Questions — OCR MEI (4301 questions)

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OCR MEI C2 Q3
3 marks Easy -1.2
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 marks Moderate -0.5
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
3 marks Moderate -0.8
5 Find the set of values of \(x\) for which \(x ^ { 2 } - 7 x\) is a decreasing function.
OCR MEI C2 Q6
2 marks Moderate -0.8
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
2 marks Easy -1.8
8 Differentiate \(6 x ^ { \frac { 5 } { 2 } } + 4\).
OCR MEI C2 Q9
5 marks Moderate -0.8
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
3 marks Moderate -0.8
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
4 marks Moderate -0.8
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
3 marks Easy -1.2
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
11 marks Moderate -0.3
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
11 marks Moderate -0.3
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.
OCR MEI C2 Q4
5 marks Moderate -0.8
4
  1. Differentiate \(x ^ { 3 } - 6 x ^ { 2 } - 15 x + 50\).
  2. Hence find the \(x\)-coordinates of the stationary points on the curve \(y = x ^ { 3 } - 6 x ^ { 2 } - 15 x + 50\).
OCR MEI C2 Q5
5 marks Moderate -0.5
5 Use calculus to find the \(x\)-coordinates of the turning points of the curve \(y = x ^ { 3 } - 6 x ^ { 2 } - 15 x\). Hence find the set of values of \(x\) for which \(x ^ { 3 } - 6 x ^ { 2 } - 15 x\) is an increasing function.
OCR MEI C2 Q1
12 marks Moderate -0.3
1 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.
OCR MEI C2 Q2
13 marks Standard +0.3
2 A cubic curve has equation \(y = x ^ { 3 } - 3 x ^ { 2 } + 1\).
  1. Use calculus to find the coordinates of the turning points on this curve. Determine the nature of these turning points.
  2. Show that the tangent to the curve at the point where \(x = - 1\) has gradient 9 . Find the coordinates of the other point, P , on the curve at which the tangent has gradient 9 and find the equation of the normal to the curve at P . Show that the area of the triangle bounded by the normal at P and the \(x\) - and \(y\)-axes is 8 square units.
OCR MEI C2 Q3
5 marks Moderate -0.8
3 A curve has equation \(y = x + \frac { 1 } { x }\).
Use calculus to show that the curve has a turning point at \(x = 1\).
Show also that this point is a minimum.
OCR MEI C2 Q4
12 marks Moderate -0.8
4 The equation of a curve is \(y = 9 x ^ { 2 } - x ^ { 4 }\).
  1. Show that the curve meets the \(x\)-axis at the origin and at \(x = \pm a\), stating the value of \(a\).
  2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\). Hence show that the origin is a minimum point on the curve. Find the \(x\)-coordinates of the maximum points.
  3. Use calculus to find the area of the region bounded by the curve and the \(x\)-axis between \(x = 0\) and \(x = a\), using the value you found for \(a\) in part (i).
OCR MEI C2 Q5
5 marks Moderate -0.3
5 Differentiate \(4 x ^ { 2 } + \frac { 1 } { x }\) and hence find the \(x\)-coordinate of the stationary point of the curve \(y = 4 x ^ { 2 } + \frac { 1 } { x }\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bba82ee6-90b2-4f03-9bb9-0371ff711a09-3_639_1027_302_542} \captionsetup{labelformat=empty} \caption{Fig. 11}
\end{figure} The equation of the curve shown in Fig. 11 is \(y = x ^ { 3 } - 6 x + 2\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Find, in exact form, the range of values of \(x\) for which \(x ^ { 3 } - 6 x + 2\) is a decreasing function.
  3. Find the equation of the tangent to the curve at the point \(( - 1,7 )\). Find also the coordinates of the point where this tangent crosses the curve again.
OCR MEI C2 Q1
12 marks Moderate -0.3
1 The equation of a curve is \(y = 7 + 6 x - x ^ { 2 }\).
  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 Q2
13 marks Moderate -0.3
2 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. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{4e8d7217-61f7-4ae4-96dd-d34e37c4d623-2_1020_940_244_679} \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 \(( 0 , - 20 )\).
