Questions — OCR MEI C2 (480 questions)

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OCR MEI C2 2008 June Q2
3 marks Easy -1.3
The first term of a geometric series is 5.4 and the common ratio is 0.1.
  1. Find the fourth term of the series. [1]
  2. Find the sum to infinity of the series. [2]
OCR MEI C2 2008 June Q3
2 marks Easy -1.2
State the transformation which maps the graph of \(y = x^2 + 5\) onto the graph of \(y = 3x^2 + 15\). [2]
OCR MEI C2 2008 June Q4
3 marks Moderate -0.3
Use calculus to find the set of values of \(x\) for which \(\text{f}(x) = 12x - x^3\) is an increasing function. [3]
OCR MEI C2 2008 June Q5
4 marks Moderate -0.8
In Fig. 5, A and B are the points on the curve \(y = 2^x\) with \(x\)-coordinates 3 and 3.1 respectively. \includegraphics{figure_5}
  1. Find the gradient of the chord AB. Give your answer correct to 2 decimal places. [2]
  2. Stating the points you use, find the gradient of another chord which will give a closer approximation to the gradient of the tangent to \(y = 2^x\) at A. [2]
OCR MEI C2 2008 June Q6
4 marks Moderate -0.8
A curve has gradient given by \(\frac{\text{d}y}{\text{d}x} = 6\sqrt{x}\). Find the equation of the curve, given that it passes through the point \((9, 105)\). [4]
OCR MEI C2 2008 June Q7
5 marks Moderate -0.8
\includegraphics{figure_7} A sector of a circle of radius 6 cm has angle 1.6 radians, as shown in Fig. 7. Find the area of the sector. Hence find the area of the shaded segment. [5]
OCR MEI C2 2008 June Q8
5 marks Moderate -0.3
The 11th term of an arithmetic progression is 1. The sum of the first 10 terms is 120. Find the 4th term. [5]
OCR MEI C2 2008 June Q9
3 marks Moderate -0.8
Use logarithms to solve the equation \(5^x = 235\), giving your answer correct to 2 decimal places. [3]
OCR MEI C2 2008 June Q10
5 marks Standard +0.3
Showing your method, solve the equation \(2\sin^2\theta = \cos\theta + 2\) for values of \(\theta\) between \(0°\) and \(360°\). [5]
OCR MEI C2 2008 June Q11
12 marks Moderate -0.3
\includegraphics{figure_11} Fig. 11 shows a sketch of the cubic curve \(y = \text{f}(x)\). The values of \(x\) where it crosses the \(x\)-axis are \(-5\), \(-2\) and \(2\), and it crosses the \(y\)-axis at \((0, -20)\).
  1. Express f(\(x\)) in factorised form. [2]
  2. Show that the equation of the curve may be written as \(y = x^3 + 5x^2 - 4x - 20\). [2]
  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. [6]
  4. State, correct to 1 decimal place, the coordinates of the maximum point on the curve \(y = \text{f}(2x)\). [2]
OCR MEI C2 2008 June Q12
12 marks Moderate -0.8
\includegraphics{figure_12} A water trough is a prism 2.5 m long. Fig. 12 shows the cross-section of the trough, with the depths in metres at 0.1 m intervals across the trough. The trough is full of water.
  1. Use the trapezium rule with 5 strips to calculate an estimate of the area of cross-section of the trough. Hence estimate the volume of water in the trough. [5]
  2. A computer program models the curve of the base of the trough, with axes as shown and units in metres, using the equation \(y = 8x^3 - 3x^2 - 0.5x - 0.15\), for \(0 \leq x \leq 0.5\). Calculate \(\int_0^{0.5} (8x^3 - 3x^2 - 0.5x - 0.15) \, \text{d}x\) and state what this represents. Hence find the volume of water in the trough as given by this model. [7]
OCR MEI C2 2008 June Q13
12 marks Moderate -0.3
The percentage of the adult population visiting the cinema in Great Britain has tended to increase since the 1980s. The table shows the results of surveys in various years.
Year1986/871991/921996/971999/002000/012001/02
Percentage of the adult population visiting the cinema314454565557
Source: Department of National Statistics, www.statistics.gov.uk This growth may be modelled by an equation of the form $$P = at^b,$$ where \(P\) is the percentage of the adult population visiting the cinema, \(t\) is the number of years after the year 1985/86 and \(a\) and \(b\) are constants to be determined.
