Questions — OCR MEI (4301 questions)

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OCR MEI C2 2005 January Q10
Moderate -0.8
10 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 2005 January Q11
Moderate -0.3
11 Answer part (iii) of this question on the insert provided.
A hot drink is made and left to cool. The table shows its temperature at ten-minute intervals after it is made.
Time (minutes)1020304050
Temperature \(\left( { } ^ { \circ } \mathrm { C } \right)\)6853423631
The room temperature is \(22 ^ { \circ } \mathrm { C }\). The difference between the temperature of the drink and room temperature at time \(t\) minutes is \(z ^ { \circ } \mathrm { C }\). The relationship between \(z\) and \(t\) is modelled by $$z = z _ { 0 } 10 ^ { - k t }$$ where \(z _ { 0 }\) and \(k\) are positive constants.
  1. Give a physical interpretation for the constant \(z _ { 0 }\).
  2. Show that \(\log _ { 10 } z = - k t + \log _ { 10 } z _ { 0 }\).
  3. On the insert, complete the table and draw the graph of \(\log _ { 10 } z\) against \(t\). Use your graph to estimate the values of \(k\) and \(z _ { 0 }\).
    Hence estimate the temperature of the drink 70 minutes after it is made.
OCR MEI C2 2006 January Q1
Easy -1.8
1 Given that \(140 ^ { \circ } = k \pi\) radians, find the exact value of \(k\).
OCR MEI C2 2006 January Q2
Easy -1.8
2 Find the numerical value of \(\sum _ { k = 2 } ^ { 5 } k ^ { 3 }\).
OCR MEI C2 2006 January Q3
Easy -1.8
3 Fig. 3 Beginning with the triangle shown in Fig. 3, prove that \(\sin 60 ^ { \circ } = \frac { \sqrt { 3 } } { 2 }\).
OCR MEI C2 2006 January Q4
Moderate -0.8
4 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{15b8f97b-c058-409f-907f-cb0a6102abc4-2_615_971_1457_539} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure} Fig. 4 shows a curve which passes through the points shown in the following table.
\(x\)11.522.533.54
\(y\)8.26.45.55.04.74.44.2
Use the trapezium rule with 6 strips to estimate the area of the region bounded by the curve, the lines \(x = 1\) and \(x = 4\), and the \(x\)-axis. State, with a reason, whether the trapezium rule gives an overestimate or an underestimate of the area of this region.
OCR MEI C2 2006 January Q5
Moderate -0.8
5
  1. Sketch the graph of \(y = \tan x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
  2. Solve the equation \(4 \sin x = 3 \cos x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
OCR MEI C2 2006 January Q6
Moderate -0.3
6 A curve has gradient given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = x ^ { 2 } - 6 x + 9\). Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
Show that the curve has a stationary point of inflection when \(x = 3\).
OCR MEI C2 2006 January Q7
Easy -1.2
7 In Fig. 7, A and B are points on the circumference of a circle with centre O . Angle \(\mathrm { AOB } = 1.2\) radians. The arc length AB is 6 cm . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{15b8f97b-c058-409f-907f-cb0a6102abc4-3_371_723_1048_833} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure}
  1. Calculate the radius of the circle.
  2. Calculate the length of the chord AB .
OCR MEI C2 2006 January Q8
Easy -1.2
8 Find \(\int \left( x ^ { \frac { 1 } { 2 } } + \frac { 6 } { x ^ { 3 } } \right) \mathrm { d } x\).
OCR MEI C2 2006 January Q9
Moderate -0.5
9 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{15b8f97b-c058-409f-907f-cb0a6102abc4-4_591_985_312_701} \captionsetup{labelformat=empty} \caption{Fig. 9}
\end{figure} The graph of \(\log _ { 10 } y\) against \(x\) is a straight line as shown in Fig. 9 .
