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Edexcel C2 Q2
8 marks Moderate -0.8
  1. find the first 4 terms, simplifying each term.
  2. Find, in its simplest form, the term independent of \(x\) in this expansion.
    [0pt] [P2 June 2004 Question 3] \item The curve \(C\) has equation \(y = \cos \left( x + \frac { \pi } { 4 } \right) , 0 \leq x \leq 2 \pi\).
  3. Sketch \(C\).
  4. Write down the exact coordinates of the points at which \(C\) meets the coordinate axes.
  5. Solve, for \(x\) in the interval \(0 \leq x \leq 2 \pi , \cos \left( x + \frac { \pi } { 4 } \right) = 0.5\), giving your answers in terms of \(\pi\). \item Given that \(\log _ { 2 } x = a\), find, in terms of \(a\), the simplest form of
  6. \(\log _ { 2 } ( 16 x )\),
  7. \(\log _ { 2 } \left( \frac { x ^ { 4 } } { 2 } \right)\).
  8. Hence, or otherwise, solve \(\log _ { 2 } ( 16 x ) - \log _ { 2 } \left( \frac { x ^ { 4 } } { 2 } \right) = \frac { 1 } { 2 }\), giving your answer in its simplest surd form. \item (a) Given that \(3 \sin x = 8 \cos x\), find the value of \(\tan x\).
  9. Find, to 1 decimal place, all the solutions of \(3 \sin x - 8 \cos x = 0\) in the interval \(0 \leq x < 360 ^ { \circ }\).
  10. Find, to 1 decimal place, all the solutions of \(3 \sin ^ { 2 } y - 8 \cos y = 0\) in the interval \(0 \leq y < 360 ^ { \circ }\). \item \end{enumerate} $$f ( x ) = \frac { \left( x ^ { 2 } - 3 \right) ^ { 2 } } { x ^ { 3 } } , x \neq 0$$
  11. Show that \(\mathrm { f } ( x ) \equiv x - 6 x ^ { - 1 } + 9 x ^ { - 3 }\).
  12. Hence, or otherwise, differentiate \(\mathrm { f } ( x )\) with respect to \(x\).
  13. Verify that the graph of \(y = \mathrm { f } ( x )\) has stationary points at \(x = \pm \sqrt { } 3\).
  14. Determine whether the stationary value at \(x = \sqrt { } 3\) is a maximum or a minimum.
OCR C2 2007 January Q9
10 marks Standard +0.3
  1. Show that the amount of coal used on the fifth trip is 1.624 tonnes, correct to 4 significant figures.
  2. There are 39 tonnes of coal available. An engineer wishes to calculate \(N\), the total number of trips possible. Show that \(N\) satisfies the inequality $$1.02 ^ { N } \leqslant 1.52$$
  3. Hence, by using logarithms, find the greatest number of trips possible.
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 2013 June Q5
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 \(x = 2\). \(6 S\) is the sum to infinity of a geometric progression with first term \(a\) and common ratio \(r\).
  3. Another geometric progression has first term \(2 a\) and common ratio \(r\). Express the sum to infinity of this progression in terms of \(S\).
  4. A third geometric progression has first term \(a\) and common ratio \(r ^ { 2 }\). Express, in its simplest form, the sum to infinity of this progression in terms of \(S\) and \(r\).
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}
  5. Express \(y\) as a function of \(x\).
  6. 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 $$A = 10 x - 2 x ^ { 2 } - \frac { 1 } { 2 } \pi x ^ { 2 }$$
  7. 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.
Edexcel C2 Q6
9 marks Moderate -0.3
  1. Find the remainder when \(\mathrm { f } ( x )\) is divided by ( \(2 x - 1\) ).
    1. Find the remainder when \(\mathrm { f } ( x )\) is divided by \(( x + 2 )\).
    2. Hence, or otherwise, solve the equation $$2 x ^ { 3 } + 3 x ^ { 2 } - 6 x - 8 = 0 ,$$ giving your answers to 2 decimal places where appropriate.
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. \end{enumerate} 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 C3 2010 January Q5
9 marks Moderate -0.3
  1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence show that the only stationary point on the curve is the point for which \(x = 0\).
  2. Find an expression for \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) and hence find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at the stationary point.