  1. Show that, according to this model, the graph of \(\log_{10} P\) against \(\log_{10} t\) should be a straight line of gradient \(b\). State, in terms of \(a\), the intercept on the vertical axis. [3]
  2. Complete the table of values on the insert, and plot \(\log_{10} P\) against \(\log_{10} t\). Draw by eye a line of best fit for the data. [4]
  3. Use your graph to find the equation for \(P\) in terms of \(t\). [4]
  4. Predict the percentage of the adult population visiting the cinema in the year 2007/2008 (i.e. when \(t = 22\)), according to this model. [1]
OCR MEI C2 2010 June Q1
2 marks Easy -1.2
You are given that $$u_1 = 1,$$ $$u_{n+1} = \frac{u_n}{1 + u_n}.$$ Find the values of \(u_2\), \(u_3\) and \(u_4\). Give your answers as fractions. [2]
OCR MEI C2 2010 June Q2
4 marks Moderate -0.8
  1. Evaluate \(\sum_{r=2}^{5} \frac{1}{r-1}\). [2]
  2. Express the series \(2 \times 3 + 3 \times 4 + 4 \times 5 + 5 \times 6 + 6 \times 7\) in the form \(\sum_{r=2}^{a} f(r)\) where \(f(r)\) and \(a\) are to be determined. [2]
OCR MEI C2 2010 June Q3
5 marks Moderate -0.8
  1. Differentiate \(x^3 - 6x^2 - 15x + 50\). [2]
  2. Hence find the \(x\)-coordinates of the stationary points on the curve \(y = x^3 - 6x^2 - 15x + 50\). [3]
OCR MEI C2 2010 June Q4
4 marks Moderate -0.8
In this question, \(f(x) = x^2 - 5x\). Fig. 4 shows a sketch of the graph of \(y = f(x)\). \includegraphics{figure_4} On separate diagrams, sketch the curves \(y = f(2x)\) and \(y = 3f(x)\), labelling the coordinates of their intersections with the axes and their turning points. [4]
OCR MEI C2 2010 June Q5
4 marks Moderate -0.8
Find \(\int_{2}^{5} \left(1 - \frac{6}{x^3}\right) dx\). [4]
OCR MEI C2 2010 June Q6
5 marks Moderate -0.5
The gradient of a curve is \(6x^2 + 12x^{\frac{1}{2}}\). The curve passes through the point \((4, 10)\). Find the equation of the curve. [5]
OCR MEI C2 2010 June Q7
2 marks Easy -1.2
Express \(\log_a x^3 + \log_a \sqrt{x}\) in the form \(k \log_a x\). [2]
OCR MEI C2 2010 June Q8
5 marks Moderate -0.3
Showing your method clearly, solve the equation \(4 \sin^2 \theta = 3 + \cos^2 \theta\), for values of \(\theta\) between \(0°\) and \(360°\). [5]
OCR MEI C2 2010 June Q9
5 marks Standard +0.3
The points \((2, 6)\) and \((3, 18)\) lie on the curve \(y = ax^n\). Use logarithms to find the values of \(a\) and \(n\), giving your answers correct to 2 decimal places. [5]
OCR MEI C2 2010 June Q10
13 marks Moderate -0.8
  1. Find the equation of the tangent to the curve \(y = x^4\) at the point where \(x = 2\). Give your answer in the form \(y = mx + c\). [4]
  2. Calculate the gradient of the chord joining the points on the curve \(y = x^4\) where \(x = 2\) and \(x = 2.1\). [2]
    1. Expand \((2 + h)^4\). [3]
    2. Simplify \(\frac{(2 + h)^4 - 2^4}{h}\). [2]
    3. Show how your result in part (iii) \((B)\) can be used to find the gradient of \(y = x^4\) at the point where \(x = 2\). [2]
OCR MEI C2 2010 June Q11
13 marks Standard +0.3
  1. \includegraphics{figure_11_1} A boat travels from P to Q and then to R. As shown in Fig. 11.1, Q is 10.6 km from P on a bearing of \(045°\). R is 9.2 km from P on a bearing of \(113°\), so that angle QPR is \(68°\). Calculate the distance and bearing of R from Q. [5]
  2. Fig. 11.2 shows the cross-section, EBC, of the rudder of a boat. \includegraphics{figure_11_2} BC is an arc of a circle with centre A and radius 80 cm. Angle CAB = \(\frac{2\pi}{3}\) radians. EC is an arc of a circle with centre D and radius \(r\) cm. Angle CDE is a right angle.
    1. Calculate the area of sector ABC. [2]
    2. Show that \(r = 40\sqrt{3}\) and calculate the area of triangle CDA. [3]
    3. Hence calculate the area of cross-section of the rudder. [3]
OCR MEI C2 2010 June Q12
10 marks Standard +0.3
\includegraphics{figure_12} A branching plant has stems, nodes, leaves and buds. • There are 7 leaves at each node. • From each node, 2 new stems grow. • At the end of each final stem, there is a bud. Fig. 12 shows one such plant with 3 stages of nodes. It has 15 stems, 7 nodes, 49 leaves and 8 buds.
  1. One of these plants has 10 stages of nodes.
    1. How many buds does it have? [2]
    2. How many stems does it have? [2]
    1. Show that the number of leaves on one of these plants with \(n\) stages of nodes is $$7(2^n - 1).$$ [2]
    2. One of these plants has \(n\) stages of nodes and more than 200000 leaves. Show that \(n\) satisfies the inequality \(n > \frac{\log_{10} 200007 - \log_{10} 7}{\log_{10} 2}\). Hence find the least possible value of \(n\). [4]
OCR MEI C2 2013 June Q1
5 marks Easy -1.8
Find \(\frac{dy}{dx}\) when
  1. \(y = 2x^{-5}\). [2]
  2. \(y = ^4\sqrt{x}\). [3]