Questions C2 (1550 questions)

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OCR MEI C2 2005 June Q4
5 marks Moderate -0.8
4 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 2005 June Q5
5 marks Easy -1.8
5
  1. Write down the value of \(\log _ { 5 } 5\).
  2. Find \(\log _ { 3 } \left( \frac { 1 } { 9 } \right)\).
  3. Express \(\log _ { a } x + \log _ { a } \left( x ^ { 5 } \right)\) as a multiple of \(\log _ { a } x\).
OCR MEI C2 2005 June Q6
5 marks Moderate -0.8
6 Sketch the graph of \(y = 2 ^ { x }\).
Solve the equation \(2 ^ { x } = 50\), giving your answer correct to 2 decimal places.
OCR MEI C2 2005 June Q7
5 marks Easy -1.2
7 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 2005 June Q8
5 marks Moderate -0.8
8
  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 2005 June Q9
13 marks Standard +0.3
9 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{faeaf2aa-ed4e-4926-b402-40c4c9aad479-3_535_790_450_630} \captionsetup{labelformat=empty} \caption{Fig. 9}
\end{figure} Fig. 9 shows a sketch of the graph of \(y = x ^ { 3 } - 10 x ^ { 2 } + 12 x + 72\).
  1. Write down \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Find the equation of the tangent to the curve at the point on the curve where \(x = 2\).
  3. Show that the curve crosses the \(x\)-axis at \(x = - 2\). Show also that the curve touches the \(x\)-axis at \(x = 6\).
  4. Find the area of the finite region bounded by the curve and the \(x\)-axis, shown shaded in Fig. 9 . [4]
OCR MEI C2 2005 June Q10
12 marks Standard +0.3
10 Arrowline Enterprises is considering two possible logos: \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{faeaf2aa-ed4e-4926-b402-40c4c9aad479-4_1123_1676_356_230} \captionsetup{labelformat=empty} \caption{Fig. 10.1}
\end{figure} Fig. 10.2
  1. Fig. 10.1 shows the first logo ABCD . It is symmetrical about AC . Find the length of AB and hence find the area of this logo.
  2. Fig. 10.2 shows a circle with centre O and radius 12.6 cm . ST and RT are tangents to the circle and angle SOR is 1.82 radians. The shaded region shows the second logo. Show that \(\mathrm { ST } = 16.2 \mathrm {~cm}\) to 3 significant figures.
    Find the area and perimeter of this logo.
OCR MEI C2 2005 June Q11
10 marks Standard +0.3
11 There is a flowerhead at the end of each stem of an oleander plant. The next year, each flowerhead is replaced by three stems and flowerheads, as shown in Fig. 11. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{faeaf2aa-ed4e-4926-b402-40c4c9aad479-5_501_1102_431_504} \captionsetup{labelformat=empty} \caption{Fig. 11}
\end{figure}
  1. How many flowerheads are there in year 5?
  2. How many flowerheads are there in year \(n\) ?
  3. As shown in Fig. 11, the total number of stems in year 2 is 4, (that is, 1 old one and 3 new ones). Similarly, the total number of stems in year 3 is 13 , (that is, \(1 + 3 + 9\) ). Show that the total number of stems in year \(n\) is given by \(\frac { 3 ^ { n } - 1 } { 2 }\).
  4. Kitty's oleander has a total of 364 stems. Find
    (A) its age,
    (B) how many flowerheads it has.
  5. Abdul's oleander has over 900 flowerheads. Show that its age, \(y\) years, satisfies the inequality \(y > \frac { \log _ { 10 } 900 } { \log _ { 10 } 3 } + 1\).
    Find the smallest integer value of \(y\) for which this is true.
OCR MEI C2 2007 June Q1
3 marks Easy -1.3
1
  1. State the exact value of \(\tan 300 ^ { \circ }\).
  2. Express \(300 ^ { \circ }\) in radians, giving your answer in the form \(k \pi\), where \(k\) is a fraction in its lowest terms.
OCR MEI C2 2007 June Q2
5 marks Easy -1.3
2 Given that \(y = 6 x ^ { \frac { 3 } { 2 } }\), 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 2007 June Q3
4 marks Easy -1.2
3 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2bdf241f-4538-4227-ba00-fe843d1b3aca-2_830_1393_959_334} \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 2007 June Q4
4 marks Easy -1.3
4
  1. Find the second and third terms of the sequence defined by the following. $$\begin{aligned} t _ { n + 1 } & = 2 t _ { n } + 5 \\ t _ { 1 } & = 3 \end{aligned}$$
  2. Find \(\sum _ { k = 1 } ^ { 3 } k ( k + 1 )\).
OCR MEI C2 2007 June Q5
5 marks Moderate -0.8
5 A sector of a circle of radius 5 cm has area \(9 \mathrm {~cm} ^ { 2 }\).
Find, in radians, the angle of the sector.
