Questions — OCR MEI C2 (454 questions)

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OCR MEI C2 Q10
10 A function \(y = \mathrm { f } ( x )\) may be modelled by the equation \(y = a x ^ { b }\).
  1. Show why, if this is so, then plotting \(\log y\) against \(\log x\) will produce a straight line graph. Explain how \(a\) and \(b\) may be determined experimentally from the graph.
  2. Values of \(x\) and \(y\) are given below. By plotting a graph of logy against log \(x\), show that the model above is appropriate for this set of data and find values of \(a\) and \(b\) given that \(a\) is an integer and \(b\) can be written as a fraction with a denominator less than 10 .
    \(x\)23456
    \(y\)4.65.05.35.55.7
  3. Use your formula from part (ii) to estimate the value of \(y\) when \(x = 2.8\).
OCR MEI C2 Q11
11 The cross-section of a brick wall built on horizontal ground is given, for \(0 \leq x \leq 6\), by the following function $$\begin{array} { l l } 0 \leq x \leq 2 & y = 1
2 \leq x \leq 4 & y = - \frac { 1 } { 2 } x ^ { 2 } + 3 x - 3
4 \leq x \leq 6 & y = 1 \end{array}$$
\includegraphics[max width=\textwidth, alt={}]{13bfa97b-ec49-4f41-b3dd-d9a31a2c30e8-4_523_1327_633_413}
Units are metres.
  1. Show that the highest point on the wall is 1.5 metres above the ground.
  2. Find the area of the cross-section of the wall.
OCR MEI C2 Q1
1 Find all the angles in the range \(0 ^ { 0 } \leq x \leq 360 ^ { 0 }\) satisfying the equation \(\sin x + \frac { 1 } { 2 } \sqrt { 3 } = 0\).
OCR MEI C2 Q2
2 Solve the equation \(3 ^ { x } = 15\), giving your answer correct to 4 decimal places.
OCR MEI C2 Q3
3 The sum to infinity of a geometric series is 5 and the first term is 2 .
Find the common ratio of the series.
OCR MEI C2 Q4
4 The first 3 terms of an arithmetical progression are 7, 5.9 and 4.8.
Find
  1. the common difference,
  2. the smallest value of \(n\) for which the sum to \(n\) terms is negative.
OCR MEI C2 Q5
5 The gradient of a curve is given by the function \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 - x\).
The curve passes through the point \(( 1,2 )\).
Find the equation of the curve.
OCR MEI C2 Q6
6 Evaluate \(\int _ { 1 } ^ { 2 } \left( x ^ { 2 } + \frac { 1 } { x ^ { 2 } } \right) \mathrm { d } x\).
OCR MEI C2 Q7
7
  1. Using the triangle, show that \(\sin ^ { 2 } x + \cos ^ { 2 } x = 1\).
  2. Hence prove that
    \includegraphics[max width=\textwidth, alt={}]{73d1c02b-1b7b-426d-a171-c762597cfed4-2_255_501_1779_1022} \(1 + \tan ^ { 2 } x = \frac { 1 } { \cos ^ { 2 } x }\).
OCR MEI C2 Q8
8 Draw two sketches of the graph of \(y = \sin x\) in the range \(0 ^ { \circ } \leq x \leq 360 ^ { \circ }\).
  1. On the first sketch, draw also a sketch of \(y = \sin ( 2 x )\).
  2. On the second sketch, draw also a sketch of \(y = 2 \sin x\).
OCR MEI C2 Q9
9 A sector of a circle has an angle of 0.8 radians. The arc length is 5 cm . Calculate the radius of the circle and the area of the sector.
OCR MEI C2 Q10
10 At 1200 the captain of a ship observes that the bearing of a lighthouse is \(340 ^ { \circ }\). His position is at A.
At 1230 he takes another bearing of the lighthouse and finds it to be \(030 ^ { \circ }\). During this time the ship moves on a constant course of \(280 ^ { \circ }\) to the point B . His plot on the chart is as shown in Fig. 11 below. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{73d1c02b-1b7b-426d-a171-c762597cfed4-3_501_1156_661_387} \captionsetup{labelformat=empty} \caption{Fig. 11}
\end{figure}
  1. Write down the size of the angles LAB and LBA .
  2. The captain believes that at A he is 5 km from L . Assuming that LA is exactly 5 km , show that LB is 4.61 km , correct to 2 decimal places, and find AB . Hence calculate the speed of the ship.
