Questions — OCR MEI C2 (480 questions)

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OCR MEI C2 2015 June Q10
13 marks Standard +0.3
10 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 4 x + 3\). The curve passes through the point ( 2,9 ).
  1. Find the equation of the tangent to the curve at the point \(( 2,9 )\).
  2. Find the equation of the curve and the coordinates of its points of intersection with the \(x\)-axis. Find also the coordinates of the minimum point of this curve.
  3. Find the equation of the curve after it has been stretched parallel to the \(x\)-axis with scale factor \(\frac { 1 } { 2 }\). Write down the coordinates of the minimum point of the transformed curve.
OCR MEI C2 2015 June Q11
12 marks Standard +0.3
11 Jill has 3 daughters and no sons. They are generation 1 of Jill's descendants.
Each of her daughters has 3 daughters and no sons. Jill's 9 granddaughters are generation 2 of her descendants. Each of her granddaughters has 3 daughters and no sons; they are descendant generation 3. Jill decides to investigate what would happen if this pattern continues, with each descendant having 3 daughters and no sons.
  1. How many of Jill's descendants would there be in generation 8 ?
  2. How many of Jill's descendants would there be altogether in the first 15 generations?
  3. After \(n\) generations, Jill would have over a million descendants altogether. Show that \(n\) satisfies the inequality $$n > \frac { \log _ { 10 } 2000003 } { \log _ { 10 } 3 } - 1 .$$ Hence find the least possible value of \(n\).
  4. How many fewer descendants would Jill have altogether in 15 generations if instead of having 3 daughters, she and each subsequent descendant has 2 daughters? \section*{END OF QUESTION PAPER}
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}
  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}
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 C2 2007 January Q13
12 marks Moderate -0.3
13 Answer part (ii) of this question on the insert provided. The table gives a firm's monthly profits for the first few months after the start of its business, rounded to the nearest \(\pounds 100\).
Number of months after start-up \(( x )\)123456
Profit for this month \(( \pounds y )\)5008001200190030004800
The firm's profits, \(\pounds y\), for the \(x\) th month after start-up are modelled by $$y = k \times 10 ^ { a x }$$ where \(a\) and \(k\) are constants.
  1. Show that, according to this model, a graph of \(\log _ { 10 } y\) against \(x\) gives a straight line of gradient \(a\) and intercept \(\log _ { 10 } k\).
  2. On the insert, complete the table and plot \(\log _ { 10 } y\) against \(x\), drawing by eye a line of best fit.
  3. Use your graph to find an equation for \(y\) in terms of \(x\) for this model.
  4. For which month after start-up does this model predict profits of about \(\pounds 75000\) ?
  5. State one way in which this model is unrealistic.
OCR MEI C2 2009 January Q5
4 marks Moderate -0.8
5 Answer this question on the insert provided. Fig. 5 shows the graph of \(y = \mathrm { f } ( x )\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{838d6b15-69a9-4e67-bc36-5bf60254a767-3_979_1077_422_536} \captionsetup{labelformat=empty} \caption{Fig. 5}
\end{figure} On the insert, draw the graph of
  1. \(y = \mathrm { f } ( x - 2 )\),
  2. \(y = 3 \mathrm { f } ( x )\).
OCR MEI C2 2009 January Q12
12 marks Moderate -0.3
12 Answer part (ii) of this question on the insert provided. The proposal for a major building project was accepted, but actual construction was delayed. Each year a new estimate of the cost was made. The table shows the estimated cost, \(\pounds y\) million, of the project \(t\) years after the project was first accepted.
Years after proposal accepted \(( t )\)12345
Cost \(( \pounds y\) million \()\)250300360440530
The relationship between \(y\) and \(t\) is modelled by \(y = a b ^ { t }\), where \(a\) and \(b\) are constants.
  1. Show that \(y = a b ^ { t }\) may be written as $$\log _ { 10 } y = \log _ { 10 } a + t \log _ { 10 } b$$
  2. On the insert, complete the table and plot \(\log _ { 10 } y\) against \(t\), drawing by eye a line of best fit.
  3. Use your graph and the results of part (i) to find the values of \(\log _ { 10 } a\) and \(\log _ { 10 } b\) and hence \(a\) and \(b\).
  4. According to this model, what was the estimated cost of the project when it was first accepted?
  5. Find the value of \(t\) given by this model when the estimated cost is \(\pounds 1000\) million. Give your answer rounded to 1 decimal place.
OCR MEI C2 2010 January Q1
3 marks Easy -1.2
Find \(\int \left(x - \frac{3}{x^2}\right) dx\). [3]
OCR MEI C2 2010 January Q2
3 marks Easy -1.3
A sequence begins $$1 \quad 3 \quad 5 \quad 3 \quad 1 \quad 3 \quad 5 \quad 3 \quad 1 \quad 3 \quad \ldots$$ and continues in this pattern.