OCR C3 2010 January Q7
7 marks Standard +0.3
  1. Find the value of the integer \(N\) for which the sequence converges to the value 1.9037 (correct to 4 decimal places).
  2. Find the value of the integer \(N\) for which, correct to 4 decimal places, \(x _ { 3 } = 2.6022\) and \(x _ { 4 } = 2.6282\). \section*{[Question 9 is printed overleaf.]}
OCR MEI C3 Q3
4 marks Easy -1.2
  1. \(\quad y = 2 \mathrm { f } ( x )\),
  2. \(y = \mathrm { f } ( 2 x )\).
OCR MEI C3 Q9
7 marks Standard +0.3
(A) \(( x - y ) \left( x ^ { 2 } + x y + y ^ { 2 } \right) = x ^ { 3 } - y ^ { 3 }\),
(B) \(\left( x + \frac { 1 } { 2 } y \right) ^ { 2 } + \frac { 3 } { 4 } y ^ { 2 } = x ^ { 2 } + x y + y ^ { 2 }\).
(ii) Hence prove that, for all real numbers \(x\) and \(y\), if \(x > y\) then \(x ^ { 3 } > y ^ { 3 }\).
(i) Verify the following statement: $$\text { ' } 2 ^ { p } - 1 \text { is a prime number for all prime numbers } p \text { less than } 11 \text { '. }$$ (ii) Calculate \(23 \times 89\), and hence disprove this statement:
' \(2 ^ { p } - 1\) is a prime number for all prime numbers \(p\) '.
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]
AQA C4 2016 June Q7
9 marks Standard +0.3
  1. Find the exact value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at the point on \(C\) where \(t = \frac { 2 } { 3 }\).
  2. Show that \(x = \frac { 4 - \mathrm { e } ^ { 2 - 6 t } } { 4 }\) can be rearranged into the form \(\mathrm { e } ^ { 3 t } = \frac { \mathrm { e } } { 2 \sqrt { ( 1 - x ) } }\).
  3. Hence find the Cartesian equation of \(C\), giving your answer in the form $$y = \frac { \mathrm { e } } { \mathrm { f } ( x ) [ 1 - \ln ( \mathrm { f } ( x ) ) ] }$$
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} \captionsetup{labelformat=empty} \caption{Fig. 8}
    \end{figure}
  5. Find the values of the parameter at A and B . Hence show that the ratio of the lengths OA and AC is \(( \pi - 1 ) : ( \pi + 1 )\).
  6. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\theta\). Find the gradient of the track at A .
  7. Show that, when the gradient of the track is \(1 , \theta\) satisfies the equation $$\cos \theta - 4 \sin \theta = 2 .$$
  8. 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 Cambridge Assessment Group; Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }\section*{ADVANCED GCE
    MATHEMATICS (MEI)} 4754B
    Applications of Advanced Mathematics (C4) Paper B: Comprehension \section*{Candidates answer on the Question Paper} OCR Supplied Materials:
    • Insert (inserted)
    • MEI Examination Formulae and Tables (MF2)
    \section*{Other Materials Required:}
    • Rough paper
    • Scientific or graphical calculator
    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
  9. 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.
  10. 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.
  11. \(\_\_\_\_\)
  12. \(\_\_\_\_\) 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. $$\begin{aligned} & n = 10 \text { for } 0 \leqslant P < 10 ^ { 6 } \\ & n = 11 \text { for } 10 ^ { 6 } \leqslant P < 1.5 \times 10 ^ { 6 } \\ & n = 12 \text { for } 1.5 \times 10 ^ { 6 } \leqslant P < 2 \times 10 ^ { 6 } \\ & n = 13 \text { for } 2 \times 10 ^ { 6 } \leqslant P < 2.5 \times 10 ^ { 6 } \\ & n = 14 \text { for } 2.5 \times 10 ^ { 6 } \leqslant P < 3 \times 10 ^ { 6 } \\ & \ldots \text { and so on } \ldots \end{aligned}$$
  13. Sketch the graph of \(n\) against \(P\).
  14. Describe, in words, the relationship between the daily number of trains and the annual number of passengers.
  15. \includegraphics[max width=\textwidth, alt={}, center]{5c149cb5-7392-4219-b285-486f4694aa6f-7_716_1249_1011_440}
  16. \(\_\_\_\_\)