Questions — OCR MEI C2 (454 questions)

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OCR MEI C2 2010 June Q7
7 Express \(\log _ { a } x ^ { 3 } + \log _ { a } \sqrt { x }\) in the form \(k \log _ { a } x\).
OCR MEI C2 2010 June Q8
8 Showing your method clearly, solve the equation \(4 \sin ^ { 2 } \theta = 3 + \cos ^ { 2 } \theta\), for values of \(\theta\) between \(0 ^ { \circ }\) and \(360 ^ { \circ }\).
OCR MEI C2 2010 June Q9
9 The points \(( 2,6 )\) and \(( 3,18 )\) lie on the curve \(y = a x ^ { n }\).
Use logarithms to find the values of \(a\) and \(n\), giving your answers correct to 2 decimal places.
OCR MEI C2 2010 June Q11
11
  1. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{e5ac28f3-d61a-4b40-8b47-28c930761a28-4_775_768_260_733} \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]{e5ac28f3-d61a-4b40-8b47-28c930761a28-4_531_1490_1509_363} \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]{e5ac28f3-d61a-4b40-8b47-28c930761a28-5_881_1378_255_379} \captionsetup{labelformat=empty} \caption{Fig. 12}
      \end{figure} A branching plant has stems, nodes, leaves and buds.
      • There are 7 leaves at each node.
  3. From each node, 2 new stems grow.
  4. At the end of each final stem, there is a bud.
    5. Fig. 12 shows one such plant with 3 stages of nodes. It has 15 stems, 7 nodes, 49 leaves and 8 buds.
    (i) One of these plants has 10 stages of nodes.
    (A) How many buds does it have?
    (B) How many stems does it have?
    (ii) (A) Show that the number of leaves on one of these plants with \(n\) stages of nodes is $$7 \left( 2 ^ { n } - 1 \right) .$$ (B) One of these plants has \(n\) stages of nodes and more than 200000 leaves. Show that \(n\) satisfies the inequality \(n > \frac { \log _ { 10 } 200007 - \log _ { 10 } 7 } { \log _ { 10 } 2 }\). Hence find the least possible value of \(n\).