Questions — OCR MEI (4301 questions)

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OCR MEI C2 Q12
12 marks Moderate -0.3
12 Fig. 12 shows a window. The base and sides are parts of a rectangle with dimensions \(2 x\) metres horizontally by \(y\) metres vertically. The top is a semicircle of radius \(x\) metres. The perimeter of the window is 10 metres. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{73d1c02b-1b7b-426d-a171-c762597cfed4-4_428_433_1638_766} \captionsetup{labelformat=empty} \caption{Fig. 12}
\end{figure}
  1. Express \(y\) as a function of \(x\).
  2. Find the total area, \(A \mathrm {~m} ^ { 2 }\), in terms of \(x\) and \(y\). Use your answer to part (i) to show that this simplifies to $$A = 10 x - 2 x ^ { 2 } - \frac { 1 } { 2 } \pi x ^ { 2 }$$
  3. Prove that for the maximum value of \(A\), \(y = x\) exactly.
    \section*{MEI STRUCTURED MATHEMATICS } \section*{CONCEPTS FOR ADVANCED MATHEMATICS, C2} \section*{Practice Paper C2-B
    Insert sheet for question 11} 11 Speed-time graph with the first two points plotted.
    \includegraphics[max width=\textwidth, alt={}, center]{73d1c02b-1b7b-426d-a171-c762597cfed4-5_768_1772_1389_205}
OCR MEI C2 Q1
5 marks Moderate -0.5
1 The common ratio of a geometric progression is - 0.5 . The sum of its first three terms is 15 . Find the first term.
Find also the sum to infinity.
OCR MEI C2 Q2
3 marks Easy -1.2
2 The diagram shows the graph of \(y = \mathrm { f } ( x )\). The graph passes through the point with coordinates \(( 0,2 )\).
\includegraphics[max width=\textwidth, alt={}, center]{1c52d6b5-84b4-455a-9620-c377ae457069-2_524_1350_775_346} On separate diagrams sketch the graphs of the following functions, indicating clearly the point of intersection with the \(y\) axis.
  1. \(\quad y = - \mathrm { f } ( x )\)
  2. \(y = f ( 3 x )\)
OCR MEI C2 Q3
4 marks Moderate -0.8
3 Given that \(A\) is the obtuse angle such that \(\sin A = \frac { 1 } { 5 }\), find the exact value of \(\cos A\).
OCR MEI C2 Q4
5 marks Moderate -0.8
4 You are given that \(y = x ^ { 3 } - 12 x\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Hence find the coordinates of the turning points of the curve.
OCR MEI C2 Q5
5 marks Moderate -0.8
5 A sequence is defined by \(a _ { k } = 5 k + 1\), for \(k = 1,2,3 \ldots\)
  1. Write down the first three terms of the sequence.
  2. Evaluate \(\sum _ { k = 1 } ^ { 100 } a _ { k }\).
OCR MEI C2 Q6
5 marks Standard +0.3
6 Find the solution to this equation, correct to 3 significant figures. $$\left( 2 ^ { x } \right) \left( 2 ^ { x + 1 } \right) = 10 .$$
OCR MEI C2 Q7
5 marks Moderate -0.8
7 The gradient of a curve \(y = \mathrm { f } ( x )\) is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } - 10 x + 6\). The curve passes through the point \(( 2,3 )\) Find the equation of the curve.
OCR MEI C2 Q8
4 marks Moderate -0.8
8 In the triangle ABC shown, \(\mathrm { AB } = 8 \mathrm {~cm}\). \(\mathrm { AC } = 12 \mathrm {~cm}\) and angle \(\mathrm { ABC } = 82 ^ { \circ }\). Find \(\theta\) correct to 3 significant figures.
\includegraphics[max width=\textwidth, alt={}, center]{1c52d6b5-84b4-455a-9620-c377ae457069-3_382_540_1492_718}
OCR MEI C2 Q9
12 marks Moderate -0.3
9 Fig. 9 shows
\(P \quad\) The line \(y = x\)
\(Q\) The curve \(y = \sqrt { \frac { 1 } { 2 } \left( x + x ^ { 2 } \right) }\)
\(R \quad\) The curve \(\quad y = \sqrt { x }\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1c52d6b5-84b4-455a-9620-c377ae457069-4_471_1103_762_374} \captionsetup{labelformat=empty} \caption{Fig. 9}
\end{figure}
  1. Write down the area of the triangle formed by the line \(y = x\), the line \(x = 1\) and the \(x\)-axis.
  2. Show that the area of the region formed by the curve \(y = \sqrt { x }\), the line \(x = 1\) and the \(x\)-axis is \(\frac { 2 } { 3 }\). An estimate is required of the Area, \(A\), of the region formed by the curve \(y = \sqrt { \frac { 1 } { 2 } \left( x + x ^ { 2 } \right) }\), the line \(x = 1\) and the \(x\)-axis.