  1. Find the 55th term of this sequence, showing your method. [1]
  2. Find the sum of the first 55 terms of the sequence. [2]
OCR MEI C2 2010 January Q3
3 marks Moderate -0.8
You are given that \(\sin \theta = \frac{\sqrt{2}}{3}\) and that \(\theta\) is an acute angle. Find the exact value of \(\tan \theta\). [3]
OCR MEI C2 2010 January Q4
3 marks Moderate -0.8
A sector of a circle has area \(8.45 \text{ cm}^2\) and sector angle \(0.4\) radians. Calculate the radius of the sector. [3]
OCR MEI C2 2010 January Q5
4 marks Easy -1.2
\includegraphics{figure_5} Fig. 5 shows a sketch of the graph of \(y = f(x)\). On separate diagrams, sketch the graphs of the following, showing clearly the coordinates of the points corresponding to P, Q and R.
  1. \(y = f(2x)\) [2]
  2. \(y = \frac{1}{2}f(x)\) [2]
OCR MEI C2 2010 January Q6
5 marks Easy -1.3
  1. Find the 51st term of the sequence given by $$u_1 = 5,$$ $$u_{n+1} = u_n + 4.$$ [3]
  2. Find the sum to infinity of the geometric progression which begins $$5 \quad 2 \quad 0.8 \quad \ldots .$$ [2]
OCR MEI C2 2010 January Q7
5 marks Moderate -0.8
\includegraphics{figure_7} Fig. 7 shows triangle ABC, with AB = 8.4 cm. D is a point on AC such that angle ADB = 79°, BD = 5.6 cm and CD = 7.8 cm. Calculate
  1. angle BAD, [2]
  2. the length BC. [3]
OCR MEI C2 2010 January Q8
5 marks Moderate -0.3
Find the equation of the tangent to the curve \(y = 6\sqrt{x}\) at the point where \(x = 16\). [5]
OCR MEI C2 2010 January Q9
5 marks Moderate -0.8
  1. Sketch the graph of \(y = 3^x\). [2]
  2. Use logarithms to solve \(3^{2x+1} = 10\), giving your answer correct to 2 decimal places. [3]
OCR MEI C2 2010 January Q10
11 marks Moderate -0.8
  1. Differentiate \(x^3 - 3x^2 - 9x\). Hence find the \(x\)-coordinates of the stationary points on the curve \(y = x^3 - 3x^2 - 9x\), showing which is the maximum and which the minimum. [6]
  2. Find, in exact form, the coordinates of the points at which the curve crosses the \(x\)-axis. [3]
  3. Sketch the curve. [2]
OCR MEI C2 2010 January Q11
12 marks Moderate -0.3
Fig. 11 shows the cross-section of a school hall, with measurements of the height in metres taken at 1.5 m intervals from O. \includegraphics{figure_11}
  1. Use the trapezium rule with 8 strips to calculate an estimate of the area of the cross-section. [4]
  2. Use 8 rectangles to calculate a lower bound for the area of the cross-section. [2]
The curve of the roof may be modelled by \(y = -0.013x^3 + 0.16x^2 - 0.082x + 2.4\), where \(x\) metres is the horizontal distance from O across the hall, and \(y\) metres is the height.
  1. Use integration to find the area of the cross-section according to this model. [4]
  2. Comment on the accuracy of this model for the height of the hall when \(x = 7.5\). [2]
OCR MEI C2 2010 January Q12
13 marks Moderate -0.3
Answer part (ii) of this question on the insert provided. Since 1945 the populations of many countries have been growing. The table shows the estimated population of 15- to 59-year-olds in Africa during the period 1955 to 2005.
Year195519651975198519952005
Population (millions)131161209277372492
Source: United Nations Such estimates are used to model future population growth and world needs of resources. One model is \(P = a10^{bt}\), where the population is \(P\) millions, \(t\) is the number of years after 1945 and \(a\) and \(b\) are constants.
  1. Show that, using this model, the graph of \(\log_{10} P\) against \(t\) is a straight line of gradient \(b\). State the intercept of this line on the vertical axis. [3]
  2. On the insert, complete the table, giving values correct to 2 decimal places, and plot the graph of \(\log_{10} P\) against \(t\). Draw, by eye, a line of best fit on your graph. [3]
  3. Use your graph to find the equation for \(P\) in terms of \(t\). [4]
  4. Use your results to estimate the population of 15- to 59-year-olds in Africa in 2050. Comment, with a reason, on the reliability of this estimate. [3]