Questions — OCR MEI (4456 questions)

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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]
OCR MEI C2 2013 January Q1
3 marks Easy -1.8
Find \(\int 30x^2 dx\). [3]
OCR MEI C2 2013 January Q2
3 marks Easy -1.3
For each of the following sequences, state with a reason whether it is convergent, periodic or neither. Each sequence continues in the pattern established by the given terms.
  1. \(3, \frac{3}{2}, \frac{3}{4}, \frac{3}{8}, \ldots\) [1]
  2. \(3, 7, 11, 15, \ldots\) [1]
  3. \(3, 5, -3, -5, 3, 5, -3, -5, \ldots\) [1]
OCR MEI C2 2013 January Q3
4 marks Moderate -0.8
  1. The point P\((4, -2)\) lies on the curve \(y = f(x)\). Find the coordinates of the image of P when the curve is transformed to \(y = f(5x)\). [2]
  2. Describe fully a single transformation which maps the curve \(y = \sin x^2\) onto the curve \(y = \sin(x - 90)^2\). [2]
OCR MEI C2 2013 January Q4
4 marks Moderate -0.8
\includegraphics{figure_4} Fig. 4 shows sector OAB with sector angle 1.2 radians and arc length 4.2 cm. It also shows chord AB.
  1. Find the radius of this sector. [2]
  2. Calculate the perpendicular distance of the chord AB from O. [2]
OCR MEI C2 2013 January Q5
3 marks Easy -1.2
A and B are points on the curve \(y = 4\sqrt{x}\). Point A has coordinates \((9, 12)\) and point B has \(x\)-coordinate \(9.5\). Find the gradient of the chord AB. The gradient of AB is an approximation to the gradient of the curve at A. State the \(x\)-coordinate of a point C on the curve such that the gradient of AC is a closer approximation. [3]
OCR MEI C2 2013 January Q6
4 marks Moderate -0.8
Differentiate \(2x^3 + 9x^2 - 24x\). Hence find the set of values of \(x\) for which the function \(f(x) = 2x^3 + 9x^2 - 24x\) is increasing. [4]
OCR MEI C2 2013 January Q7
5 marks Moderate -0.3
Fig. 7 shows a sketch of a village green ABC which is bounded by three straight roads. AB = 92 m, BC = 75 m and AC = 105 m. \includegraphics{figure_7} Calculate the area of the village green. [5]
OCR MEI C2 2013 January Q8
5 marks Moderate -0.8
  1. Sketch the graph of \(y = 3^x\). [2]
  2. Solve the equation \(3^{3x-1} = 500000\). [3]
OCR MEI C2 2013 January Q9
5 marks Moderate -0.3
  1. Show that the equation \(\frac{\tan \theta}{\cos \theta} = 1\) may be rewritten as \(\sin \theta = 1 - \sin^2 \theta\). [2]
  2. Hence solve the equation \(\frac{\tan \theta}{\cos \theta} = 1\) for \(0° \leq \theta \leq 360°\). [3]
OCR MEI C2 2013 January Q10
11 marks Standard +0.3
Fig. 10 shows a sketch of the curve \(y = x^2 - 4x + 3\). The point A on the curve has \(x\)-coordinate 4. At point B the curve crosses the \(x\)-axis. \includegraphics{figure_10}
  1. Use calculus to find the equation of the normal to the curve at A and show that this normal intersects the \(x\)-axis at C\((16, 0)\). [6]
  2. Find the area of the region ABC bounded by the curve, the normal at A and the \(x\)-axis. [5]
OCR MEI C2 2013 January Q11
12 marks Moderate -0.3
  1. An arithmetic progression has first term \(A\) and common difference \(D\). The sum of its first two terms is 25 and the sum of its first four terms is 250.
    1. Find the values of \(A\) and \(D\). [4]
    2. Find the sum of the 21st to 50th terms inclusive of this sequence. [3]
  2. A geometric progression has first term \(a\) and common ratio \(r\), with \(r \neq \pm 1\). The sum of its first two terms is 25 and the sum of its first four terms is 250. Use the formula for the sum of a geometric progression to show that \(\frac{r^4 - 1}{r^2 - 1} = 10\) and hence or otherwise find algebraically the possible values of \(r\) and the corresponding values of \(a\). [5]
OCR MEI C2 2013 January Q12
13 marks Moderate -0.3
The table shows population data for a country.
Year19691979198919992009
Population in millions (\(p\))58.8180.35105.27134.79169.71
The data may be represented by an exponential model of growth. Using \(t\) as the number of years after 1960, a suitable model is \(p = a \times 10^{kt}\).
  1. Derive an equation for \(\log_{10} p\) in terms of \(a\), \(k\) and \(t\). [2]
  2. Complete the table and draw the graph of \(\log_{10} p\) against \(t\), drawing a line of best fit by eye. [3]
  3. Use your line of best fit to express \(\log_{10} p\) in terms of \(t\) and hence find \(p\) in terms of \(t\). [4]
  4. According to the model, what was the population in 1960? [1]
  5. According to the model, when will the population reach 200 million? [3]
OCR MEI C2 2006 June Q1
2 marks Easy -1.8
Write down the values of \(\log_a a\) and \(\log_a (a^3)\). [2]
OCR MEI C2 2006 June Q2
3 marks Moderate -0.8
The first term of a geometric series is 8. The sum to infinity of the series is 10. Find the common ratio. [3]
OCR MEI C2 2006 June Q3
3 marks Moderate -0.8
\(\theta\) is an acute angle and \(\sin \theta = \frac{1}{4}\). Find the exact value of \(\tan \theta\). [3]