  3. Use results to parts (i) and (ii) to complete the statement $$\ldots \ldots \ldots \ldots . . < A < \ldots \ldots \ldots \ldots \ldots . .$$
  4. Use the Trapezium Rule with 4 strips to find an estimate for \(A\).
  5. Draw a sketch of Fig. 9. Use it to illustrate the area found as the trapezium rule estimate for \(A\).
    Explain how your diagram shows that the trapezium rule estimate must be:
    consistent with the answer to part (iv);
    an under-estimate for A .
OCR MEI C2 Q10
12 marks Moderate -0.5
10 A culture of bacteria is observed during an experiment. The number of bacteria is denoted by \(N\) and the time in hours after the start of the experiment by \(t\).
The table gives observations of \(t\) and \(N\).
Time \(( t\) hours \()\)12345
Number of bacteria \(( N )\)120170250370530
  1. Plot the points \(( t , N )\) on graph paper and join them with a smooth curve.
  2. Explain why the curve suggests why the relationship connecting \(t\) and \(N\) may be of the form \(N = a b ^ { t }\).
  3. Explain how, by using logarithms, the curve given by plotting \(N\) against \(t\) can be transformed into a straight line.
    State the gradient of this straight line and its intercept with the vertical axis in terms of \(a\) and \(b\).
  4. Complete a table of values for \(\log _ { 10 } N\) and plot the points \(\left( t , \log _ { 10 } N \right)\) on graph paper. Draw the best fit line through the points and use it to estimate the values of \(a\) and \(b\).
OCR MEI C2 Q11
12 marks Challenging +1.2
11 The sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots \ldots\) where \(a _ { 1 }\) is a given real number is defined by \(a _ { n + 1 } = 1 - \frac { 1 } { a _ { n } }\).
  1. For the case when \(a _ { 1 } = 2\), find \(a _ { 2 } , a _ { 3 }\) and \(a _ { 4 }\). Describe the behaviour of this sequence
  2. For the case when \(a _ { 1 } = k\), where \(k\) is an integer greater than 1 , find \(a _ { 2 }\) in terms of \(k\) as a single fraction.
    Find also \(a _ { 3 }\) in its simplest form and hence deduce that \(a _ { 4 } = k\).
  3. Show that \(a _ { 2 } a _ { 3 } a _ { 4 } = - 1\) for any integer \(k\).
  4. When \(a _ { 1 } = 2\) evaluate \(\sum _ { i = 1 } ^ { 99 } a _ { i }\).
OCR MEI C2 Q1
5 marks Moderate -0.8
1
  1. Find \(\int \left( x ^ { 3 } - 2 x \right) \mathrm { d } x\). The graph below shows part of the curve \(y = x ^ { 3 } - 2 x\) for \(0 \leq x \leq 2\).
    \includegraphics[max width=\textwidth, alt={}, center]{c55a5f04-3573-4f36-a12c-3755bdd4a45b-2_528_1019_520_321}
  2. Show that the area of the shaded region \(P\) is the same as the area of the shaded region \(Q\).
OCR MEI C2 Q2
4 marks Moderate -0.8
2 The growth in population \(P\) of a certain town after time \(t\) years can be modelled by the equation \(P = 11000 \times 10 ^ { k t }\) where \(k\) is a constant.
  1. State the initial population of the town.
  2. After three years the population of the town is 24000 . Use this information to find the value of \(k\) correct to two decimal places.
OCR MEI C2 Q3
4 marks Moderate -0.8
3
  1. Write \(\log _ { 10 } ( x + 4 ) - 2 \log _ { 10 } x + \log _ { 10 } ( x + 16 )\) as a single logarithm.
  2. Without using your calculator, verify that \(x = 4\) is a root of the equation $$\log _ { 10 } ( x + 4 ) - 2 \log _ { 10 } x + \log _ { 10 } ( x + 16 ) = 1$$
OCR MEI C2 Q4
3 marks Moderate -0.3
4 Find the values of \(\theta\) such that \(0 ^ { \circ } \leq \theta \leq 360 ^ { \circ }\) which satisfy the equation $$\cos \theta \tan \theta = \frac { \sqrt { 3 } } { 2 }$$
OCR MEI C2 Q5
5 marks Moderate -0.8
5 The diagram shows the curve \(y = \mathrm { f } ( x )\) where \(a\) is a positive constant.
\includegraphics[max width=\textwidth, alt={}, center]{c55a5f04-3573-4f36-a12c-3755bdd4a45b-3_551_962_255_476} Sketch the following curves on separate diagrams, in each case stating the coordinates of points where they meet the \(x\) - and \(y\)-axes.
  1. \(\quad y = - \mathrm { f } ( x )\)
  2. \(\quad y = \mathrm { f } ( - x )\)
OCR MEI C2 Q6
5 marks Standard +0.3
6 A and B are points on the same side of a straight river. A and B are 180 metres apart. The angles made with a jetty J on the opposite side of the river \(78 ^ { \circ }\) and \(56 ^ { \circ }\) respectively as shown.