  1. Find the equation for \(\log _ { 10 } y\) in terms of \(x\).
  2. Find the equation for \(y\) in terms of \(x\). Section B (36 marks)
OCR MEI C2 2006 January Q10
Moderate -0.8
10 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 \(y = 12\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{15b8f97b-c058-409f-907f-cb0a6102abc4-5_643_1034_331_513} \captionsetup{labelformat=empty} \caption{Fig. 11}
    \end{figure} The equation of the curve shown in Fig. 11 is \(y = x ^ { 3 } - 6 x + 2\).
OCR MEI C2 2007 January Q1
Easy -1.8
1 Differentiate \(6 x ^ { \frac { 5 } { 2 } } + 4\).
OCR MEI C2 2007 January Q2
Moderate -0.8
2 A geometric progression has 6 as its first term. Its sum to infinity is 5 .
Calculate its common ratio.
OCR MEI C2 2007 January Q3
Moderate -0.8
3 Given that \(\cos \theta = \frac { 1 } { 3 }\) and \(\theta\) is acute, find the exact value of \(\tan \theta\).
OCR MEI C2 2007 January Q4
Moderate -0.8
4 Sequences \(\mathrm { A } , \mathrm { B }\) and C are shown below. They each continue in the pattern established by the given terms. $$\begin{array} { l l l l l l l l l } \text { A: } & 1 , & 2 , & 4 , & 16 , & 32 , & \ldots & \\ \text { B: } & 20 , & - 10 , & 5 , & - 2.5 , & 1.25 , & - 0.625 , & \ldots \\ \text { C: } & 20 , & 5 , & 1 , & 20 , & 5 , & 1 , & \ldots \end{array}$$
  1. Which of these sequences is periodic?
  2. Which of these sequences is convergent?
  3. Find, in terms of \(n\), the \(n\)th term of sequence A .
OCR MEI C2 2007 January Q5
Moderate -0.8
5 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 2007 January Q6
Easy -1.2
6 Sketch the curve \(y = \sin x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
Solve the equation \(\sin x = - 0.68\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
OCR MEI C2 2007 January Q7
Moderate -0.8
7 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 2007 January Q8
Moderate -0.3
8 The 7th term of an arithmetic progression is 6. The sum of the first 10 terms of the progression is 30. Find the 5th term of the progression.
OCR MEI C2 2007 January Q9
Easy -1.2
9 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 2007 January Q10
Easy -1.2
10
  1. Express \(\log _ { a } x ^ { 4 } + \log _ { a } \left( \frac { 1 } { x } \right)\) as a multiple of \(\log _ { a } x\).
  2. Given that \(\log _ { 10 } b + \log _ { 10 } c = 3\), find \(b\) in terms of \(c\).
OCR MEI C2 2007 January Q11
Standard +0.3
11 Fig. 11.1 shows a village green which is bordered by 3 straight roads \(A B , B C\) and \(C A\). The road AC runs due North and the measurements shown are in metres. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b4c0b4b0-f13c-49a9-9f98-f86f28d1f577-4_460_1143_486_591} \captionsetup{labelformat=empty} \caption{Fig. 11.1}
\end{figure}
  1. Calculate the bearing of B from C , giving your answer to the nearest \(0.1 ^ { \circ }\).
  2. Calculate the area of the village green. The road AB is replaced by a new road, as shown in Fig. 11.2. The village green is extended up to the new road. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{b4c0b4b0-f13c-49a9-9f98-f86f28d1f577-4_440_1002_1436_737} \captionsetup{labelformat=empty} \caption{Fig. 11.2}
    \end{figure} The new road is an arc of a circle with centre O and radius 130 m .
  3. (A) Show that angle AOB is 1.63 radians, correct to 3 significant figures.
    (B) Show that the area of land added to the village green is \(5300 \mathrm {~m} ^ { 2 }\) correct to 2 significant figures.
OCR MEI C2 2007 January Q12
Moderate -0.8
12 Fig. 12 is a sketch of the curve \(y = 2 x ^ { 2 } - 11 x + 12\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b4c0b4b0-f13c-49a9-9f98-f86f28d1f577-5_478_951_333_792} \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 2008 January Q1
Easy -1.8
1 Differentiate \(10 x ^ { 4 } + 12\).