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 MEI C1 2009 January Q11
14 marks Moderate -0.3
  1. Show that the equation of the circle with AB as diameter may be written as $$( x - 5 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 40$$
  2. Find the coordinates of the points of intersection of this circle with the \(y\)-axis. Give your answer in the form \(a \pm \sqrt { b }\).
  3. Find the equation of the tangent to the circle at B . Hence find the coordinates of the points of intersection of this tangent with the axes.
OCR MEI C1 Q10
12 marks Moderate -0.3
  1. Write down the equations of the circles A and B .
  2. Find the \(x\) coordinates of the points where the two curves intersect.
  3. Find the \(y\) coordinates of these points, giving your answers in surd form.
OCR C1 Q9
10 marks Standard +0.3
  1. Find an equation for the straight line \(l\) which passes through \(P\) and \(Q\). Give your answer in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers. The straight line \(m\) has gradient 8 and passes through the origin, \(O\).
  2. Write down an equation for \(m\). The lines \(l\) and \(m\) intersect at the point \(R\).
  3. Show that \(O P = O R\).
AQA C2 2011 January Q7
16 marks Moderate -0.3
  1. Given that \(y = x + 3 + \frac { 8 } { x ^ { 4 } }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Find an equation of the tangent at the point on the curve \(C\) where \(x = 1\).
  3. The curve \(C\) has a minimum point \(M\). Find the coordinates of \(M\).
    1. Find \(\int \left( x + 3 + \frac { 8 } { x ^ { 4 } } \right) \mathrm { d } x\).
    2. Hence find the area of the region bounded by the curve \(C\), the \(x\)-axis and the lines \(x = 1\) and \(x = 2\).
  5. The curve \(C\) is translated by \(\left[ \begin{array} { l } 0 \\ k \end{array} \right]\) to give the curve \(y = \mathrm { f } ( x )\). Given that the \(x\)-axis is a tangent to the curve \(y = \mathrm { f } ( x )\), state the value of the constant \(k\).
    (1 mark)
AQA C2 2011 June Q6
10 marks Moderate -0.3
  1. The area of the shaded region is given by \(\int _ { 0 } ^ { 2 } \sin x \mathrm {~d} x\), where \(x\) is in radians. Use the trapezium rule with five ordinates (four strips) to find an approximate value for the area of the shaded region, giving your answer to three significant figures.
  2. Describe the geometrical transformation that maps the graph of \(y = \sin x\) onto the graph of \(y = 2 \sin x\).
  3. Use a trigonometrical identity to solve the equation $$2 \sin x = \cos x$$ in the interval \(0 \leqslant x \leqslant 2 \pi\), giving your solutions in radians to three significant figures.
OCR MEI C2 2006 January Q11
11 marks Standard +0.3
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Find, in exact form, the range of values of \(x\) for which \(x ^ { 3 } - 6 x + 2\) is a decreasing function.
  3. Find the equation of the tangent to the curve at the point \(( - 1,7 )\). Find also the coordinates of the point where this tangent crosses the curve again.
OCR MEI C2 2011 January Q11
11 marks Moderate -0.3
  1. Use calculus to find \(\int _ { 1 } ^ { 3 } \left( x ^ { 3 } - 3 x ^ { 2 } - x + 3 \right) \mathrm { d } x\) and state what this represents.
  2. Find the \(x\)-coordinates of the turning points of the curve \(y = x ^ { 3 } - 3 x ^ { 2 } - x + 3\), giving your answers in surd form. Hence state the set of values of \(x\) for which \(y = x ^ { 3 } - 3 x ^ { 2 } - x + 3\) is a decreasing function.
OCR MEI C2 2009 June Q10
12 marks Moderate -0.5
  1. On the insert, complete the table and plot \(h\) against \(\log _ { 10 } t\), drawing by eye a line of best fit.
  2. Use your graph to find an equation for \(h\) in terms of \(\log _ { 10 } t\) for this model.
  3. Find the height of the tree at age 100 years, as predicted by this model.
  4. Find the age of the tree when it reaches a height of 29 m , according to this model.
  5. Comment on the suitability of the model when the tree is very young.
OCR MEI C2 Q11
12 marks Moderate -0.8
  1. The speed-time graph on the insert sheet provides the axes and the first two points plotted. Plot the remainder of these points and join them with a smooth curve. The area between this curve and the \(t\)-axis represents the distance travelled by the car in this time.
