Questions — OCR MEI C2 (454 questions)

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OCR MEI C2 Q2
2 The growth in population \(P\) of a certain town after time \(t\) years can be modelled by the equation \(P = 11000 \times 10 ^ { k t }\) where \(k\) is a constant.
  1. State the initial population of the town.
  2. After three years the population of the town is 24000 . Use this information to find the value of \(k\) correct to two decimal places.
OCR MEI C2 Q3
3
  1. Write \(\log _ { 10 } ( x + 4 ) - 2 \log _ { 10 } x + \log _ { 10 } ( x + 16 )\) as a single logarithm.
  2. Without using your calculator, verify that \(x = 4\) is a root of the equation $$\log _ { 10 } ( x + 4 ) - 2 \log _ { 10 } x + \log _ { 10 } ( x + 16 ) = 1$$
OCR MEI C2 Q4
4 Find the values of \(\theta\) such that \(0 ^ { \circ } \leq \theta \leq 360 ^ { \circ }\) which satisfy the equation $$\cos \theta \tan \theta = \frac { \sqrt { 3 } } { 2 }$$
OCR MEI C2 Q5
5 The diagram shows the curve \(y = \mathrm { f } ( x )\) where \(a\) is a positive constant.
\includegraphics[max width=\textwidth, alt={}, center]{c55a5f04-3573-4f36-a12c-3755bdd4a45b-3_551_962_255_476} Sketch the following curves on separate diagrams, in each case stating the coordinates of points where they meet the \(x\) - and \(y\)-axes.
  1. \(\quad y = - \mathrm { f } ( x )\)
  2. \(\quad y = \mathrm { f } ( - x )\)
OCR MEI C2 Q6
6 A and B are points on the same side of a straight river. A and B are 180 metres apart. The angles made with a jetty J on the opposite side of the river \(78 ^ { \circ }\) and \(56 ^ { \circ }\) respectively as shown.
\includegraphics[max width=\textwidth, alt={}, center]{c55a5f04-3573-4f36-a12c-3755bdd4a45b-3_332_681_1451_565} Not to scale Calculate the width of the river correct to the nearest metre.
OCR MEI C2 Q7
7 For each of the following sequences, write down sufficient terms of the sequence in order to be able to describe its behaviour as divergent, periodic or convergent. For any convergent sequence, state its limit.
  1. \(a _ { 1 } = - 1 ; \quad a _ { k + 1 } = \frac { 4 } { a _ { k } }\)
  2. \(\quad a _ { 1 } = 1 ; \quad a _ { k } = 2 - 2 \times \left( \frac { 1 } { 2 } \right) ^ { k }\)
  3. \(\quad a _ { 1 } = 0 \quad a _ { k + 1 } = \left( 1 + a _ { k } \right) ^ { 2 }\).
OCR MEI C2 Q8
8 Fig. 8 shows a sector of a circle with centre O and radius 6 cm and a chord AB which subtends an angle of 1.8 radians at O . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c55a5f04-3573-4f36-a12c-3755bdd4a45b-4_341_485_310_771} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure}
  1. Calculate the area of the sector OAXB .
  2. Calculate the area of the triangle OAB and hence find the area of the shaded segment AXB.
OCR MEI C2 Q9
9 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } - 12 x + 9\). The curve passes through the point \(( 2 , - 2 )\).
  1. Find the equation of the curve.
  2. Show that the curve touches the \(x\)-axis at one point (A) and cuts it at another (B). State the coordinates of A and B.
  3. The curve cuts the \(y\)-axis at C . Show that the tangent at C is perpendicular to the normal at B.
OCR MEI C2 Q10
10 Fig. 10 shows the curve with equation \(y = x ^ { 2 } + \frac { 16 } { x }\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c55a5f04-3573-4f36-a12c-3755bdd4a45b-5_522_1019_403_394} \captionsetup{labelformat=empty} \caption{Fig. 10}
\end{figure}
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Hence calculate the coordinates of the stationary point on the curve.
  3. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) and explain why this confirms that he stationary point is a minimum.
  4. Using the trapezium rule with 4 intervals, estimate the area between the curve and the \(x\) axis between \(x = 2\) and \(x = 4\).
  5. State, giving a reason, whether this estimate of the area under-estimates or over-estimates the true area beneath the curve.
OCR MEI C2 Q11
11 When Fred joined a computer firm his salary was \(\pounds 28000\) per annum. In each subsequent year he received an annual increase of \(12 \%\) of his previous year's salary.
  1. State Fred's salary for each of his first 3 years with the company. State also the common ratio of the geometric sequence formed by his salaries.
  2. How much did Fred earn in the tenth year?
  3. Show that the total amount Fred earned over the ten years was between \(\pounds 400000\) and £500000.
  4. When Fred joined the computer firm, his brother Archie set up a plumbing business. He earned \(\pounds 35000\) in his first year and each year earned \(\pounds d\) more than in the previous year. At the end of ten years, he had earned exactly the same total amount as Fred. Calculate the value of \(d\).
OCR MEI C2 Q1
1
  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 Q2
2 The equation of a cubic curve is \(y = 2 x ^ { 3 } - 9 x ^ { 2 } + 12 x - 2\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and show that the tangent to the curve when \(x = 3\) passes through the point \(( - 1 , - 41 )\).
  2. Use calculus to find the coordinates of the turning points of the curve. You need not distinguish between the maximum and minimum.