  4. Express \(\mathrm { f } ( x )\) in factorised form.
  5. Show that the equation of the curve may be written as \(y = x ^ { 3 } + 5 x ^ { 2 } - 4 x - 20\).
  6. Use calculus to show that, correct to 1 decimal place, the \(x\)-coordinate of the minimum point on the curve is 0.4 . Find also the coordinates of the maximum point on the curve, giving your answers correct to 1 decimal place.
  7. State, correct to 1 decimal place, the coordinates of the maximum point on the curve \(y = \mathrm { f } ( 2 x )\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{4e8d7217-61f7-4ae4-96dd-d34e37c4d623-3_768_1023_223_598} \captionsetup{labelformat=empty} \caption{Fig. 11}
    \end{figure} Fig. 11 shows the curve \(y = x ^ { 3 } - 3 x ^ { 2 } - x + 3\).
  8. Use calculus to find \(\int _ { 1 } ^ { 3 } \left( x ^ { 3 } - 3 x ^ { 2 } - x + 3 \right) \mathrm { d } x\) and state what this represents.
  9. Find the \(x\)-coordinates of the turning points of the curve \(y = x ^ { 3 } - 3 x ^ { 2 } - x + 3\), giving your answers in surd form. Hence state the set of values of \(x\) for which \(y = x ^ { 3 } - 3 x ^ { 2 } - x + 3\) is a decreasing function.
  10. 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.
  11. Find, in exact form, the coordinates of the points at which the curve crosses the \(x\)-axis.
  12. Sketch the curve. A curve has equation \(y = x ^ { 3 } - 6 x ^ { 2 } + 12\).
  13. Use calculus to find the coordinates of the turning points of this curve. Determine also the nature of these turning points.
  14. Find, in the form \(y = m x + c\), the equation of the normal to the curve at the point \(( 2 , - 4 )\).
OCR MEI C2 Q1
13 marks Standard +0.3
1 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 Q2
5 marks Moderate -0.8
2 Find the equation of the normal to the curve \(y = 8 x ^ { 4 } + 4\) at the point where \(x = \frac { 1 } { 2 }\).
  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 \(x = 2\).
OCR MEI C2 Q4
12 marks Moderate -0.3
4
  1. Calculate the gradient of the chord joining the points on the curve \(y = x ^ { 2 } - 7\) for which \(x = 3\) and \(x = 3.1\).
  2. Given that \(\mathrm { f } ( x ) = x ^ { 2 } - 7\), find and simplify \(\frac { \mathrm { f } ( 3 + h ) - \mathrm { f } ( 3 ) } { h }\).
  3. Use your result in part (ii) to find the gradient of \(y = x ^ { 2 } - 7\) at the point where \(x = 3\), showing your reasoning.
  4. Find the equation of the tangent to the curve \(y = x ^ { 2 } - 7\) at the point where \(x = 3\).
  5. 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.
OCR MEI C2 Q5
4 marks Easy -1.2
5 In Fig. 5, A and B are the points on the curve \(y = 2 ^ { x }\) with \(x\)-coordinates 3 and 3.1 respectively. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e15637e4-8532-4b8b-9c95-7a4359503c7b-2_703_728_1199_754} \captionsetup{labelformat=empty} \caption{Fig. 5}
\end{figure}
  1. Find the gradient of the chord AB . Give your answer correct to 2 decimal places.
  2. Stating the points you use, find the gradient of another chord which will give a closer approximation to the gradient of the tangent to \(y = 2 ^ { x }\) at A .
OCR MEI C2 Q2
11 marks Standard +0.3
2 Fig. 10 shows a sketch of the curve \(y = x ^ { 2 } - 4 x + 3\). The point A on the curve has \(x\)-coordinate 4 . At point B the curve crosses the \(x\)-axis. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{65dd0efe-5c99-4814-b741-16e368c3469e-2_770_738_337_698} \captionsetup{labelformat=empty} \caption{Fig. 10}
\end{figure}
  1. Use calculus to find the equation of the normal to the curve at A and show that this normal intersects the \(x\)-axis at \(\mathrm { C } ( 16,0 )\).
  2. Find the area of the region ABC bounded by the curve, the normal at A and the \(x\)-axis.