Questions — OCR MEI C2 (454 questions)

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OCR MEI C2 2012 June Q6
6 Fig. 6 shows the relationship between \(\log _ { 10 } x\) and \(\log _ { 10 } y\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8f7413d8-2814-4d5c-bec0-ce118fec80eb-3_497_787_287_644} \captionsetup{labelformat=empty} \caption{Fig. 6}
\end{figure} Find \(y\) in terms of \(x\).
OCR MEI C2 2012 June Q7
7 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 x ^ { \frac { 1 } { 2 } } - 5\). Given also that the curve passes through the point (4, 20), find the equation of the curve.
OCR MEI C2 2012 June Q8
8 Solve the equation \(\sin 2 \theta = 0.7\) for values of \(\theta\) between 0 and \(2 \pi\), giving your answers in radians correct to 3 significant figures.
OCR MEI C2 2012 June Q9
9 A farmer digs ditches for flood relief. He experiments with different cross-sections. Assume that the surface of the ground is horizontal.
  1. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{8f7413d8-2814-4d5c-bec0-ce118fec80eb-4_437_640_470_715} \captionsetup{labelformat=empty} \caption{Fig. 9.1}
    \end{figure} Fig. 9.1 shows the cross-section of one ditch, with measurements in metres. The width of the ditch is 1.2 m and Fig. 9.1 shows the depth every 0.2 m across the ditch. Use the trapezium rule with six intervals to estimate the area of cross-section. Hence estimate the volume of water that can be contained in a 50-metre length of this ditch.
  2. Another ditch is 0.9 m wide, with cross-section as shown in Fig. 9.2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{8f7413d8-2814-4d5c-bec0-ce118fec80eb-4_574_808_1402_632} \captionsetup{labelformat=empty} \caption{Fig. 9.2}
    \end{figure} With \(x\) - and \(y\)-axes as shown in Fig. 9.2, the curve of the ditch may be modelled closely by \(y = 3.8 x ^ { 4 } - 6.8 x ^ { 3 } + 7.7 x ^ { 2 } - 4.2 x\).
    (A) The actual ditch is 0.6 m deep when \(x = 0.2\). Calculate the difference between the depth given by the model and the true depth for this value of \(x\).
    (B) Find \(\int \left( 3.8 x ^ { 4 } - 6.8 x ^ { 3 } + 7.7 x ^ { 2 } - 4.2 x \right) \mathrm { d } x\). Hence estimate the volume of water that can be contained in a 50 -metre length of this ditch.
OCR MEI C2 2012 June Q10
10
  1. Use calculus to find, correct to 1 decimal place, the coordinates of the turning points of the curve \(y = x ^ { 3 } - 5 x\). [You need not determine the nature of the turning points.]
  2. Find the coordinates of the points where the curve \(y = x ^ { 3 } - 5 x\) meets the axes and sketch the curve.
  3. Find the equation of the tangent to the curve \(y = x ^ { 3 } - 5 x\) at the point \(( 1 , - 4 )\). Show that, where this tangent meets the curve again, the \(x\)-coordinate satisfies the equation $$x ^ { 3 } - 3 x + 2 = 0$$ Hence find the \(x\)-coordinate of the point where this tangent meets the curve again.
OCR MEI C2 2012 June Q11
11 A geometric progression has first term \(a\) and common ratio \(r\). The second term is 6 and the sum to infinity is 25 .
  1. Write down two equations in \(a\) and \(r\). Show that one possible value of \(a\) is 10 and find the other possible value of \(a\). Write down the corresponding values of \(r\).
  2. Show that the ratio of the \(n\)th terms of the two geometric progressions found in part (i) can be written as \(2 ^ { n - 2 } : 3 ^ { n - 2 }\).
OCR MEI C2 2013 June Q1
1 Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when
  1. \(y = 2 x ^ { - 5 }\),
  2. \(y = \sqrt [ 3 ] { x }\).
OCR MEI C2 2013 June Q3
3 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 18 } { x ^ { 3 } } + 2\). The curve passes through the point \(( 3,6 )\). Find the equation of the curve.