  3. The speed of the ship is actually 10 kilometres per hour. Given that the bearings of \(340 ^ { \circ }\) and \(030 ^ { \circ }\) and the ship's course of \(280 ^ { \circ }\) are all accurate, calculate the true value of the distance LA.
OCR MEI C2 Q12
12 Fig. 12 shows a window. The base and sides are parts of a rectangle with dimensions \(2 x\) metres horizontally by \(y\) metres vertically. The top is a semicircle of radius \(x\) metres. The perimeter of the window is 10 metres. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{73d1c02b-1b7b-426d-a171-c762597cfed4-4_428_433_1638_766} \captionsetup{labelformat=empty} \caption{Fig. 12}
\end{figure}
  1. Express \(y\) as a function of \(x\).
  2. Find the total area, \(A \mathrm {~m} ^ { 2 }\), in terms of \(x\) and \(y\). Use your answer to part (i) to show that this simplifies to $$A = 10 x - 2 x ^ { 2 } - \frac { 1 } { 2 } \pi x ^ { 2 }$$
  3. Prove that for the maximum value of \(A\), \(y = x\) exactly.
    \section*{MEI STRUCTURED MATHEMATICS } \section*{CONCEPTS FOR ADVANCED MATHEMATICS, C2} \section*{Practice Paper C2-B
    Insert sheet for question 11} 11 Speed-time graph with the first two points plotted.
    \includegraphics[max width=\textwidth, alt={}, center]{73d1c02b-1b7b-426d-a171-c762597cfed4-5_768_1772_1389_205}
OCR MEI C2 Q1
1 The common ratio of a geometric progression is - 0.5 . The sum of its first three terms is 15 . Find the first term.
Find also the sum to infinity.
OCR MEI C2 Q2
2 The diagram shows the graph of \(y = \mathrm { f } ( x )\). The graph passes through the point with coordinates \(( 0,2 )\).
\includegraphics[max width=\textwidth, alt={}, center]{1c52d6b5-84b4-455a-9620-c377ae457069-2_524_1350_775_346} On separate diagrams sketch the graphs of the following functions, indicating clearly the point of intersection with the \(y\) axis.
  1. \(\quad y = - \mathrm { f } ( x )\)
  2. \(y = f ( 3 x )\)
OCR MEI C2 Q3
3 Given that \(A\) is the obtuse angle such that \(\sin A = \frac { 1 } { 5 }\), find the exact value of \(\cos A\).
OCR MEI C2 Q4
4 You are given that \(y = x ^ { 3 } - 12 x\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Hence find the coordinates of the turning points of the curve.
OCR MEI C2 Q5
5 A sequence is defined by \(a _ { k } = 5 k + 1\), for \(k = 1,2,3 \ldots\)
  1. Write down the first three terms of the sequence.
  2. Evaluate \(\sum _ { k = 1 } ^ { 100 } a _ { k }\).
OCR MEI C2 Q6
6 Find the solution to this equation, correct to 3 significant figures. $$\left( 2 ^ { x } \right) \left( 2 ^ { x + 1 } \right) = 10 .$$
OCR MEI C2 Q7
7 The gradient of a curve \(y = \mathrm { f } ( x )\) is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } - 10 x + 6\). The curve passes through the point \(( 2,3 )\) Find the equation of the curve.
OCR MEI C2 Q8
8 In the triangle ABC shown, \(\mathrm { AB } = 8 \mathrm {~cm}\). \(\mathrm { AC } = 12 \mathrm {~cm}\) and angle \(\mathrm { ABC } = 82 ^ { \circ }\). Find \(\theta\) correct to 3 significant figures.
