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OCR C3 Q3
8 marks Standard +0.3
3. The curve \(C\) has the equation \(y = 2 \mathrm { e } ^ { x } - 6 \ln x\) and passes through the point \(P\) with \(x\)-coordinate 1.
  1. Find an equation for the tangent to \(C\) at \(P\). The tangent to \(C\) at \(P\) meets the coordinate axes at the points \(Q\) and \(R\).
  2. Show that the area of triangle \(O Q R\), where \(O\) is the origin, is \(\frac { 9 } { 3 - \mathrm { e } }\).
OCR C3 Q4
8 marks Moderate -0.3
4. The finite region \(R\) is bounded by the curve with equation \(y = \frac { 1 } { 2 x - 1 }\), the \(x\)-axis and the lines \(x = 1\) and \(x = 2\).
  1. Find the exact area of \(R\).
  2. Show that the volume of the solid formed when \(R\) is rotated through four right angles about the \(x\)-axis is \(\frac { 1 } { 3 } \pi\).
OCR C3 Q5
8 marks Standard +0.3
5. \includegraphics[max width=\textwidth, alt={}, center]{5dd332a5-56d9-407a-8ff6-fa59294b358d-2_520_787_246_479} The diagram shows the graph of \(y = \mathrm { f } ( x )\). The graph has a minimum at \(\left( \frac { \pi } { 2 } , - 1 \right)\), a maximum at \(\left( \frac { 3 \pi } { 2 } , - 5 \right)\) and an asymptote with equation \(x = \pi\).
  1. Showing the coordinates of any stationary points, sketch the graph of \(y = | \mathrm { f } ( x ) |\). Given that $$\mathrm { f } : x \rightarrow a + b \operatorname { cosec } x , \quad x \in \mathbb { R } , \quad 0 < x < 2 \pi , \quad x \neq \pi$$
  2. find the values of the constants \(a\) and \(b\),
  3. find, to 2 decimal places, the \(x\)-coordinates of the points where the graph of \(y = \mathrm { f } ( x )\) crosses the \(x\)-axis.
OCR C3 Q6
11 marks Standard +0.8
6.
  1. Prove the identity $$2 \cot 2 x + \tan x \equiv \cot x , \quad x \neq \frac { n } { 2 } \pi , \quad n \in \mathbb { Z }$$
  2. Solve, for \(0 \leq x < \pi\), the equation $$2 \cot 2 x + \tan x = \operatorname { cosec } ^ { 2 } x - 7$$ giving your answers to 2 decimal places.
OCR C3 Q7
11 marks Moderate -0.3
7. The function \(f\) is defined by $$\mathrm { f } : x \rightarrow 3 \mathrm { e } ^ { x - 1 } , \quad x \in \mathbb { R }$$
  1. State the range of f .
  2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state its domain. The function \(g\) is defined by $$g : x \rightarrow 5 x - 2 , \quad x \in \mathbb { R }$$ Find, in terms of e,
  3. the value of \(\mathrm { gf } ( \ln 2 )\),
  4. the solution of the equation $$\mathrm { f } ^ { - 1 } \mathrm {~g} ( x ) = 4$$
OCR C3 Q8
13 marks Standard +0.3
A curve has the equation \(y = x ^ { 2 } - \sqrt { 4 + \ln x }\).
  1. Show that the tangent to the curve at the point where \(x = 1\) has the equation $$7 x - 4 y = 11$$ The curve has a stationary point with \(x\)-coordinate \(\alpha\).
  2. Show that \(0.3 < \alpha < 0.4\)
  3. Show that \(\alpha\) is a solution of the equation $$x = \frac { 1 } { 2 } ( 4 + \ln x ) ^ { - \frac { 1 } { 4 } }$$
  4. Use the iterative formula $$x _ { n + 1 } = \frac { 1 } { 2 } \left( 4 + \ln x _ { n } \right) ^ { - \frac { 1 } { 4 } }$$ with \(x _ { 0 } = 0.35\), to find \(\alpha\) correct to 5 decimal places.
    You should show the result of each iteration.
OCR MEI C2 Q1
12 marks Standard +0.3
1 Fig. 12 is a sketch of the curve \(y = 2 x ^ { 2 } - 11 x + 12\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{44b860fb-040f-4d3f-94d8-42eae41c0e2d-1_468_940_285_830} \captionsetup{labelformat=empty} \caption{Fig. 12}
\end{figure}
  1. Show that the curve intersects the \(x\)-axis at \(( 4,0 )\) and find the coordinates of the other point of intersection of the curve and the \(x\)-axis.
