Questions — OCR MEI C2 (454 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
OCR MEI C2 Q13
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
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
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
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
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 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
7 Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(y = \sqrt { x } + \frac { 3 } { x }\).
OCR MEI C2 Q8
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
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
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 \(\frac { 3 } { 4 }\).
OCR MEI C2 Q11
11 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 - x ^ { 2 }\). The curve passes through the point \(( 6,1 )\). Find the equation of the curve.
OCR MEI C2 Q12
12 Given tha \(y = 6 x ^ { 3 } + \sqrt { x } + 3\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
OCR MEI C2 Q1
1 Differentiate \(x + \sqrt { x ^ { 3 } }\).
OCR MEI C2 Q2
2 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 6 } { x ^ { 3 } }\). The curve passes through \(( 1,4 )\).
Find the equation of the curve.
OCR MEI C2 Q3
3 A and B are points on the curve \(y = 4 \sqrt { x }\). Point A has coordinates \(( 9,12 )\) and point B has \(x\)-coordinate 9.5. Find the gradient of the chord AB . The gradient of AB is an approximation to the gradient of the curve at A . State the \(x\)-coordinate of a point C on the curve such that the gradient of AC is a closer approximation.
OCR MEI C2 Q4
4 Differentiate \(2 x ^ { 3 } + 9 x ^ { 2 } - 24 x\). Hence find the set of values of \(x\) for which the function \(\mathrm { f } ( x ) = 2 x ^ { 3 } + 9 x ^ { 2 } - 24 x\) is increasing.
OCR MEI C2 Q5
5 Find the set of values of \(x\) for which \(x ^ { 2 } - 7 x\) is a decreasing function.
OCR MEI C2 Q6
6 Differentiate \(10 x ^ { 4 } + 12\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{936dd0c9-0776-47c5-9eb8-613752bbf286-2_507_494_217_839} \captionsetup{labelformat=empty} \caption{Fig. 10}
\end{figure} Fig. 10 shows a solid cuboid with square base of side \(x \mathrm {~cm}\) and height \(h \mathrm {~cm}\). Its volume is \(120 \mathrm {~cm} ^ { 3 }\).
  1. Find \(h\) in terms of \(x\). Hence show that the surface area, \(A \mathrm {~cm} ^ { 2 }\), of the cuboid is given by \(A = 2 x ^ { 2 } + \frac { 480 } { x }\).
  2. Find \(\frac { \mathrm { d } A } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } A } { \mathrm {~d} x ^ { 2 } }\).
  3. Hence find the value of \(x\) which gives the minimum surface area. Find also the value of the surface area in this case.
OCR MEI C2 Q8
8 Differentiate \(6 x ^ { \frac { 5 } { 2 } } + 4\).
OCR MEI C2 Q9
9 A is the point \(( 2,1 )\) on the curve \(y = \frac { 4 } { x ^ { 2 } }\). B is the point on the same curve with \(x\)-coordinate 2.1.
  1. Calculate the gradient of the chord AB of the curve. Give your answer correct to 2 decimal places.
  2. Give the \(x\)-coordinate of a point C on the curve for which the gradient of chord AC is a better approximation to the gradient of the curve at A .
  3. Use calculus to find the gradient of the curve at A .
OCR MEI C2 Q10
10 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = x ^ { 2 } - 6 x\). Find the set of values of \(x\) for which \(y\) is an increasing function of \(x\).
OCR MEI C2 Q11
11 A curve has gradient given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 x ^ { 2 } + 8 x\). The curve passes through the point \(( 1,5 )\). Find the equation of the curve.
OCR MEI C2 Q2
2 Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(y = x ^ { 6 } + \sqrt { x }\).
  1. Find the equation of the tangent to the curve \(y = x ^ { 4 }\) at the point where \(x = 2\). Give your answer in the form \(y = m x + c\).
  2. Calculate the gradient of the chord joining the points on the curve \(y = x ^ { 4 }\) where \(x = 2\) and \(x = 2.1\).
  3. (A) Expand \(( 2 + h ) ^ { 4 }\).
    (B) Simplify \(\frac { ( 2 + h ) ^ { 4 } - 2 ^ { 4 } } { h }\).
    (C) Show how your result in part (iii) (B) can be used to find the gradient of \(y = x ^ { 4 }\) at the point where \(x = 2\).
  4. Calculate the gradient of the chord joining the points on the curve \(y = x ^ { 2 } - 7\) for which \(x = 3\) and \(x = 3.1\).
  5. Given that \(\mathrm { f } ( x ) = x ^ { 2 } - 7\), find and simplify \(\frac { \mathrm { f } ( 3 + h ) - \mathrm { f } ( 3 ) } { h }\).
  6. Use your result in part (ii) to find the gradient of \(y = x ^ { 2 } - 7\) at the point where \(x = 3\), showing your reasoning.
  7. Find the equation of the tangent to the curve \(y = x ^ { 2 } - 7\) at the point where \(x = 3\).
  8. This tangent crosses the \(x\)-axis at the point P . The curve crosses the positive \(x\)-axis at the point Q . Find the distance PQ , giving your answer correct to 3 decimal places.
  9. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{8ff8b67d-1489-4cb1-bcd2-b32db674e29f-3_651_770_242_737} \captionsetup{labelformat=empty} \caption{Fig. 12}
    \end{figure} Fig. 12 shows part of the curve \(y = x ^ { 4 }\) and the line \(y = 8 x\), which intersect at the origin and the point P .
    (A) Find the coordinates of P , and show that the area of triangle OPQ is 16 square units.
    (B) Find the area of the region bounded by the line and the curve.
  10. You are given that \(\mathrm { f } ( x ) = x ^ { 4 }\).
    (A) Complete this identity for \(\mathrm { f } ( x + h )\). $$\mathrm { f } ( x + h ) = ( x + h ) ^ { 4 } = x ^ { 4 } + 4 x ^ { 3 } h + \ldots$$ (B) Simplify \(\frac { \mathrm { f } ( x + h ) - \mathrm { f } ( x ) } { h }\).
    (C) Find \(\lim _ { h \rightarrow 0 } \frac { \mathrm { f } ( x + h ) - \mathrm { f } ( x ) } { h }\).
    (D) State what this limit represents.
OCR MEI C2 Q2
2 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]{4a9ca68f-f980-4a8f-b387-80dbdca33dfe-2_782_1168_319_451} \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 Q3
3
  1. The standard formulae for the volume \(V\) and total surface area \(A\) of a solid cylinder of radius \(r\) and height \(h\) are $$V = \pi r ^ { 2 } h \quad \text { and } \quad A = 2 \pi r ^ { 2 } + 2 \pi r h .$$ Use these to show that, for a cylinder with \(A = 200\), $$V = 100 r - \pi r ^ { 3 }$$
  2. Find \(\frac { \mathrm { d } V } { \mathrm {~d} r }\) and \(\frac { \mathrm { d } ^ { 2 } V } { \mathrm {~d} r ^ { 2 } }\).
  3. Use calculus to find the value of \(r\) that gives a maximum value for \(V\) and hence find this maximum value, giving your answers correct to 3 significant figures.