleration of \(0.1 \mathrm {~ms} ^ { - 2 }\).
  1. Complete the force diagram in the Printed Answer Booklet to show all the forces acting on the engine and each of the trucks.
  2. Calculate the value of \(D\).
  3. The tension in the coupling between the engine and truck A is larger than that in the coupling between the trucks. Determine how much larger.
OCR MEI AS Paper 1 2019 June Q7
11 marks Moderate -0.3
7 In this question you must show detailed reasoning.
  1. Nigel is asked to determine whether \(( x + 7 )\) is a factor of \(x ^ { 3 } - 37 x + 84\). He substitutes \(x = 7\) and calculates \(7 ^ { 3 } - 37 \times 7 + 84\). This comes to 168 , so Nigel concludes that ( \(x + 7\) ) is not a factor. Nigel's conclusion is wrong.
OCR MEI AS Paper 1 2019 June Q10
7 marks Moderate -0.3
10 In this question you must show detailed reasoning.
  1. Sketch the gradient function for the curve \(y = 24 x - 3 x ^ { 2 } - x ^ { 3 }\).
  2. Determine the set of values of \(x\) for which \(24 x - 3 x ^ { 2 } - x ^ { 3 }\) is decreasing.
OCR MEI AS Paper 1 2019 June Q11
11 marks Moderate -0.8
11 David puts a block of ice into a cool-box. He wishes to model the mass \(m \mathrm {~kg}\) of the remaining block of ice at time \(t\) hours later. He finds that when \(t = 5 , m = 2.1\), and when \(t = 50 , m = 0.21\).
  1. David at first guesses that the mass may be inversely proportional to time. Show that this model fits his measurements.
  2. Explain why this model
    1. is not suitable for small values of \(t\),
    2. cannot be used to find the time for the block to melt completely. David instead proposes a linear model \(m = a t + b\), where \(a\) and \(b\) are constants.
  3. Find the values of the constants for which the model fits the mass of the block when \(t = 5\) and \(t = 50\).
  4. Interpret these values of \(a\) and \(b\).
  5. Find the time according to this model for the block of ice to melt completely.
OCR MEI AS Paper 1 2022 June Q1
3 marks Easy -1.2
1 Rationalise the denominator of the fraction \(\frac { 2 + \sqrt { n } } { 3 + \sqrt { n } }\), where \(n\) is a positive integer.