  2. Find the equation of the normal to the curve at the point \(( 4,0 )\). Show also that the area of the triangle bounded by this normal and the axes is 1.6 units \({ } ^ { 2 }\).
  3. Find the area of the region bounded by the curve and the \(x\)-axis.
OCR MEI C2 Q2
11 marks Moderate -0.8
2 A curve has equation \(y = x ^ { 3 } - 6 x ^ { 2 } + 12\).
  1. Use calculus to find the coordinates of the turning points of this curve. Determine also the nature of these turning points.
  2. Find, in the form \(y = m x + c\), the equation of the normal to the curve at the point \(( 2 , - 4 )\).
OCR MEI C2 Q3
13 marks Standard +0.3
3 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 Q4
13 marks Moderate -0.3
4 Fig. 10 shows a sketch of the graph of \(y = 7 x - x ^ { 2 } - 6\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{12e190fc-437f-499d-9c27-da49a7546755-2_604_912_1100_638} \captionsetup{labelformat=empty} \caption{Fig. 10}
\end{figure}
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence find the equation of the tangent to the curve at the point on the curve where \(x = 2\). Show that this tangent crosses the \(x\)-axis where \(x = \frac { 2 } { 3 }\).
  2. Show that the curve crosses the \(x\)-axis where \(x = 1\) and find the \(x\)-coordinate of the other point of intersection of the curve with the \(x\)-axis.
  3. Find \(\int _ { 1 } ^ { 2 } \left( 7 x - x ^ { 2 } - 6 \right) \mathrm { d } x\). Hence find the area of the region bounded by the curve, the tangent and the \(x\)-axis, shown shaded on Fig. 10. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{12e190fc-437f-499d-9c27-da49a7546755-3_643_1034_267_549} \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
10 marks Moderate -0.8
1 Oskar is designing a building. Fig. 12 shows his design for the end wall and the curve of the roof. The units for \(x\) and \(y\) are metres. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e97df57f-3b69-4bec-bc58-9730873dea53-1_735_1246_335_441} \captionsetup{labelformat=empty} \caption{Fig. 12}
\end{figure}
  1. Use the trapezium rule with 5 strips to estimate the area of the end wall of the building.
  2. Oskar now uses the equation \(y = - 0.001 x ^ { 3 } - 0.025 x ^ { 2 } + 0.6 x + 9\), for \(0 \leqslant x \leqslant 15\), to model the curve of the roof.
    (A) Calculate the difference between the height of the roof when \(x = 12\) given by this model and the data shown in Fig. 12.
    (B) Use integration to find the area of the end wall given by this model.
OCR MEI C2 Q2
4 marks Easy -1.2
2 Fig. 7 shows a curve and the coordinates of some points on it. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e97df57f-3b69-4bec-bc58-9730873dea53-2_639_1037_294_517} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure} Use the trapezium rule with 6 strips to estimate the area of the region bounded by the curve and the positive \(x\) - and \(y\)-axes.
OCR MEI C2 Q3
12 marks Moderate -0.3
3 A farmer digs ditches for flood relief. He experiments with different cross-sections. Assume that the surface of the ground is horizontal.
  1. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{e97df57f-3b69-4bec-bc58-9730873dea53-3_432_640_410_745} \captionsetup{labelformat=empty} \caption{Fig. 9.1}
    \end{figure} Fig. 9.1 shows the cross-section of one ditch, with measurements in metres. The width of the ditch is 1.2 m and Fig. 9.1 shows the depth every 0.2 m across the ditch. Use the trapezium rule with six intervals to estimate the area of cross-section. Hence estimate the volume of water that can be contained in a 50 -metre length of this ditch.
  2. Another ditch is 0.9 m wide, with cross-section as shown in Fig. 9.2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{e97df57f-3b69-4bec-bc58-9730873dea53-3_579_813_1336_656} \captionsetup{labelformat=empty} \caption{Fig. 9.2}
    \end{figure} With \(x\) - and \(y\)-axes as shown in Fig. 9.2, the curve of the ditch may be modelled closely by \(y = 3.8 x ^ { 4 } - 6.8 x ^ { 3 } + 7.7 x ^ { 2 } - 4.2 x\).
    (A) The actual ditch is 0.6 m deep when \(x = 0.2\). Calculate the difference between the depth given by the model and the true depth for this value of \(x\).
    (B) Find \(\int \left( 3.8 x ^ { 4 } - 6.8 x ^ { 3 } + 7.7 x ^ { 2 } - 4.2 x \right) \mathrm { d } x\). Hence estimate the volume of water that can be contained in a 50 -metre length of this ditch.