  3. Sketch the curve, given that the only real root of \(2 x ^ { 3 } - 9 x ^ { 2 } + 12 x - 2 = 0\) is \(x = 0.2\) correct to 1 decimal place. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{b6ea89e3-a8a4-41a2-8ed5-eed6c2dfda7e-2_1017_935_285_638} \captionsetup{labelformat=empty} \caption{Fig. 11}
    \end{figure} Fig. 11 shows a sketch of the cubic curve \(y = \mathrm { f } ( x )\). The values of \(x\) where it crosses the \(x\)-axis are - 5 , - 2 and 2 , and it crosses the \(y\)-axis at \(( 0 , - 20 )\).
OCR MEI C2 Q4
4
  1. Differentiate \(x ^ { 3 } - 3 x ^ { 2 } - 9 x\). Hence find the \(x\)-coordinates of the stationary points on the curve \(y = x ^ { 3 } - 3 x ^ { 2 } - 9 x\), showing which is the maximum and which the minimum.
  2. Find, in exact form, the coordinates of the points at which the curve crosses the \(x\)-axis.
  3. Sketch the curve.
OCR MEI C2 Q5
5 The equation of a curve is \(\quad y = 7 + 6 x - x ^ { 2 }\).
  1. Use calculus to find the coordinates of the turning point on this curve. Find also the coordinates of the points of intersection of this curve with the axes, and sketch the curve.
  2. Find \(\int _ { 1 } ^ { 5 } \left( 7 + 6 x - x ^ { 2 } \right) \mathrm { d } x\), showing your working.
  3. The curve and the line \(y = 12\) intersect at \(( 1,12 )\) and \(( 5,12 )\). Using your answer to part (ii), find the area of the finite region between the curve and the line \(y = 12\).
OCR MEI C2 Q1
1 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 Q2
2 Fig. 8 shows the graph of \(y = \mathrm { g } ( x )\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{669be128-491c-4152-8f3a-e37a34dd9383-1_800_1401_781_385} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure} Draw the graph of
  1. \(y = \mathrm { g } ( 2 x )\),
  2. \(y = 3 \mathrm {~g} ( x )\).
OCR MEI C2 Q3
3 The point \(\mathrm { P } ( 6,3 )\) lies on the curve \(y = \mathrm { f } ( x )\). State the coordinates of the image of P after the transformation which maps \(y = \mathrm { f } ( x )\) onto
  1. \(y = 3 \mathrm { f } ( x )\),
  2. \(y = \mathrm { f } ( 4 x )\).
OCR MEI C2 Q4
4 In this question, \(\mathrm { f } ( x ) = x ^ { 2 } - 5 x\). Fig. 4 shows a sketch of the graph of \(y = \mathrm { f } ( x )\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{669be128-491c-4152-8f3a-e37a34dd9383-2_795_898_824_654} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure} On separate diagrams, sketch the curves \(y = \mathrm { f } ( 2 x )\) and \(y = 3 \mathrm { f } ( x )\), labelling the coordinates of their intersections with the axes and their turning points.
OCR MEI C2 Q5
5 State the transformation which maps the graph of \(y = x ^ { 2 } + 5\) onto the graph of \(y = 3 x ^ { 2 } + 15\).
OCR MEI C2 Q6
6 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{669be128-491c-4152-8f3a-e37a34dd9383-3_819_1370_271_383} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} Fig. 3 shows sketches of three graphs, A, B and C. The equation of graph A is \(y = \mathrm { f } ( x )\). State the equation of
  1. graph B ,
  2. graph C .
OCR MEI C2 Q7
7
  1. Solve the equation \(\cos x = 0.4\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
  2. Describe the transformation which maps the graph of \(y = \cos x\) onto the graph of \(y = \cos 2 x\).
OCR MEI C2 Q8
8
  1. The point \(\mathrm { P } ( 4 , - 2 )\) lies on the curve \(y = \mathrm { f } ( x )\). Find the coordinates of the image of P when the curve is transformed to \(y = \mathrm { f } ( 5 x )\).
  2. Describe fully a single transformation which maps the curve \(y = \sin x ^ { \circ }\) onto the curve \(y = \sin ( x - 90 ) ^ { \circ }\).
OCR MEI C2 Q9
9 Figs. 5.1 and 5.2 show the graph of \(y = \sin x\) for values of \(x\) from \(0 ^ { \circ }\) to \(360 ^ { \circ }\) and two transformations of this graph. State the equation of each graph after it has been transformed.
  1. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{669be128-491c-4152-8f3a-e37a34dd9383-4_511_941_828_586} \captionsetup{labelformat=empty} \caption{Fig. 5.1}
    \end{figure}
  2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{669be128-491c-4152-8f3a-e37a34dd9383-4_517_937_1508_584} \captionsetup{labelformat=empty} \caption{Fig. 5.2}
    \end{figure}
OCR MEI C2 Q10
10 The curve \(y = \mathrm { f } ( x )\) has a minimum point at \(( 3,5 )\).
State the coordinates of the corresponding minimum point on the graph of
  1. \(y = 3 \mathrm { f } ( x )\),
  2. \(y = \mathrm { f } ( 2 x )\).
OCR MEI C2 Q11
11 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{669be128-491c-4152-8f3a-e37a34dd9383-5_546_989_828_596} \captionsetup{labelformat=empty} \caption{Fig. 5}
\end{figure} Fig. 5 shows a sketch of the graph of \(y = \mathrm { f } ( x )\). On separate diagrams, sketch the graphs of the following, showing clearly the coordinates of the points corresponding to \(\mathrm { P } , \mathrm { Q }\) and R .
  1. \(y = \mathrm { f } ( 2 x )\)
  2. \(y = \frac { 1 } { 4 } \mathrm { f } ( x )\)