\includegraphics[max width=\textwidth, alt={}, center]{1c52d6b5-84b4-455a-9620-c377ae457069-3_382_540_1492_718}
OCR MEI C2 Q9
9 Fig. 9 shows
\(P \quad\) The line \(y = x\)
\(Q\) The curve \(y = \sqrt { \frac { 1 } { 2 } \left( x + x ^ { 2 } \right) }\)
\(R \quad\) The curve \(\quad y = \sqrt { x }\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1c52d6b5-84b4-455a-9620-c377ae457069-4_471_1103_762_374} \captionsetup{labelformat=empty} \caption{Fig. 9}
\end{figure}
  1. Write down the area of the triangle formed by the line \(y = x\), the line \(x = 1\) and the \(x\)-axis.
  2. Show that the area of the region formed by the curve \(y = \sqrt { x }\), the line \(x = 1\) and the \(x\)-axis is \(\frac { 2 } { 3 }\). An estimate is required of the Area, \(A\), of the region formed by the curve \(y = \sqrt { \frac { 1 } { 2 } \left( x + x ^ { 2 } \right) }\), the line \(x = 1\) and the \(x\)-axis.
  3. Use results to parts (i) and (ii) to complete the statement $$\ldots \ldots \ldots \ldots . . < A < \ldots \ldots \ldots \ldots \ldots . .$$
  4. Use the Trapezium Rule with 4 strips to find an estimate for \(A\).
  5. Draw a sketch of Fig. 9. Use it to illustrate the area found as the trapezium rule estimate for \(A\).
    Explain how your diagram shows that the trapezium rule estimate must be:
    consistent with the answer to part (iv);
    an under-estimate for A .
OCR MEI C2 Q10
10 A culture of bacteria is observed during an experiment. The number of bacteria is denoted by \(N\) and the time in hours after the start of the experiment by \(t\).
The table gives observations of \(t\) and \(N\).
Time \(( t\) hours \()\)12345
Number of bacteria \(( N )\)120170250370530
  1. Plot the points \(( t , N )\) on graph paper and join them with a smooth curve.
  2. Explain why the curve suggests why the relationship connecting \(t\) and \(N\) may be of the form \(N = a b ^ { t }\).
  3. Explain how, by using logarithms, the curve given by plotting \(N\) against \(t\) can be transformed into a straight line.
    State the gradient of this straight line and its intercept with the vertical axis in terms of \(a\) and \(b\).
  4. Complete a table of values for \(\log _ { 10 } N\) and plot the points \(\left( t , \log _ { 10 } N \right)\) on graph paper. Draw the best fit line through the points and use it to estimate the values of \(a\) and \(b\).
OCR MEI C2 Q11
11 The sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots \ldots\) where \(a _ { 1 }\) is a given real number is defined by \(a _ { n + 1 } = 1 - \frac { 1 } { a _ { n } }\).
  1. For the case when \(a _ { 1 } = 2\), find \(a _ { 2 } , a _ { 3 }\) and \(a _ { 4 }\). Describe the behaviour of this sequence
  2. For the case when \(a _ { 1 } = k\), where \(k\) is an integer greater than 1 , find \(a _ { 2 }\) in terms of \(k\) as a single fraction.
    Find also \(a _ { 3 }\) in its simplest form and hence deduce that \(a _ { 4 } = k\).
  3. Show that \(a _ { 2 } a _ { 3 } a _ { 4 } = - 1\) for any integer \(k\).
  4. When \(a _ { 1 } = 2\) evaluate \(\sum _ { i = 1 } ^ { 99 } a _ { i }\).
OCR MEI C2 Q1
1
  1. Find \(\int \left( x ^ { 3 } - 2 x \right) \mathrm { d } x\). The graph below shows part of the curve \(y = x ^ { 3 } - 2 x\) for \(0 \leq x \leq 2\).
    \includegraphics[max width=\textwidth, alt={}, center]{c55a5f04-3573-4f36-a12c-3755bdd4a45b-2_528_1019_520_321}
  2. Show that the area of the shaded region \(P\) is the same as the area of the shaded region \(Q\).