Questions — OCR MEI C2 (480 questions)

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OCR MEI C2 Q1
14 marks Moderate -0.3
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
12 marks Moderate -0.3
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
11 marks Moderate -0.8
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
12 marks Moderate -0.8
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
4 marks Easy -1.2
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
4 marks Moderate -0.8
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
4 marks Easy -1.2
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 marks Moderate -0.3
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
2 marks Easy -1.2
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
4 marks Moderate -0.8
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
5 marks Moderate -0.8
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
4 marks Moderate -0.8
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
3 marks Moderate -0.8
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
4 marks Moderate -0.8
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
4 marks Moderate -0.8
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. \(y = \frac { 1 } { 4 } \mathrm { f } ( x )\)
OCR MEI C2 Q13
4 marks Moderate -0.8
13 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{669be128-491c-4152-8f3a-e37a34dd9383-7_618_867_267_679} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure} Fig. 4 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 \(\mathrm { A } , \mathrm { B }\) and C .
  1. \(y = 2 \mathrm { f } ( x )\)
  2. \(y = \mathrm { f } ( x + 3 )\)
OCR MEI C2 Q14
5 marks Moderate -0.8
14
  1. On the same axes, sketch the graphs of \(y = \cos x\) and \(y = \cos 2 x\) for values of \(x\) from 0 to \(2 \pi\).
  2. Describe the transformation which maps the graph of \(y = \cos x\) onto the graph of \(y = 3 \cos x\).
OCR MEI C2 Q2
5 marks Moderate -0.3
2 Use calculus to find the set of values of \(x\) for which \(x ^ { 3 } - 6 x\) is an increasing function.
OCR MEI C2 Q3
2 marks Easy -1.3
3 The points \(\mathrm { P } ( 2,3.6 )\) and \(\mathrm { Q } ( 2.2,2.4 )\) lie on the curve \(y = \mathrm { f } ( x )\). Use P and Q to estimate the gradient of the curve at the point where \(x = 2\).
OCR MEI C2 Q4
5 marks Easy -1.8
4 Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when
  1. \(y = 2 x ^ { - 5 }\),
  2. \(y = \sqrt [ 3 ] { x }\).
OCR MEI C2 Q5
5 marks Easy -1.2
5 The equation of a curve is \(y = \sqrt { 1 + 2 x }\).
  1. Calculate the gradient of the chord joining the points on the curve where \(x = 4\) and \(x = 4\). Give your answer correct to 4 decimal places.
  2. Showing the points you use, calculate the gradient of another chord of the curve which is a closer approximation to the gradient of the curve when \(x = 4\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{f540b962-ee6b-409a-a2a1-cd7ad4945514-2_1031_1113_273_499} \captionsetup{labelformat=empty} \caption{Fig. 5}
    \end{figure} Fig. 5 shows the graph of \(y = 2 ^ { x }\).
OCR MEI C2 Q7
3 marks Easy -1.2
7 Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(y = \sqrt { x } + \frac { 3 } { x }\).
OCR MEI C2 Q8
5 marks Moderate -0.8
8 The gradient of a curve is \(6 x ^ { 2 } + 12 x ^ { \frac { 1 } { 2 } }\). The curve passes through the point \(( 4,10 )\). Find the equation of the curve.
OCR MEI C2 Q9
3 marks Moderate -0.5
9 Use calculus to find the set of values of \(x\) for which \(\mathrm { f } ( x ) = 12 x - x ^ { 3 }\) is an increasing function.
OCR MEI C2 Q10
5 marks Easy -1.3
10 Given tha \(y = 6 x ^ { \frac { 3 } { 2 } }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
Show, without using a calculator, that when \(x = 36\) the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) is \(\frac { 3 } { 4 }\).