Questions — OCR MEI (4456 questions)

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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]
OCR MEI C2 2013 June Q2
5 marks Easy -1.3
The \(n\)th term of a sequence, \(u_n\), is given by $$u_n = 12 - \frac{1}{2}n.$$
  1. Write down the values of \(u_1\), \(u_2\) and \(u_3\). State what type of sequence this is. [2]
  2. Find \(\sum_{n=1}^{30} u_n\). [3]
OCR MEI C2 2013 June Q3
5 marks Moderate -0.8
The gradient of a curve is given by \(\frac{dy}{dx} = \frac{18}{x^3} + 2\). The curve passes through the point \((3, 6)\). Find the equation of the curve. [5]
OCR MEI C2 2013 June Q4
5 marks Moderate -0.8
  1. Starting with an equilateral triangle, prove that \(\cos 30° = \frac{\sqrt{3}}{2}\). [2]
  2. Solve the equation \(2 \sin \theta = -1\) for \(0 \leq \theta \leq 2\pi\), giving your answers in terms of \(\pi\). [3]
OCR MEI C2 2013 June Q5
5 marks Moderate -0.8
\includegraphics{figure_5} Fig. 5 shows the graph of \(y = 2^x\).
  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 \(x = 2\). [3]
  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 \(x = 2\). [2]
OCR MEI C2 2013 June Q6
3 marks Moderate -0.8
\(S\) is the sum to infinity of a geometric progression with first term \(a\) and common ratio \(r\).
  1. Another geometric progression has first term \(2a\) and common ratio \(r\). Express the sum to infinity of this progression in terms of \(S\). [1]
  2. A third geometric progression has first term \(a\) and common ratio \(r^2\). Express, in its simplest form, the sum to infinity of this progression in terms of \(S\) and \(r\). [2]
OCR MEI C2 2013 June Q7
4 marks Easy -1.2
Fig. 7 shows a curve and the coordinates of some points on it. \includegraphics{figure_7} Use the trapezium rule with 6 strips to estimate the area of the region bounded by the curve and the positive \(x\)- and \(y\)-axes. [4]
OCR MEI C2 2013 June Q8
4 marks Easy -1.3
Fig. 8 shows the graph of \(y = g(x)\). \includegraphics{figure_8} Draw the graph of
  1. \(y = g(2x)\), [2]
  2. \(y = 3g(x)\). [2]
OCR MEI C2 2013 June Q9
11 marks Standard +0.3
Fig. 9 shows a sketch of the curve \(y = x^3 - 3x^2 - 22x + 24\) and the line \(y = 6x + 24\). \includegraphics{figure_9}
  1. Differentiate \(y = x^3 - 3x^2 - 22x + 24\) and hence find the \(x\)-coordinates of the turning points of the curve. Give your answers to 2 decimal places. [4]
  2. You are given that the line and the curve intersect when \(x = 0\) and when \(x = -4\). Find algebraically the \(x\)-coordinate of the other point of intersection. [3]
  3. Use calculus to find the area of the region bounded by the curve and the line \(y = 6x + 24\) for \(-4 \leq x \leq 0\), shown shaded on Fig. 9. [4]
OCR MEI C2 2013 June Q10
14 marks Standard +0.3
Fig. 10.1 shows Jean's back garden. This is a quadrilateral ABCD with dimensions as shown. \includegraphics{figure_10.1}
    1. Calculate AC and angle ACB. Hence calculate AD. [6]
    2. Calculate the area of the garden. [3]
  2. The shape of the fence panels used in the garden is shown in Fig. 10.2. EH is the arc of a sector of a circle with centre at the midpoint, M, of side FG, and sector angle 1.1 radians, as shown. FG = 1.8 m. \includegraphics{figure_10.2} Calculate the area of one of these fence panels. [5]
OCR MEI C2 2013 June Q11
11 marks Moderate -0.3
A hot drink when first made has a temperature which is \(65°C\) higher than room temperature. The temperature difference, \(d °C\), between the drink and its surroundings decreases by \(1.7\%\) each minute.
  1. Show that 3 minutes after the drink is made, \(d = 61.7\) to 3 significant figures. [2]
  2. Write down an expression for the value of \(d\) at time \(n\) minutes after the drink is made, where \(n\) is an integer. [1]
  3. Show that when \(d < 3\), \(n\) must satisfy the inequality $$n > \frac{\log_{10} 3 - \log_{10} 65}{\log_{10} 0.983}.$$ Hence find the least integer value of \(n\) for which \(d < 3\). [4]
  4. The temperature difference at any time \(t\) minutes after the drink is made can also be expressed as \(d = 65 \times 10^{-kt}\), for some constant \(k\). Use the value of \(d\) for 1 minute after the drink is made to calculate the value of \(k\). Hence find the temperature difference 25.3 minutes after the drink is made. [4]
OCR MEI C2 2014 June Q1
3 marks Easy -1.8
Find \(\int 7x^3 \, dx\). [3]
OCR MEI C2 2014 June Q2
5 marks Moderate -0.8
  1. Find \(\sum_{r=1}^{5} \frac{21}{r+2}\). [2]
  2. A sequence is defined by $$u_1 = a, \text{ where } a \text{ is an unknown constant,}$$ $$u_{n+1} = u_n + 5.$$ Find, in terms of \(a\), the tenth term and the sum of the first ten terms of this sequence. [3]
OCR MEI C2 2014 June Q3
2 marks Easy -1.2
The points P\((2, 3.6)\) and Q\((2.2, 2.4)\) lie on the curve \(y = f(x)\). Use P and Q to estimate the gradient of the curve at the point where \(x = 2\). [2]
OCR MEI C2 2014 June Q4
4 marks Moderate -0.8
The point R\((6, -3)\) is on the curve \(y = f(x)\).
  1. Find the coordinates of the image of R when the curve is transformed to \(y = \frac{1}{2}f(x)\). [2]
  2. Find the coordinates of the image of R when the curve is transformed to \(y = f(3x)\). [2]
OCR MEI C2 2014 June Q5
3 marks Moderate -0.8
\includegraphics{figure_5} Fig. 5 shows triangle ABC, where angle ABC = \(72°\), AB = \(5.9\) cm and BC = \(8.5\) cm. Calculate the length of AC. [3]