OCR MEI C2 2013 June Q4
4
  1. Starting with an equilateral triangle, prove that \(\cos 30 ^ { \circ } = \frac { \sqrt { 3 } } { 2 }\).
  2. Solve the equation \(2 \sin \theta = - 1\) for \(0 \leqslant \theta \leqslant 2 \pi\), giving your answers in terms of \(\pi\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{ee79022b-b9a6-4076-8db7-67b9788ac28a-3_1032_1113_264_466} \captionsetup{labelformat=empty} \caption{Fig. 5}
    \end{figure} Fig. 5 shows the graph of \(y = 2 ^ { x }\).
OCR MEI C2 2013 June Q7
7 Fig. 7 shows a curve and the coordinates of some points on it. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ee79022b-b9a6-4076-8db7-67b9788ac28a-4_631_1031_315_495} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure} 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.
OCR MEI C2 2013 June Q8
8 Fig. 8 shows the graph of \(y = \mathrm { g } ( x )\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ee79022b-b9a6-4076-8db7-67b9788ac28a-4_800_1402_1382_328} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure} Draw the graph of
  1. \(y = \mathrm { g } ( 2 x )\),
  2. \(y = 3 \mathrm {~g} ( x )\). Section B (36 marks)
OCR MEI C2 2013 June Q9
9 Fig. 9 shows a sketch of the curve \(y = x ^ { 3 } - 3 x ^ { 2 } - 22 x + 24\) and the line \(y = 6 x + 24\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ee79022b-b9a6-4076-8db7-67b9788ac28a-5_780_1171_422_424} \captionsetup{labelformat=empty} \caption{Fig. 9}
\end{figure}
  1. Differentiate \(y = x ^ { 3 } - 3 x ^ { 2 } - 22 x + 24\) and hence find the \(x\)-coordinates of the turning points of the curve. Give your answers to 2 decimal places.
  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. Use calculus to find the area of the region bounded by the curve and the line \(y = 6 x + 24\) for \(- 4 \leqslant x \leqslant 0\), shown shaded on Fig. 9.
OCR MEI C2 2013 June Q10
10 Fig. 10.1 shows Jean's back garden. This is a quadrilateral ABCD with dimensions as shown. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ee79022b-b9a6-4076-8db7-67b9788ac28a-6_707_1015_310_507} \captionsetup{labelformat=empty} \caption{Fig. 10.1}
\end{figure}
  1. (A) Calculate AC and angle ACB . Hence calculate AD .
    (B) Calculate the area of the garden.
  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. \(\mathrm { FG } = 1.8 \mathrm {~m}\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{ee79022b-b9a6-4076-8db7-67b9788ac28a-6_565_986_1528_520} \captionsetup{labelformat=empty} \caption{Fig. 10.2}
    \end{figure} Calculate the area of one of these fence panels.
OCR MEI C2 2013 June Q11
11 A hot drink when first made has a temperature which is \(65 ^ { \circ } \mathrm { C }\) higher than room temperature. The temperature difference, \(d ^ { \circ } \mathrm { C }\), 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. Write down an expression for the value of \(d\) at time \(n\) minutes after the drink is made, where \(n\) is an integer.
  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. The temperature difference at any time \(t\) minutes after the drink is made can also be expressed as \(d = 65 \times 10 ^ { - k t }\), 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.
OCR MEI C2 2014 June Q1
1 Find \(\int 7 x ^ { \frac { 5 } { 2 } } \mathrm {~d} x\).
  1. Find \(\sum _ { r = 1 } ^ { 5 } \frac { 21 } { r + 2 }\).
  2. A sequence is defined by $$\begin{aligned} u _ { 1 } & = a , \text { where } a \text { is an unknown constant, }
    u _ { n + 1 } & = u _ { n } + 5 . \end{aligned}$$ Find, in terms of \(a\), the tenth term and the sum of the first ten terms of this sequence.
OCR MEI C2 2014 June Q4
4 The point \(\mathrm { R } ( 6 , - 3 )\) is on the curve \(y = \mathrm { f } ( x )\).
  1. Find the coordinates of the image of R when the curve is transformed to \(y = \frac { 1 } { 2 } \mathrm { f } ( x )\).
  2. Find the coordinates of the image of R when the curve is transformed to \(y = \mathrm { f } ( 3 x )\).
OCR MEI C2 2014 June Q5
5 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{aded99ef-873e-42fb-ade5-f6f385e7e549-2_510_652_1471_708} \captionsetup{labelformat=empty} \caption{Fig. 5}
\end{figure} Fig. 5 shows triangle ABC , where angle \(\mathrm { ABC } = 72 ^ { \circ } , \mathrm { AB } = 5.9 \mathrm {~cm}\) and \(\mathrm { BC } = 8.5 \mathrm {~cm}\). Calculate the length of AC.
