Questions C2 (1410 questions)

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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 \(n\).
OCR C2 Q1
  1. Giving your answers in terms of \(\pi\), solve the equation
$$3 \tan ^ { 2 } \theta - 1 = 0 ,$$ for \(\theta\) in the interval \(- \pi \leq \theta \leq \pi\).
OCR C2 Q2
2. Given that \(p = \log _ { 2 } 3\) and \(q = \log _ { 2 } 5\), find expressions in terms of \(p\) and \(q\) for
  1. \(\quad \log _ { 2 } 45\),
  2. \(\log _ { 2 } 0.3\)
OCR C2 Q3
3. For the binomial expansion in ascending powers of \(x\) of \(\left( 1 + \frac { 1 } { 4 } x \right) ^ { n }\), where \(n\) is an integer and \(n \geq 2\),
  1. find and simplify the first three terms,
  2. find the value of \(n\) for which the coefficient of \(x\) is equal to the coefficient of \(x ^ { 2 }\).
OCR C2 Q4
4.
\includegraphics[max width=\textwidth, alt={}, center]{30d4e6e5-8235-44b0-ad8e-c4c0b313677f-1_572_803_1336_461} The diagram shows the curves with equations \(y = 7 - 2 x - 3 x ^ { 2 }\) and \(y = \frac { 2 } { x }\).
The two curves intersect at the points \(P , Q\) and \(R\).
  1. Show that the \(x\)-coordinates of \(P , Q\) and \(R\) satisfy the equation $$3 x ^ { 3 } + 2 x ^ { 2 } - 7 x + 2 = 0$$ Given that \(P\) has coordinates \(( - 2 , - 1 )\),
  2. find the coordinates of \(Q\) and \(R\).
OCR C2 Q5
5. The curve \(y = \mathrm { f } ( x )\) passes through the point \(P ( - 1,3 )\) and is such that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 4 } { x ^ { 3 } } , \quad x \neq 0$$
  1. Find \(\mathrm { f } ( x )\).
  2. Show that the area of the finite region bounded by the curve \(y = \mathrm { f } ( x )\), the \(x\)-axis and the lines \(x = 1\) and \(x = 4\) is \(4 \frac { 1 } { 2 }\).
OCR C2 Q6
6.
\includegraphics[max width=\textwidth, alt={}, center]{30d4e6e5-8235-44b0-ad8e-c4c0b313677f-2_577_970_799_360} The diagram shows triangle \(A B C\) in which \(A C = 14 \mathrm {~cm} , B C = 8 \mathrm {~cm}\) and \(\angle A B C = 1.7\) radians.
  1. Find the size of \(\angle A C B\) in radians. The point \(D\) lies on \(A C\) such that \(B D\) is an arc of a circle, centre \(C\).
  2. Find the perimeter of the shaded region bounded by the arc \(B D\) and the straight lines \(A B\) and \(A D\).
OCR C2 Q7
7. (a) Given that \(y = 3 ^ { x }\), find expressions in terms of \(y\) for
  1. \(3 ^ { x + 1 }\),
  2. \(3 ^ { 2 x - 1 }\).
    (b) Hence, or otherwise, solve the equation $$3 ^ { x + 1 } - 3 ^ { 2 x - 1 } = 6$$
OCR C2 Q8
  1. (i) Given that
$$\int _ { 1 } ^ { 3 } \left( x ^ { 2 } - 2 x + k \right) d x = 8 \frac { 2 } { 3 }$$ find the value of the constant \(k\).
(ii) Evaluate $$\int _ { 2 } ^ { \infty } \frac { 6 } { x ^ { \frac { 5 } { 2 } } } \mathrm {~d} x$$ giving your answer in its simplest form.
OCR C2 Q9
9. The second and fifth terms of a geometric series are - 48 and 6 respectively.
  1. Find the first term and the common ratio of the series.
  2. Find the sum to infinity of the series.
  3. Show that the difference between the sum of the first \(n\) terms of the series and its sum to infinity is given by \(2 ^ { 6 - n }\).