OCR MEI C2 Q4
13 marks Moderate -0.3
4
  1. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{e97df57f-3b69-4bec-bc58-9730873dea53-4_765_757_203_764} \captionsetup{labelformat=empty} \caption{Fig. 11.1}
    \end{figure} A boat travels from P to Q and then to R . As shown in Fig. 11.1, Q is 10.6 km from P on a bearing of \(045 ^ { \circ }\). R is 9.2 km from P on a bearing of \(113 ^ { \circ }\), so that angle QPR is \(68 ^ { \circ }\). Calculate the distance and bearing of R from Q .
  2. Fig. 11.2 shows the cross-section, EBC, of the rudder of a boat. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{e97df57f-3b69-4bec-bc58-9730873dea53-4_527_1474_1452_404} \captionsetup{labelformat=empty} \caption{Fig. 11.2}
    \end{figure} BC is an arc of a circle with centre A and radius 80 cm . Angle \(\mathrm { CAB } = \frac { 2 \pi } { 3 }\) radians.
    EC is an arc of a circle with centre D and radius \(r \mathrm {~cm}\). Angle CDE is a right angle.
    1. Calculate the area of sector ABC .
    2. Show that \(r = 40 \sqrt { 3 }\) and calculate the area of triangle CDA.
    3. Hence calculate the area of cross-section of the rudder. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{e97df57f-3b69-4bec-bc58-9730873dea53-5_695_1012_271_600} \captionsetup{labelformat=empty} \caption{Fig. 12}
      \end{figure} A water trough is a prism 2.5 m long. Fig. 12 shows the cross-section of the trough, with the depths in metres at 0.1 m intervals across the trough. The trough is full of water.
      1. Use the trapezium rule with 5 strips to calculate an estimate of the area of cross-section of the trough. Hence estimate the volume of water in the trough.
      2. A computer program models the curve of the base of the trough, with axes as shown and units in metres, using the equation \(y = 8 x ^ { 3 } - 3 x ^ { 2 } - 0.5 x - 0.15\), for \(0 \leqslant x \leqslant 0.5\). Calculate \(\int _ { 0 } ^ { 0.5 } \left( 8 x ^ { 3 } - 3 x ^ { 2 } - 0.5 x - 0.15 \right) \mathrm { d } x\) and state what this represents.
        Hence find the volume of water in the trough as given by this model.
OCR C3 Q1
5 marks Standard +0.3
  1. Solve the inequality $$| x - 0.2 | < 0.03$$
  2. Hence, find all integers \(n\) such that $$\left| 0.95 ^ { n } - 0.2 \right| < 0.03$$
OCR C3 Q2
5 marks Standard +0.3
2.
\includegraphics[max width=\textwidth, alt={}]{d1cf3850-964a-4ff1-ae25-f1bc60a6aded-1_474_823_685_482}
The diagram shows the curve with equation \(y = x \sqrt { 2 - x } , 0 \leq x \leq 2\).
Find, in terms of \(\pi\), the volume of the solid formed when the region bounded by the curve and the \(x\)-axis is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
OCR C3 Q3
6 marks Standard +0.3
3. Solve, for \(0 \leq y \leq 360\), the equation $$2 \cot ^ { 2 } y ^ { \circ } + 5 \operatorname { cosec } y ^ { \circ } + \operatorname { cosec } ^ { 2 } y ^ { \circ } = 0$$
OCR C3 Q4
7 marks Standard +0.3
A curve has the equation \(x = y \sqrt { 1 - 2 y }\).
  1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { \sqrt { 1 - 2 y } } { 1 - 3 y } .$$ The point \(A\) on the curve has \(y\)-coordinate - 1 .
  2. Show that the equation of tangent to the curve at \(A\) can be written in the form $$\sqrt { 3 } x + p y + q = 0$$ where \(p\) and \(q\) are integers to be found.
OCR C3 Q5
7 marks Moderate -0.3
5. The function \(f\) is defined by $$\mathrm { f } ( x ) \equiv 4 - \ln 3 x , \quad x \in \mathbb { R } , \quad x > 0$$
  1. Solve the equation \(\mathrm { f } ( x ) = 0\).
  2. Sketch the curve \(y = \mathrm { f } ( x )\). The function g is defined by $$\mathrm { g } ( x ) \equiv \mathrm { e } ^ { 2 - x } , \quad x \in \mathbb { R }$$
  3. Show that $$\operatorname { fg } ( x ) = x + a - \ln b$$ where \(a\) and \(b\) are integers to be found.
OCR C3 Q6
8 marks Moderate -0.5
6. Find the value of each of the following integrals in exact, simplified form.
  1. \(\quad \int _ { - 1 } ^ { 0 } \mathrm { e } ^ { 1 - 2 x } \mathrm {~d} x\)
  2. \(\int _ { 2 } ^ { 4 } \frac { 3 x ^ { 2 } - 2 } { x } \mathrm {~d} x\)
OCR C3 Q7
10 marks Standard +0.3
7 $$f ( x ) = 2 + \cos x + 3 \sin x$$
  1. Express \(\mathrm { f } ( x )\) in the form $$\mathrm { f } ( x ) = a + b \cos ( x - c )$$ where \(a , b\) and \(c\) are constants, \(b > 0\) and \(0 < c < \frac { \pi } { 2 }\).