  2. Using the trapezium rule with 6 values of \(t\) estimate the area under the curve to give the distance travelled. Illustrate on your graph the area found.
  3. John's teacher suggests that the equation of the curve could be \(v = 6 t - \frac { 1 } { 2 } t ^ { 2 }\). Find, by calculus, the area between this curve and the \(t\) axis.
  4. Plot this curve on your graph. Comment on whether the estimates obtained in parts (ii) and (iii) are overestimates or underestimates. 12 Fig. 12 shows a window. The base and sides are parts of a rectangle with dimensions \(2 x\) metres horizontally by \(y\) metres vertically. The top is a semicircle of radius \(x\) metres. The perimeter of the window is 10 metres. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{73d1c02b-1b7b-426d-a171-c762597cfed4-4_428_433_1638_766} \captionsetup{labelformat=empty} \caption{Fig. 12}
    \end{figure}
  1. Express \(y\) as a function of \(x\).
  2. Find the total area, \(A \mathrm {~m} ^ { 2 }\), in terms of \(x\) and \(y\). Use your answer to part (i) to show that this simplifies to $$A = 10 x - 2 x ^ { 2 } - \frac { 1 } { 2 } \pi x ^ { 2 }$$
  3. Prove that for the maximum value of \(A\), \(y = x\) exactly.
    \section*{MEI STRUCTURED MATHEMATICS } \section*{CONCEPTS FOR ADVANCED MATHEMATICS, C2} \section*{Practice Paper C2-B
    Insert sheet for question 11}
Edexcel C2 Q7
11 marks Moderate -0.3
  1. Find the coordinates of the points where the curve and line intersect.
  2. Find the area of the shaded region bounded by the curve and line.
OCR C2 Q5
8 marks Moderate -0.3
  1. Find the number of sit-ups that Habib will do in the fifth week.
  2. Show that he will do a total of 1512 sit-ups during the first eight weeks. In the \(n\)th week of training, the number of sit-ups that Habib does is greater than 300 for the first time.
  3. Find the value of \(n\).
OCR C2 Q7
10 marks Standard +0.3
  1. Show that the common difference is 5 .
  2. Find the 12th term. Another arithmetic sequence has first term -12 and common difference 7 .
    Given that the sums of the first \(n\) terms of these two sequences are equal,
  3. find the value of \(n\).
OCR MEI C2 Q3
12 marks Standard +0.3
  1. Express \(\mathrm { f } ( x )\) in factorised form.
  2. Show that the equation of the curve may be written as \(y = x ^ { 3 } + 5 x ^ { 2 } - 4 x - 20\).
  3. Use calculus to show that, correct to 1 decimal place, the \(x\)-coordinate of the minimum point on the curve is 0.4 . Find also the coordinates of the maximum point on the curve, giving your answers correct to 1 decimal place.
  4. State, correct to 1 decimal place, the coordinates of the maximum point on the curve \(y = \mathrm { f } ( 2 x )\).
OCR MEI C2 Q12
4 marks Easy -1.2
  1. \(y = \mathrm { f } ( x - 2 )\),
  2. \(y = 3 \mathrm { f } ( x )\).
OCR MEI C2 Q6
5 marks Moderate -0.8
  1. On the copy of Fig. 5, draw by eye a tangent to the curve at the point where \(x = 2\). Hence find an estimate of the gradient of \(y = 2 ^ { x }\) when \(x = 2\).
  2. Calculate the \(y\)-values on the curve when \(x = 1.8\) and \(x = 2.2\). Hence calculate another approximation to the gradient of \(y = 2 ^ { x }\) when \(x = 2\).
