Edexcel P2 (Pure Mathematics 2) 2020 January

1. The table below shows corresponding values of \(x\) and \(y\) for \(y = \log _ { 2 } ( 2 x )\)

The values of \(y\) are given to 2 decimal places as appropriate. Using the trapezium rule with all the values of \(y\) in the given table,

1. obtain an estimate for \(\int _ { 2 } ^ { 14 } \log _ { 2 } ( 2 x ) \mathrm { d } x\), giving your answer to one decimal place. Using your answer to part (a) and making your method clear, estimate
2. 1. \(\quad \int _ { 2 } ^ { 14 } \frac { \log _ { 2 } \left( 4 x ^ { 2 } \right) } { 5 } \mathrm {~d} x\)
   2. \(\int _ { 2 } ^ { 14 } \log _ { 2 } \left( \frac { 2 } { x } \right) \mathrm { d } x\)

      \(x\)	2	5	8	11	14
      \(y\)	2	3.32	4	4.46	4.81

2. One of the terms in the binomial expansion of \(( 3 + a x ) ^ { 6 }\), where \(a\) is a constant, is \(540 x ^ { 4 }\)

1. Find the possible values of \(a\).
2. Hence find the term independent of \(x\) in the expansion of $$\left( \frac { 1 } { 81 } + \frac { 1 } { x ^ { 6 } } \right) ( 3 + a x ) ^ { 6 }$$

3. $$f ( x ) = 6 x ^ { 3 } + 17 x ^ { 2 } + 4 x - 12$$

1. Use the factor theorem to show that ( \(2 x + 3\) ) is a factor of \(\mathrm { f } ( x )\).
2. Hence, using algebra, write \(\mathrm { f } ( x )\) as a product of three linear factors.
3. Solve, for \(\frac { \pi } { 2 } < \theta < \pi\), the equation $$6 \tan ^ { 3 } \theta + 17 \tan ^ { 2 } \theta + 4 \tan \theta - 12 = 0$$ giving your answers to 3 significant figures.

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation $$y = 2 x ^ { 2 } + 7 \quad x \geqslant 0$$ The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the \(y\)-axis and the line with equation \(y = 17\) Find the exact area of \(R\).

5. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. A colony of bees is being studied. The number of bees in the colony at the start of the study was 30000 Three years after the start of the study, the number of bees in the colony is 34000 A model predicts that the number of bees in the colony will increase by \(p \%\) each year, so that the number of bees in the colony at the end of each year of study forms a geometric sequence. Assuming the model,

1. find the value of \(p\), giving your answer to 2 decimal places. According to the model, at the end of \(N\) years of study the number of bees in the colony exceeds 75000
2. Find, showing all steps in your working, the smallest integer value of \(N\).

6. The circle \(C\) has equation $$x ^ { 2 } + y ^ { 2 } + 6 x - 4 y - 14 = 0$$

1. Find
   1. the coordinates of the centre of \(C\),
   2. the exact radius of \(C\). The line with equation \(y = k\), where \(k\) is a constant, is a tangent to \(C\).
2. Find the possible values of \(k\). The line with equation \(y = p\), where \(p\) is a negative constant, is a chord of \(C\).
   Given that the length of this chord is 4 units,
3. find the value of \(p\).
   

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7. (a) Show that the equation $$8 \tan \theta = 3 \cos \theta$$ may be rewritten in the form $$3 \sin ^ { 2 } \theta + 8 \sin \theta - 3 = 0$$ (b) Hence solve, for \(0 \leqslant x \leqslant 90 ^ { \circ }\), the equation $$8 \tan 2 x = 3 \cos 2 x$$ giving your answers to 2 decimal places.

8. (i) An arithmetic series has first term \(a\) and common difference \(d\). Prove that the sum to \(n\) terms of this series is $$\frac { n } { 2 } \{ 2 a + ( n - 1 ) d \}$$ (ii) A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is given by $$u _ { n } = 5 n + 3 ( - 1 ) ^ { n }$$ Find the value of

1. \(u _ { 5 }\)
2. \(\sum _ { n = 1 } ^ { 59 } u _ { n }\)
   

9. (a) Sketch the curve with equation $$y = 3 \times 4 ^ { x }$$ showing the coordinates of any points of intersection with the coordinate axes. The curve with equation \(y = 6 ^ { 1 - x }\) meets the curve with equation \(y = 3 \times 4 ^ { x }\) at the point \(P\).
(b) Show that the \(x\) coordinate of \(P\) is \(\frac { \log _ { 10 } 2 } { \log _ { 10 } 24 }\)

10. A curve \(C\) has equation $$y = 4 x ^ { 3 } - 9 x + \frac { k } { x } \quad x > 0$$ where \(k\) is a constant.
The point \(P\) with \(x\) coordinate \(\frac { 1 } { 2 }\) lies on \(C\).
Given that \(P\) is a stationary point of \(C\),

1. show that \(k = - \frac { 3 } { 2 }\)
2. Determine the nature of the stationary point at \(P\), justifying your answer. The curve \(C\) has a second stationary point.
3. Using algebra, find the \(x\) coordinate of this second stationary point.
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