Questions — OCR MEI C2 (480 questions)

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OCR MEI C2 2016 June Q3
5 marks Standard +0.3
An arithmetic progression (AP) and a geometric progression (GP) have the same first and fourth terms as each other. The first term of both is 1.5 and the fourth term of both is 12. Calculate the difference between the tenth terms of the AP and the GP. [5]
OCR MEI C2 2016 June Q4
5 marks Moderate -0.3
\includegraphics{figure_4} Fig. 4 shows triangle ABC, where AB = 7.2 cm, AC = 5.6 cm and angle BAC = 68°. Calculate the size of angle ACB. [5]
OCR MEI C2 2016 June Q5
4 marks Easy -1.3
  1. Fig. 5 shows the graph of a sine function. \includegraphics{figure_5} State the equation of this curve. [2]
  2. Sketch the graph of \(y = \sin x - 3\) for \(0° \leq x \leq 450°\). [2]
OCR MEI C2 2016 June Q6
4 marks Moderate -0.3
A sector of a circle has radius \(r\) cm and sector angle \(\theta\) radians. It is divided into two regions, A and B. Region A is an isosceles triangle with the equal sides being of length \(a\) cm, as shown in Fig. 6. \includegraphics{figure_6}
  1. Express the area of B in terms of \(a\), \(r\) and \(\theta\). [2]
  2. Given that \(r = 12\) and \(\theta = 0.8\), find the value of \(a\) for which the areas of A and B are equal. Give your answer correct to 3 significant figures. [2]
OCR MEI C2 2016 June Q7
5 marks Moderate -0.8
  1. Show that, when \(x\) is an acute angle, \(\tan x \sqrt{1 - \sin^2 x} = \sin x\). [2]
  2. Solve \(4 \sin^2 y = \sin y\) for \(0° \leq y \leq 360°\). [3]
OCR MEI C2 2016 June Q8
5 marks Moderate -0.8
  1. Simplify \(\log_a 1 - \log_a (a^m)^3\). [2]
  2. Use logarithms to solve the equation \(3^{2x+1} = 1000\). Give your answer correct to 3 significant figures. [3]
OCR MEI C2 2016 June Q9
11 marks Standard +0.3
Fig. 9 shows the cross-section of a straight, horizontal tunnel. The \(x\)-axis from 0 to 6 represents the floor of the tunnel. \includegraphics{figure_9} With axes as shown, and units in metres, the roof of the tunnel passes through the points shown in the table.
\(x\)0123456
\(y\)04.04.95.04.94.00
The length of the tunnel is 50 m.
  1. Use the trapezium rule with 6 strips to estimate the area of cross-section of the tunnel. Hence estimate the volume of earth removed in digging the tunnel. [4]
  2. An engineer models the height of the roof of the tunnel using the curve \(y = \frac{x}{81}(108x - 54x^2 + 12x^3 - x^4)\). This curve is symmetrical about \(x = 3\).
    1. Show that, according to this model, a vehicle of rectangular cross-section which is 3.6 m wide and 4.4 m high would not be able to pass through the tunnel. [2]
    2. Use integration to calculate the area of the cross-section given by this model. Hence obtain another estimate of the volume of earth removed in digging the tunnel. [5]
OCR MEI C2 2016 June Q10
13 marks Moderate -0.8
  1. Calculate the gradient of the chord of the curve \(y = x^2 - 2x\) joining the points at which the values of \(x\) are 5 and 5.1. [2]
  2. Given that \(\mathrm{f}(x) = x^2 - 2x\), find and simplify \(\frac{\mathrm{f}(5 + h) - \mathrm{f}(5)}{h}\). [4]
  3. Use your result in part (ii) to find the gradient of the curve \(y = x^2 - 2x\) at the point where \(x = 5\), showing your reasoning. [2]
  4. Find the equation of the tangent to the curve \(y = x^2 - 2x\) at the point where \(x = 5\). Find the area of the triangle formed by this tangent and the coordinate axes. [5]
OCR MEI C2 2016 June Q11
12 marks Moderate -0.3
There are many different flu viruses. The numbers of flu viruses detected in the first few weeks of the 2012–2013 flu epidemic in the UK were as follows.
Week12345678910
Number of flu viruses710243240386396234480
These data may be modelled by an equation of the form \(y = a \times 10^{bt}\), where \(y\) is the number of flu viruses detected in week \(t\) of the epidemic, and \(a\) and \(b\) are constants to be determined.