OCR MEI C2 Q11
5 marks Moderate -0.8
  1. Solve the equation \(\cos x = 0.4\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
  2. Describe the transformation which maps the graph of \(y = \cos x\) onto the graph of \(y = \cos 2 x\).
OCR MEI C3 Q3
4 marks Easy -1.2
  1. \(\quad y = 2 \mathrm { f } ( x )\),
  2. \(y = \mathrm { f } ( 2 x )\).
AQA C4 2014 June Q7
9 marks Moderate -0.3
    1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
    2. Hence find the exact value of the gradient of the curve at \(A\).
  2. The normal at \(A\) crosses the \(y\)-axis at the point \(B\). Find the exact value of the \(y\)-coordinate of \(B\).
    [0pt] [2 marks]
OCR C4 2008 January Q7
8 marks Standard +0.3
  1. Given that $$A ( \sin \theta + \cos \theta ) + B ( \cos \theta - \sin \theta ) \equiv 4 \sin \theta$$ find the values of the constants \(A\) and \(B\).
  2. Hence find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \frac { 4 \sin \theta } { \sin \theta + \cos \theta } \mathrm { d } \theta$$ giving your answer in the form \(a \pi - \ln b\).
OCR C4 2008 June Q7
8 marks Moderate -0.3
  1. Show that, if \(y = \operatorname { cosec } x\), then \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) can be expressed as \(- \operatorname { cosec } x \cot x\).
  2. Solve the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = - \sin x \tan x \cot t$$ given that \(x = \frac { 1 } { 6 } \pi\) when \(t = \frac { 1 } { 2 } \pi\).
OCR C4 2009 June Q7
9 marks Moderate -0.3
  1. The vector \(\mathbf { u } = \frac { 3 } { 13 } \mathbf { i } + b \mathbf { j } + c \mathbf { k }\) is perpendicular to the vector \(4 \mathbf { i } + \mathbf { k }\) and to the vector \(4 \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k }\). Find the values of \(b\) and \(c\), and show that \(\mathbf { u }\) is a unit vector.
  2. Calculate, to the nearest degree, the angle between the vectors \(4 \mathbf { i } + \mathbf { k }\) and \(4 \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k }\).
OCR MEI C4 2010 June Q5
8 marks Standard +0.3
  1. Verify that \(\overrightarrow { \mathrm { AB } } = \left( \begin{array} { l } 300 \\ 100 \\ 100 \end{array} \right)\) and find the length of the pipeline.
  2. Write down a vector equation of the line AB , and calculate the angle it makes with the vertical. A thin flat layer of hard rock runs through the mountain. The equation of the plane containing this layer is \(x + 2 y + 3 z = 320\).
  3. Find the coordinates of the point where the pipeline meets the layer of rock.
  4. By calculating the angle between the line AB and the normal to the plane of the layer, find the angle at which the pipeline cuts through the layer. 8 Part of the track of a roller-coaster is modelled by a curve with the parametric equations $$x = 2 \theta - \sin \theta , \quad y = 4 \cos \theta \quad \text { for } 0 \leqslant \theta \leqslant 2 \pi$$ This is shown in Fig. 8. B is a minimum point, and BC is vertical. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{5c149cb5-7392-4219-b285-486f4694aa6f-4_602_1447_488_351} \caption{Fig. 8}
    \end{figure}
  1. Find the values of the parameter at A and B . Hence show that the ratio of the lengths OA and AC is \(( \pi - 1 ) : ( \pi + 1 )\).
  2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\theta\). Find the gradient of the track at A .
  3. Show that, when the gradient of the track is \(1 , \theta\) satisfies the equation $$\cos \theta - 4 \sin \theta = 2 .$$
  4. Express \(\cos \theta - 4 \sin \theta\) in the form \(R \cos ( \theta + \alpha )\). Hence solve the equation \(\cos \theta - 4 \sin \theta = 2\) for \(0 \leqslant \theta \leqslant 2 \pi\). {www.ocr.org.uk}) after the live examination series.
