Questions C2 (1410 questions)

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OCR MEI C2 2013 June Q8
4 marks Moderate -0.8
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
11 marks Moderate -0.3
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
14 marks Standard +0.3
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 marks Standard +0.3
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
3 marks Moderate -0.8
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 marks Easy -1.3
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
3 marks Easy -1.3
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
4 marks Moderate -0.3
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
5 marks Moderate -0.3
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
3 marks Easy -1.2
8 Simplify \(\frac { \sqrt { 1 - \cos ^ { 2 } \theta } } { \tan \theta }\), where \(\theta\) is an acute angle.
OCR MEI C2 2014 June Q9
3 marks Moderate -0.8
9 Solve the equation \(\tan 2 \theta = 3\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).
OCR MEI C2 2014 June Q10
4 marks Easy -1.2
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
10 marks Moderate -0.3
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 marks Moderate -0.3
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
5 marks Easy -1.8
1
  1. Differentiate \(12 \sqrt [ 3 ] { x }\).
  2. Integrate \(\frac { 6 } { x ^ { 3 } }\).
OCR MEI C2 2015 June Q2
3 marks Moderate -0.8
2 A sequence is defined by \(u _ { 1 } = 2\) and \(u _ { k + 1 } = \frac { 10 } { u _ { k } ^ { 2 } }\).
Calculate \(\sum _ { k = 1 } ^ { 4 } u _ { k }\).
OCR MEI C2 2015 June Q3
5 marks Easy -1.2
3 An arithmetic progression has tenth term 11.1 and fiftieth term 7.1. Find the first term and the common difference. Find also the sum of the first fifty terms of the progression.
OCR MEI C2 2015 June Q4
4 marks Moderate -0.5
4 A sector of a circle has angle 1.5 radians and area \(27 \mathrm {~cm} ^ { 2 }\). Find the perimeter of the sector.
OCR MEI C2 2015 June Q5
5 marks Moderate -0.3
5 Use calculus to find the set of values of \(x\) for which \(x ^ { 3 } - 6 x\) is an increasing function.
OCR MEI C2 2015 June Q6
5 marks Moderate -0.8
6
  1. On the same axes, sketch the curves \(y = 3 ^ { x }\) and \(y = 3 ^ { 2 x }\), identifying clearly which is which.
  2. Given that \(3 ^ { 2 x } = 729\), find in either order the values of \(3 ^ { x }\) and \(x\).
OCR MEI C2 2015 June Q7
5 marks Moderate -0.3
7 Show that the equation \(\sin ^ { 2 } x = 3 \cos x - 2\) can be expressed as a quadratic equation in \(\cos x\) and hence solve the equation for values of \(x\) between 0 and \(2 \pi\).
OCR MEI C2 2015 June Q8
4 marks Moderate -0.8
8 Fig. 8 shows the graph of \(\log _ { 10 } y\) against \(\log _ { 10 } x\). It is a straight line passing through the points \(( 2,8 )\) and \(( 0,2 )\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5c7ac296-a911-451b-ad18-5ade3ac23e74-2_460_634_1868_717} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure} Find the equation relating \(\log _ { 10 } y\) and \(\log _ { 10 } x\) and hence find the equation relating \(y\) and \(x\).
OCR MEI C2 2015 June Q9
11 marks Moderate -0.8
9 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5c7ac296-a911-451b-ad18-5ade3ac23e74-3_253_1486_328_292} \captionsetup{labelformat=empty} \caption{Fig. 9.1}
\end{figure}
  1. Jean is designing a model aeroplane. Fig. 9.1 shows her first sketch of the wing's cross-section. Calculate angle A and the area of the cross-section.
  2. Jean then modifies her design for the wing. Fig. 9.2 shows the new cross-section, with 1 unit for each of \(x\) and \(y\) representing one centimetre. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{5c7ac296-a911-451b-ad18-5ade3ac23e74-3_431_1682_970_194} \captionsetup{labelformat=empty} \caption{Fig. 9.2}
    \end{figure} Here are some of the coordinates that Jean used to draw the new cross-section.
    Upper surfaceLower surface
    \(x\)\(y\)\(x\)\(y\)
    0000
    41.454-0.85
    81.568-0.76
    121.2712-0.55
    161.0416-0.30
    200200
    Use the trapezium rule with trapezia of width 4 cm to calculate an estimate of the area of this cross-section.
OCR MEI C2 2015 June Q10
13 marks Standard +0.3
10 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 4 x + 3\). The curve passes through the point ( 2,9 ).
  1. Find the equation of the tangent to the curve at the point \(( 2,9 )\).
  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.
  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.
OCR MEI C2 2015 June Q11
12 marks Standard +0.3
11 Jill has 3 daughters and no sons. They are generation 1 of Jill's descendants.
Each of her daughters has 3 daughters and no sons. Jill's 9 granddaughters are generation 2 of her descendants. Each of her granddaughters has 3 daughters and no sons; they are descendant generation 3. Jill decides to investigate what would happen if this pattern continues, with each descendant having 3 daughters and no sons.
  1. How many of Jill's descendants would there be in generation 8 ?
  2. How many of Jill's descendants would there be altogether in the first 15 generations?
  3. After \(n\) generations, Jill would have over a million descendants altogether. Show that \(n\) satisfies the inequality $$n > \frac { \log _ { 10 } 2000003 } { \log _ { 10 } 3 } - 1 .$$ Hence find the least possible value of \(n\).
  4. How many fewer descendants would Jill have altogether in 15 generations if instead of having 3 daughters, she and each subsequent descendant has 2 daughters? \section*{END OF QUESTION PAPER}