  2. Solve the equation \(\mathrm { f } ( x ) = 0\) for \(x\) in the interval \(0 \leq x \leq 2 \pi\).
  3. Use Simpson's rule with four strips, each of width 0.5 , to find an approximate value for $$\int _ { 0 } ^ { 2 } f ( x ) d x$$
OCR C3 Q8
11 marks Moderate -0.3
8. $$\mathrm { f } ( x ) \equiv 2 x ^ { 2 } + 4 x + 2 , \quad x \in \mathbb { R } , \quad x \geq - 1$$
  1. Express \(\mathrm { f } ( x )\) in the form \(a ( x + b ) ^ { 2 } + c\).
  2. Describe fully two transformations that would map the graph of \(y = x ^ { 2 } , x \geq 0\) onto the graph of \(y = \mathrm { f } ( x )\).
  3. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state its domain.
  4. Sketch the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\) on the same diagram and state the relationship between them.
OCR C3 Q9
13 marks Standard +0.3
9. \includegraphics[max width=\textwidth, alt={}, center]{d1cf3850-964a-4ff1-ae25-f1bc60a6aded-3_501_1111_877_413} The diagram shows a graph of the temperature of a room, \(T ^ { \circ } \mathrm { C }\), at time \(t\) minutes.
The temperature is controlled by a thermostat such that when the temperature falls to \(12 ^ { \circ } \mathrm { C }\), a heater is turned on until the temperature reaches \(18 ^ { \circ } \mathrm { C }\). The room then cools until the temperature again falls to \(12 ^ { \circ } \mathrm { C }\). For \(t\) in the interval \(10 \leq t \leq 60 , T\) is given by $$T = 5 + A \mathrm { e } ^ { - k t } ,$$ where \(A\) and \(k\) are constants.
Given that \(T = 18\) when \(t = 10\) and that \(T = 12\) when \(t = 60\),
  1. show that \(k = 0.0124\) to 3 significant figures and find the value of \(A\),
  2. find the rate at which the temperature of the room is decreasing when \(t = 20\). The temperature again reaches \(18 ^ { \circ } \mathrm { C }\) when \(t = 70\) and the graph for \(70 \leq t \leq 120\) is a translation of the graph for \(10 \leq t \leq 60\).
  3. Find the value of the constant \(B\) such that for \(70 \leq t \leq 120\) $$T = 5 + B \mathrm { e } ^ { - k t }$$
OCR MEI C2 Q1
12 marks Moderate -0.3
1 Fig. 10 shows the speed of a car, in metres per second, during one minute, measured at 10 -second intervals. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f56da008-e7f5-45b9-8db8-e2ba09ab0161-1_732_753_302_700} \captionsetup{labelformat=empty} \caption{Fig. 10}
\end{figure} The measured speeds are shown below.
Time \(( t\) seconds \()\)0102030405060
Speed \(\left( v \mathrm {~m} \mathrm {~s} ^ { 1 } \right)\)28191411121622
  1. Use the trapezium rule with 6 strips to find an estimate of the area of the region bounded by the curve, the line \(t = 60\) and the axes. [This area represents the distance travelled by the car.]
  2. Explain why your calculation in part (i) gives an overestimate for this area. Use appropriate rectangles to calculate an underestimate for this area. The speed of the car may be modelled by \(v = 28 - t + 0.015 t ^ { 2 }\).
  3. Show that the difference between the value given by the model when \(t = 10\) and the measured value is less than \(3 \%\) of the measured value.
  4. According to this model, the distance travelled by the car is $$\int _ { 0 } ^ { 60 } \left( 28 \quad t + 0.015 t ^ { 2 } \right) \mathrm { d } t$$ Find this distance.
OCR MEI C2 Q2
3 marks Easy -1.2
2 At a place where a river is 7.5 m wide, its depth is measured every 1.5 m across the river. The table shows the results.
Distance across river \(( \mathrm { m } )\)01.534.567.5
Depth of river \(( \mathrm { m } )\)0.62.33.12.81.80.7
Use the trapezium rule with 5 strips to estimate the area of cross-section of the river.