Questions — OCR MEI (4301 questions)

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OCR MEI C2 2008 January Q2
Easy -1.8
2 A sequence begins $$\begin{array} { l l l l l l l l l l l l } 1 & 2 & 3 & 4 & 5 & 1 & 2 & 3 & 4 & 5 & 1 & \ldots \end{array}$$ and continues in this pattern.
  1. Find the 48th term of this sequence.
  2. Find the sum of the first 48 terms of this sequence.
OCR MEI C2 2008 January Q3
Moderate -0.8
3 You are given that \(\tan \theta = \frac { 1 } { 2 }\) and the angle \(\theta\) is acute. Show, without using a calculator, that \(\cos ^ { 2 } \theta = \frac { 4 } { 5 }\).
OCR MEI C2 2008 January Q4
Moderate -0.8
4 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{15872003-2e41-47e9-a5bd-34e533768f8a-2_625_869_1155_639} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure} Fig. 4 shows a sketch of the graph of \(y = \mathrm { f } ( x )\). On separate diagrams, sketch the graphs of the following, showing clearly the coordinates of the points corresponding to \(\mathrm { A } , \mathrm { B }\) and C .
  1. \(y = 2 \mathrm { f } ( x )\)
  2. \(y = \mathrm { f } ( x + 3 )\)
OCR MEI C2 2008 January Q5
Easy -1.3
5 Find \(\int \left( 12 x ^ { 5 } + \sqrt [ 3 ] { x } + 7 \right) \mathrm { d } x\).
OCR MEI C2 2008 January Q6
Easy -1.2
6
  1. Sketch the graph of \(y = \sin \theta\) for \(0 \leqslant \theta \leqslant 2 \pi\).
  2. Solve the equation \(2 \sin \theta = - 1\) for \(0 \leqslant \theta \leqslant 2 \pi\). Give your answers in the form \(k \pi\).
OCR MEI C2 2008 January Q7
Easy -1.3
7
  1. Find \(\sum _ { k = 2 } ^ { 5 } 2 ^ { k }\).
  2. Find the value of \(n\) for which \(2 ^ { n } = \frac { 1 } { 64 }\).
  3. Sketch the curve with equation \(y = 2 ^ { x }\).
OCR MEI C2 2008 January Q8
Moderate -0.5
8 The second term of a geometric progression is 18 and the fourth term is 2 . The common ratio is positive. Find the sum to infinity of this progression.
OCR MEI C2 2008 January Q9
Moderate -0.3
9 You are given that \(\log _ { 10 } y = 3 x + 2\).
  1. Find the value of \(x\) when \(y = 500\), giving your answer correct to 2 decimal places.
  2. Find the value of \(y\) when \(x = - 1\).
  3. Express \(\log _ { 10 } \left( y ^ { 4 } \right)\) in terms of \(x\).
  4. Find an expression for \(y\) in terms of \(x\). Section B (36 marks)
OCR MEI C2 2008 January Q10
Moderate -0.3
10 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{15872003-2e41-47e9-a5bd-34e533768f8a-3_501_493_1434_826} \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 2008 January Q11
Standard +0.3
11
  1. The course for a yacht race is a triangle, as shown in Fig. 11.1. The yachts start at A , then travel to B , then to C and finally back to A . \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{15872003-2e41-47e9-a5bd-34e533768f8a-4_661_869_404_680} \captionsetup{labelformat=empty} \caption{Fig. 11.1}
    \end{figure} (A) Calculate the total length of the course for this race.
    (B) Given that the bearing of the first stage, AB , is \(175 ^ { \circ }\), calculate the bearing of the second stage, BC.
  2. Fig. 11.2 shows the course of another yacht race. The course follows the arc of a circle from P to \(Q\), then a straight line back to \(P\). The circle has radius 120 m and centre \(O\); angle \(P O Q = 136 ^ { \circ }\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{15872003-2e41-47e9-a5bd-34e533768f8a-4_709_821_1603_703} \captionsetup{labelformat=empty} \caption{Fig. 11.2}
    \end{figure} Calculate the total length of the course for this race.
OCR MEI C2 2008 January Q12
Moderate -0.8
12
  1. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{15872003-2e41-47e9-a5bd-34e533768f8a-5_652_764_269_733} \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.
  2. You are given that \(\mathrm { f } ( x ) = x ^ { 4 }\).
    (A) Complete this identity for \(\mathrm { f } ( x + h )\). $$f ( x + h ) = ( x + h ) ^ { 4 } = x ^ { 4 } + 4 x ^ { 3 } h + \ldots$$ (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 2005 June Q1
Easy -1.2
1 Differentiate \(x + \sqrt { x ^ { 3 } }\).
OCR MEI C2 2005 June Q2
Easy -1.2
2 The \(n\)th term of an arithmetic progression is \(6 + 5 n\). Find the sum of the first 20 terms.
OCR MEI C2 2005 June Q3
Moderate -0.8
3 Given that \(\sin \theta = \frac { \sqrt { 3 } } { 4 }\), find in surd form the possible values of \(\cos \theta\).
OCR MEI C2 2005 June Q4
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
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
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
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
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
4 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
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
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 2006 June Q1
Easy -2.0
1 Write down the values of \(\log _ { a } a\) and \(\log _ { a } \left( a ^ { 3 } \right)\).
OCR MEI C2 2006 June Q2
Easy -1.2
2 The first term of a geometric series is 8 . The sum to infinity of the series is 10 .
Find the common ratio.
\(3 \theta\) is an acute angle and \(\sin \theta = \frac { 1 } { 4 }\). Find the exact value of \(\tan \theta\).
OCR MEI C2 2006 June Q4
Easy -1.2
4 Find \(\int _ { 1 } ^ { 2 } \left( x ^ { 4 } - \frac { 3 } { x ^ { 2 } } + 1 \right) \mathrm { d } x\), showing your working.