    If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity. For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
    OCR is part of the \section*{ADVANCED GCE
    MATHEMATICS (MEI)} 4754B
    Applications of Advanced Mathematics (C4) Paper B: Comprehension \section*{Candidates answer on the Question Paper} OCR Supplied Materials:
    \section*{Other Materials Required:}
    Wednesday 9 June 2010 Afternoon \includegraphics[max width=\textwidth, alt={}, center]{5c149cb5-7392-4219-b285-486f4694aa6f-5_264_456_881_1361} 1 The train journey from Swansea to London is 307 km and that by road is 300 km . Carry out the calculations performed on the First Great Western website to estimate how much lower the carbon dioxide emissions are when travelling by rail rather than road.
    2 The equation of the curve in Fig. 3 is $$y = \frac { 1 } { 10 ^ { 4 } } \left( x ^ { 3 } - 100 x ^ { 2 } - 10000 x + 2100100 \right)$$ Calculate the speed at which the car has its lowest carbon dioxide emissions and the value of its emissions at that speed.
    [0pt] [An answer obtained from the graph will be given no marks.]
3
  1. In line 109 the carbon dioxide emissions for a particular train journey from Exeter to London are estimated to be 3.7 tonnes. Obtain this figure.
  2. The text then goes on to state that the emissions per extra passenger on this journey are less than \(\frac { 1 } { 2 } \mathrm {~kg}\). Justify this figure.
  1. \(\_\_\_\_\)
  2. \(\_\_\_\_\) 4 The daily number of trains, \(n\), on a line in another country may be modelled by the function defined below, where \(P\) is the annual number of passengers. $$\begin{aligned} & n = 10 \text { for } 0 \leqslant P < 10 ^ { 6 } \\ & n = 11 \text { for } 10 ^ { 6 } \leqslant P < 1.5 \times 10 ^ { 6 } \\ & n = 12 \text { for } 1.5 \times 10 ^ { 6 } \leqslant P < 2 \times 10 ^ { 6 } \\ & n = 13 \text { for } 2 \times 10 ^ { 6 } \leqslant P < 2.5 \times 10 ^ { 6 } \\ & n = 14 \text { for } 2.5 \times 10 ^ { 6 } \leqslant P < 3 \times 10 ^ { 6 } \\ & \ldots \text { and so on } \ldots \end{aligned}$$
  1. Sketch the graph of \(n\) against \(P\).
  2. Describe, in words, the relationship between the daily number of trains and the annual number of passengers.
  1. \includegraphics[max width=\textwidth, alt={}, center]{5c149cb5-7392-4219-b285-486f4694aa6f-7_716_1249_1011_440}
  2. \(\_\_\_\_\)
Edexcel C4 Q4
11 marks Challenging +1.2
  1. Show that the volume of the solid formed is \(\frac { 1 } { 4 } \pi ( \pi + 2 )\).
  2. Find a cartesian equation for the curve.
OCR S1 2011 January Q5
10 marks Moderate -0.8
  1. The number of free gifts that Jan receives in a week is denoted by \(X\). Name a suitable probability distribution with which to model \(X\), giving the value(s) of any parameter(s). State any assumption(s) necessary for the distribution to be a valid model. Assume now that your model is valid.
  2. Find
    1. \(\mathrm { P } ( X \leqslant 2 )\),
    2. \(\mathrm { P } ( X = 2 )\).
    3. Find the probability that, in the next 7 weeks, there are exactly 3 weeks in which Jan receives exactly 2 free gifts. 6
    4. The diagram shows 7 cards, each with a digit printed on it. The digits form a 7 -digit number. {}
      3
      2
    7. \multirow[t]{21}{*}{3
    8. }
      9. \multirow[t]{4}{*}{3
    9. }
      10. 3
    10. {}
OCR S1 2012 January Q7
8 marks Moderate -0.8
  1. State a suitable distribution that can be used as a model for \(X\), giving the value(s) of any parameter(s). State also any necessary condition(s) for this distribution to be a good model. Use the distribution stated in part (i) to find
  2. \(\mathrm { P } ( X = 4 )\),
  3. \(\mathrm { P } ( X \geqslant 4 )\).