  1. Explain why this model leads to a straight-line graph of \(\log_{10} y\) against \(t\). State the gradient and intercept of this graph in terms of \(a\) and \(b\). [3]
  2. Complete the values of \(\log_{10} y\) in the table, draw the graph of \(\log_{10} y\) against \(t\), and draw by eye a line of best fit for the data. Hence determine the values of \(a\) and \(b\) and the equation for \(y\) in terms of \(t\) for this model. [8]
During the decline of the epidemic, an appropriate model was $$y = 921 \times 10^{-0.137w},$$ where \(y\) is the number of flu viruses detected in week \(w\) of the decline.
  1. Use this to find the number of viruses detected in week 4 of the decline. [1]
OCR MEI C2 Q1
4 marks Easy -1.2
Differentiate \(x + \sqrt{x^3}\). [4]
OCR MEI C2 Q2
5 marks Moderate -0.8
The gradient of a curve is given by \(\frac{dy}{dx} = \frac{6}{x^3}\). The curve passes through \((1, 4)\). Find the equation of the curve. [5]
OCR MEI C2 Q3
3 marks Moderate -0.8
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 Q4
4 marks Moderate -0.3
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 Q5
3 marks Moderate -0.8
Find the set of values of \(x\) for which \(x^2 - 7x\) is a decreasing function. [3]
OCR MEI C2 Q6
2 marks Easy -1.8
Differentiate \(10x^4 + 12\). [2]
OCR MEI C2 Q7
12 marks Moderate -0.3
\includegraphics{figure_7} Fig. 10 shows a solid cuboid with square base of side \(x\) cm and height \(h\) cm. Its volume is \(120\) cm\(^3\).
  1. Find \(h\) in terms of \(x\). Hence show that the surface area, \(A\) cm\(^2\), of the cuboid is given by $$A = 2x^2 + \frac{480}{x}.$$ [3]
  2. Find \(\frac{dA}{dx}\) and \(\frac{d^2A}{dx^2}\). [4]
  3. Hence find the value of \(x\) which gives the minimum surface area. Find also the value of the surface area in this case. [5]
OCR MEI C2 Q8
2 marks Easy -1.2
Differentiate \(6x^{\frac{5}{2}} + 4\). [2]
OCR MEI C2 Q9
5 marks Moderate -0.8
A is the point \((2, 1)\) on the curve \(y = \frac{4}{x^2}\). B is the point on the same curve with \(x\)-coordinate \(2.1\).
  1. Calculate the gradient of the chord AB of the curve. Give your answer correct to 2 decimal places. [2]
  2. Give the \(x\)-coordinate of a point C on the curve for which the gradient of chord AC is a better approximation to the gradient of the curve at A. [1]
  3. Use calculus to find the gradient of the curve at A. [2]
OCR MEI C2 Q10
3 marks Moderate -0.5
The gradient of a curve is given by \(\frac{dy}{dx} = x^2 - 6x\). Find the set of values of \(x\) for which \(y\) is an increasing function of \(x\). [3]
OCR MEI C2 Q11
4 marks Moderate -0.8
A curve has gradient given by \(\frac{dy}{dx} = 6x^2 + 8x\). The curve passes through the point \((1, 5)\). Find the equation of the curve. [4]
OCR MEI C2 Q1
13 marks Moderate -0.3
The gradient of a curve is given by \(\frac{dy}{dx} = 4x + 3\). The curve passes through the point \((2, 9)\).
  1. Find the equation of the tangent to the curve at the point \((2, 9)\). [3]
  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. [7]
  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. [3]
OCR MEI C2 Q2
5 marks Moderate -0.3
Find the equation of the normal to the curve \(y = 8x^4 + 4\) at the point where \(x = \frac{1}{2}\). [5]
OCR MEI C2 Q3
13 marks Moderate -0.3
  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 Q4
12 marks Moderate -0.8
  1. Calculate the gradient of the chord joining the points on the curve \(y = x^2 - 7\) for which \(x = 3\) and \(x = 3.1\). [2]
  2. Given that \(f(x) = x^2 - 7\), find and simplify \(\frac{f(3 + h) - f(3)}{h}\). [3]
  3. Use your result in part (ii) to find the gradient of \(y = x^2 - 7\) at the point where \(x = 3\), showing your reasoning. [2]
  4. Find the equation of the tangent to the curve \(y = x^2 - 7\) at the point where \(x = 3\). [2]
  5. This tangent crosses the \(x\)-axis at the point P. The curve crosses the positive \(x\)-axis at the point Q. Find the distance PQ, giving your answer correct to 3 decimal places. [3]
OCR MEI C2 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]