Find also the perimeter of the sector.
OCR MEI C2 2007 June Q6
5 marks Moderate -0.8
6
  1. Write down the values of \(\log _ { a } 1\) and \(\log _ { a } a\), where \(a > 1\).
  2. Show that \(\log _ { a } x ^ { 10 } - 2 \log _ { a } \left( \frac { x ^ { 3 } } { 4 } \right) = 4 \log _ { a } ( 2 x )\).
OCR MEI C2 2007 June Q7
5 marks Easy -1.2
7
  1. Sketch the graph of \(y = 3 ^ { x }\).
  2. Use logarithms to solve the equation \(3 ^ { x } = 20\). Give your answer correct to 2 decimal places.
OCR MEI C2 2007 June Q8
5 marks Moderate -0.8
8
  1. Show that the equation \(2 \cos ^ { 2 } \theta + 7 \sin \theta = 5\) may be written in the form $$2 \sin ^ { 2 } \theta - 7 \sin \theta + 3 = 0$$
  2. By factorising this quadratic equation, solve the equation for values of \(\theta\) between \(0 ^ { \circ }\) and \(180 ^ { \circ }\). Section B (36 marks)
OCR MEI C2 2007 June Q9
12 marks Moderate -0.3
9 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 2007 June Q10
12 marks Moderate -0.3
10 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]{2bdf241f-4538-4227-ba00-fe843d1b3aca-4_732_748_379_657} \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 $$\int _ { 0 } ^ { 60 } \left( 28 - t + 0.015 t ^ { 2 } \right) \mathrm { d } t$$ Find this distance.
OCR MEI C2 2007 June Q11
12 marks Moderate -0.3
11
  1. André is playing a game where he makes piles of counters. He puts 3 counters in the first pile. Each successive pile he makes has 2 more counters in it than the previous one.
    1. How many counters are there in his sixth pile?
    2. André makes ten piles of counters. How many counters has he used altogether?
  2. In another game, played with an ordinary fair die and counters, Betty needs to throw a six to start. The probability \(\mathrm { P } _ { n }\) of Betty starting on her \(n\)th throw is given by $$P _ { n } = \frac { 1 } { 6 } \times \left( \frac { 5 } { 6 } \right) ^ { n - 1 }$$
    1. Calculate \(\mathrm { P } _ { 4 }\). Give your answer as a fraction.
    2. The values \(\mathrm { P } _ { 1 } , \mathrm { P } _ { 2 } , \mathrm { P } _ { 3 } , \ldots\) form an infinite geometric progression. State the first term and the common ratio of this progression. Hence show that \(\mathrm { P } _ { 1 } + \mathrm { P } _ { 2 } + \mathrm { P } _ { 3 } + \ldots = 1\).
    3. Given that \(\mathrm { P } _ { n } < 0.001\), show that \(n\) satisfies the inequality $$n > \frac { \log _ { 10 } 0.006 } { \log _ { 10 } \left( \frac { 5 } { 6 } \right) } + 1$$ Hence find the least value of \(n\) for which \(\mathrm { P } _ { n } < 0.001\).
OCR MEI C2 2009 June Q1
2 marks Easy -1.8
1 Use an isosceles right-angled triangle to show that \(\cos 45 ^ { \circ } = \frac { 1 } { \sqrt { 2 } }\).
OCR MEI C2 2009 June Q2
4 marks Easy -1.2
2 Find \(\int _ { 1 } ^ { 2 } \left( 12 x ^ { 5 } + 5 \right) \mathrm { d } x\).
OCR MEI C2 2009 June Q3
3 marks Moderate -0.8
3
  1. Find \(\sum _ { k = 3 } ^ { 8 } \left( k ^ { 2 } - 1 \right)\).
  2. State whether the sequence with \(k\) th term \(k ^ { 2 } - 1\) is convergent or divergent, giving a reason for your answer.
OCR MEI C2 2009 June Q4
4 marks Easy -1.2
4 A sector of a circle of radius 18.0 cm has arc length 43.2 cm .
  1. Find in radians the angle of the sector.
  2. Find this angle in degrees, giving your answer to the nearest degree.
OCR MEI C2 2009 June Q5
5 marks Moderate -0.8
5
  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 2009 June Q6
5 marks Moderate -0.8
6 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.