OCR MEI C2 2014 June Q6
6 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{aded99ef-873e-42fb-ade5-f6f385e7e549-3_712_662_255_689} \captionsetup{labelformat=empty} \caption{Fig. 6}
\end{figure} A circle with centre O has radius 12.4 cm . A segment of the circle is shown shaded in Fig. 6. The segment is bounded by the arc AB and the chord AB , where the angle AOB is 2.1 radians. Calculate the area of the segment.
OCR MEI C2 2014 June Q7
7 The second term of a geometric progression is 24 . The sum to infinity of this progression is 150 . Write down two equations in \(a\) and \(r\), where \(a\) is the first term and \(r\) is the common ratio. Solve your equations to find the possible values of \(a\) and \(r\).
OCR MEI C2 2014 June Q8
8 Simplify \(\frac { \sqrt { 1 - \cos ^ { 2 } \theta } } { \tan \theta }\), where \(\theta\) is an acute angle.
OCR MEI C2 2014 June Q9
9 Solve the equation \(\tan 2 \theta = 3\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).
OCR MEI C2 2014 June Q10
10 Use logarithms to solve the equation \(3 ^ { x + 1 } = 5 ^ { 2 x }\). Give your answer correct to 3 decimal places. Section B (36 marks) \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{aded99ef-873e-42fb-ade5-f6f385e7e549-4_876_812_338_625} \captionsetup{labelformat=empty} \caption{Fig. 11}
\end{figure} Fig. 11 shows a sketch of the curve with equation \(y = x - \frac { 4 } { x ^ { 2 } }\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and show that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = - \frac { 24 } { x ^ { 4 } }\).
  2. Hence find the coordinates of the stationary point on the curve. Verify that the stationary point is a maximum.
  3. Find the equation of the normal to the curve when \(x = - 1\). Give your answer in the form \(a x + b y + c = 0\).
OCR MEI C2 2014 June Q12
12 Oskar is designing a building. Fig. 12 shows his design for the end wall and the curve of the roof. The units for \(x\) and \(y\) are metres. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{aded99ef-873e-42fb-ade5-f6f385e7e549-5_734_1244_340_413} \captionsetup{labelformat=empty} \caption{Fig. 12}
\end{figure}
  1. Use the trapezium rule with 5 strips to estimate the area of the end wall of the building.
  2. Oskar now uses the equation \(y = - 0.001 x ^ { 3 } - 0.025 x ^ { 2 } + 0.6 x + 9\), for \(0 \leqslant x \leqslant 15\), to model the curve of the roof.
    (A) Calculate the difference between the height of the roof when \(x = 12\) given by this model and the data shown in Fig. 12.
    (B) Use integration to find the area of the end wall given by this model. \section*{Question 13 begins on page 6}
OCR MEI C2 2014 June Q13
13 The thickness of a glacier has been measured every five years from 1960 to 2010. The table shows the reduction in thickness from its measurement in 1960.
Year1965197019751980198519901995200020052010
Number of years since \(1960 ( t )\)5101520253035404550
Reduction in thickness since \(1960 ( h \mathrm {~m} )\)0.71.01.72.33.64.76.08.21215.9
An exponential model may be used for these data, assuming that the relationship between \(h\) and \(t\) is of the form \(h = a \times 10 ^ { b t }\), where \(a\) and \(b\) are constants to be determined.
  1. Show that this relationship may be expressed in the form \(\log _ { 10 } h = m t + c\), stating the values of \(m\) and \(c\) in terms of \(a\) and \(b\).
  2. Complete the table of values in the answer book, giving your answers correct to 2 decimal places, and plot the graph of \(\log _ { 10 } h\) against \(t\), drawing by eye a line of best fit.
  3. Use your graph to find \(h\) in terms of \(t\) for this model.
  4. Calculate by how much the glacier will reduce in thickness between 2010 and 2020, according to the model.
  5. Give one reason why this model will not be suitable in the long term. \section*{END OF QUESTION PAPER}
OCR MEI C2 2015 June Q1
1
  1. Differentiate \(12 \sqrt [ 3 ] { x }\).
  2. Integrate \(\frac { 6 } { x ^ { 3 } }\).