\includegraphics[max width=\textwidth, alt={}, center]{c55a5f04-3573-4f36-a12c-3755bdd4a45b-3_332_681_1451_565} Not to scale Calculate the width of the river correct to the nearest metre.
OCR MEI C2 Q7
5 marks Standard +0.3
7 For each of the following sequences, write down sufficient terms of the sequence in order to be able to describe its behaviour as divergent, periodic or convergent. For any convergent sequence, state its limit.
  1. \(a _ { 1 } = - 1 ; \quad a _ { k + 1 } = \frac { 4 } { a _ { k } }\)
  2. \(\quad a _ { 1 } = 1 ; \quad a _ { k } = 2 - 2 \times \left( \frac { 1 } { 2 } \right) ^ { k }\)
  3. \(\quad a _ { 1 } = 0 \quad a _ { k + 1 } = \left( 1 + a _ { k } \right) ^ { 2 }\).
OCR MEI C2 Q8
5 marks Moderate -0.8
8 Fig. 8 shows a sector of a circle with centre O and radius 6 cm and a chord AB which subtends an angle of 1.8 radians at O . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c55a5f04-3573-4f36-a12c-3755bdd4a45b-4_341_485_310_771} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure}
  1. Calculate the area of the sector OAXB .
  2. Calculate the area of the triangle OAB and hence find the area of the shaded segment AXB.
OCR MEI C2 Q9
12 marks Standard +0.3
9 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } - 12 x + 9\). The curve passes through the point \(( 2 , - 2 )\).
  1. Find the equation of the curve.
  2. Show that the curve touches the \(x\)-axis at one point (A) and cuts it at another (B). State the coordinates of A and B.
  3. The curve cuts the \(y\)-axis at C . Show that the tangent at C is perpendicular to the normal at B.
OCR MEI C2 Q10
12 marks Moderate -0.3
10 Fig. 10 shows the curve with equation \(y = x ^ { 2 } + \frac { 16 } { x }\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c55a5f04-3573-4f36-a12c-3755bdd4a45b-5_522_1019_403_394} \captionsetup{labelformat=empty} \caption{Fig. 10}
\end{figure}
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Hence calculate the coordinates of the stationary point on the curve.
  3. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) and explain why this confirms that he stationary point is a minimum.
  4. Using the trapezium rule with 4 intervals, estimate the area between the curve and the \(x\) axis between \(x = 2\) and \(x = 4\).
  5. State, giving a reason, whether this estimate of the area under-estimates or over-estimates the true area beneath the curve.
OCR MEI C2 Q11
12 marks Moderate -0.8
11 When Fred joined a computer firm his salary was \(\pounds 28000\) per annum. In each subsequent year he received an annual increase of \(12 \%\) of his previous year's salary.
  1. State Fred's salary for each of his first 3 years with the company. State also the common ratio of the geometric sequence formed by his salaries.
  2. How much did Fred earn in the tenth year?
  3. Show that the total amount Fred earned over the ten years was between \(\pounds 400000\) and £500000.
  4. When Fred joined the computer firm, his brother Archie set up a plumbing business. He earned \(\pounds 35000\) in his first year and each year earned \(\pounds d\) more than in the previous year. At the end of ten years, he had earned exactly the same total amount as Fred. Calculate the value of \(d\).
OCR MEI M1 2005 January Q1
7 marks Moderate -0.8
1 The position vector, \(\mathbf { r }\), of a particle of mass 4 kg at time \(t\) is given by $$\mathbf { r } = t ^ { 2 } \mathbf { i } + \left( 5 t - 2 t ^ { 2 } \right) \mathbf { j } ,$$ where \(\mathbf { i }\) and \(\mathbf { j }\) are the standard unit vectors, lengths are in metres and time is in seconds.
  1. Find an expression for the acceleration of the particle. The particle is subject to a force \(\mathbf { F }\) and a force \(12 \mathbf { j } \mathbf { N }\).
  2. Find \(\mathbf { F }\).
OCR MEI M1 2005 January Q2
8 marks Standard +0.3
2 Particles of mass 2 kg and 4 kg are attached to the ends \(X\) and \(Y\) of a light, inextensible string. The string passes round fixed, smooth pulleys at \(\mathrm { P } , \mathrm { Q }\) and R , as shown in Fig. 2. The system is released from rest with the string taut. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c84a748a-a6f4-48c5-b864-fe543569bdf5-2_478_397_1211_872} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure}
  1. State what information in the question tells you that
    (A) the tension is the same throughout the string,
    (B) the magnitudes of the accelerations of the particles at X and Y are the same. The tension in the string is \(T \mathrm {~N}\) and the magnitude of the acceleration of the particles is \(a \mathrm {~ms} ^ { - 2 }\).
  2. Draw a diagram showing the forces acting at X and a diagram showing the forces acting at Y .
  3. Write down equations of motion for the particles at X and at Y . Hence calculate the values of \(T\) and \(a\).