Questions — OCR MEI C2 (480 questions)

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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]
OCR MEI C2 2006 June Q4
5 marks Moderate -0.8
Find \(\int_1^2 \left( x^4 - \frac{3}{x^2} + 1 \right) dx\), showing your working. [5]
OCR MEI C2 2006 June Q5
4 marks Moderate -0.8
The gradient of a curve is given by \(\frac{dy}{dx} = 3 - x^2\). The curve passes through the point \((6, 1)\). Find the equation of the curve. [4]
OCR MEI C2 2006 June Q6
5 marks Moderate -0.8
A sequence is given by the following. $$u_1 = 3$$ $$u_{n+1} = u_n + 5$$
  1. Write down the first 4 terms of this sequence. [1]
  2. Find the sum of the 51st to the 100th terms, inclusive, of the sequence. [4]
OCR MEI C2 2006 June Q7
5 marks Easy -1.3
  1. Sketch the graph of \(y = \cos x\) for \(0° \leq x \leq 360°\). On the same axes, sketch the graph of \(y = \cos 2x\) for \(0° \leq x \leq 360°\). Label each graph clearly. [3]
  2. Solve the equation \(\cos 2x = 0.5\) for \(0° \leq x \leq 360°\). [2]
OCR MEI C2 2006 June Q8
5 marks Easy -1.2
Given that \(y = 6x^3 + \sqrt{x} + 3\), find \(\frac{dy}{dx}\) and \(\frac{d^2y}{dx^2}\). [5]
OCR MEI C2 2006 June Q9
4 marks Moderate -0.8
Use logarithms to solve the equation \(5^{3x} = 100\). Give your answer correct to 3 decimal places. [4]
OCR MEI C2 2006 June Q10
11 marks Moderate -0.3
  1. \includegraphics{figure_10_1} At a certain time, ship S is 5.2 km from lighthouse L on a bearing of 048°. At the same time, ship T is 6.3 km from L on a bearing of 105°, as shown in Fig. 10.1. For these positions, calculate
    1. the distance between ships S and T, [3]
    2. the bearing of S from T. [3]
  2. \includegraphics{figure_10_2} Ship S then travels at 24 km h\(^{-1}\) anticlockwise along the arc of a circle, keeping 5.2 km from the lighthouse L, as shown in Fig. 10.2. Find, in radians, the angle \(\theta\) that the line LS has turned through in 26 minutes. Hence find, in degrees, the bearing of ship S from the lighthouse at this time. [5]
OCR MEI C2 2006 June Q11
13 marks Moderate -0.3
A cubic curve has equation \(y = x^3 - 3x^2 + 1\).
  1. Use calculus to find the coordinates of the turning points on this curve. Determine the nature of these turning points. [5]
  2. Show that the tangent to the curve at the point where \(x = -1\) has gradient 9. Find the coordinates of the other point, P, on the curve at which the tangent has gradient 9 and find the equation of the normal to the curve at P. Show that the area of the triangle bounded by the normal at P and the \(x\)- and \(y\)-axes is 8 square units. [8]
OCR MEI C2 2006 June Q12
12 marks Moderate -0.8
Answer the whole of this question on the insert provided. A colony of bats is increasing. The population, \(P\), is modelled by \(P = a \times 10^{bt}\), where \(t\) is the time in years after 2000.
  1. Show that, according to this model, the graph of \(\log_{10} P\) against \(t\) should be a straight line of gradient \(b\). State, in terms of \(a\), the intercept on the vertical axis. [3]
  2. The table gives the data for the population from 2001 to 2005.
    Year20012002200320042005
    \(t\)12345
    \(P\)79008800100001130012800
    Complete the table of values on the insert, and plot \(\log_{10} P\) against \(t\). Draw a line of best fit for the data. [3]
  3. Use your graph to find the equation for \(P\) in terms of \(t\). [4]
  4. Predict the population in 2008 according to this model. [2]
OCR MEI C2 2008 June Q1
2 marks Easy -1.8
Express \(\frac{7\pi}{6}\) radians in degrees. [2]