Edexcel P2 (Pure Mathematics 2) 2021 October

1. The first three terms, in ascending powers of \(x\), of the binomial expansion of \(( 1 + k x ) ^ { 16 }\) are

$$1 , - 4 x \text { and } p x ^ { 2 }$$ where \(k\) and \(p\) are constants.

1. Find, in simplest form,
   1. the value of \(k\)
   2. the value of \(p\) $$g ( x ) = \left( 2 + \frac { 16 } { x } \right) ( 1 + k x ) ^ { 16 }$$ Using the value of \(k\) found in part (a),
2. find the term in \(x ^ { 2 }\) in the expansion of \(\mathrm { g } ( x )\). $$\begin{aligned} u _ { 1 } & = 6
   u _ { n + 1 } & = k u _ { n } + 3 \end{aligned}$$ where \(k\) is a positive constant.
3. Find, in terms of \(k\), an expression for \(u _ { 3 }\) Given that \(\sum _ { n = 1 } ^ { 3 } u _ { n } = 117\)
4. find the value of \(k\).

2. A sequence is defined by

3. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = \log _ { 10 } x\)
The region \(R\), shown shaded in Figure 1, is bounded by the curve, the line with equation \(x = 2\), the \(x\)-axis and the line with equation \(x = 14\) Using the trapezium rule with four strips of equal width,

1. show that the area of \(R\) is approximately 10.10
2. Explain how the trapezium rule could be used to obtain a more accurate estimate for the area of \(R\).
3. Using the answer to part (a) and making your method clear, estimate the value of
   1. \(\quad \int _ { 2 } ^ { 14 } \log _ { 10 } \sqrt { x } \mathrm {~d} x\)
   2. \(\int _ { 2 } ^ { 14 } \log _ { 10 } 100 x ^ { 3 } \mathrm {~d} x\)

4. $$f ( x ) = \left( x ^ { 2 } - 2 \right) ( 2 x - 3 ) - 21$$

1. State the value of the remainder when \(\mathrm { f } ( x )\) is divided by ( \(2 x - 3\) )
2. Use the factor theorem to show that \(( x - 3 )\) is a factor of \(\mathrm { f } ( x )\)
3. Hence,
   1. factorise \(\mathrm { f } ( x )\)
   2. show that the equation \(\mathrm { f } ( x ) = 0\) has only one real root.

5. A company that owned a silver mine

• extracted 480 tonnes of silver from the mine in year 1
• extracted 465 tonnes of silver from the mine in year 2
• extracted 450 tonnes of silver from the mine in year 3
  and so on, forming an arithmetic sequence.
  1. Find the mass of silver extracted in year 14

After a total of 7770 tonnes of silver was extracted, the company stopped mining. Given that this occurred at the end of year \(N\),

show that $$N ^ { 2 } - 65 N + 1036 = 0$$

Hence, state the value of \(N\).

6. (i) The circle \(C _ { 1 }\) has equation $$x ^ { 2 } + y ^ { 2 } + 10 x - 12 y = k \quad \text { where } k \text { is a constant }$$

1. Find the coordinates of the centre of \(C _ { 1 }\)
2. State the possible range in values for \(k\).
   (ii) The point \(P ( p , 0 )\), the point \(Q ( - 2,10 )\) and the point \(R ( 8 , - 14 )\) lie on a different circle, \(C _ { 2 }\) Given that
   • \(p\) is a positive constant
   • \(Q R\) is a diameter of \(C _ { 2 }\)
     find the exact value of \(p\).
     

7. (i) A geometric sequence has first term 4 and common ratio 6 Given that the \(n ^ { \text {th } }\) term is greater than \(10 ^ { 100 }\), find the minimum possible value of \(n\).
(ii) A different geometric sequence has first term \(a\) and common ratio \(r\). Given that

• the second term of the sequence is - 6
• the sum to infinity of the series is 25
  1. show that

$$25 r ^ { 2 } - 25 r - 6 = 0$$

Write down the solutions of $$25 r ^ { 2 } - 25 r - 6 = 0$$ Hence,

state the value of \(r\), giving a reason for your answer,

find the sum of the first 4 terms of the series. \includegraphics[max width=\textwidth, alt={}, center]{124ee19f-8a49-42df-9f4b-5a1cc2139be9-23_70_37_2617_1914}

8. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the curve \(C\) with equation $$y = \frac { 4 } { 3 } x ^ { 3 } - 11 x ^ { 2 } + k x \quad \text { where } k \text { is a constant }$$ The point \(M\) is the maximum turning point of \(C\) and is shown in Figure 2.
Given that the \(x\) coordinate of \(M\) is 2

1. show that \(k = 28\)
2. Determine the range of values of \(x\) for which \(y\) is increasing. The line \(l\) passes through \(M\) and is parallel to the \(x\)-axis.
   The region \(R\), shown shaded in Figure 2, is bounded by the curve \(C\), the line \(l\) and the \(y\)-axis.
3. Find, by algebraic integration, the exact area of \(R\).

9. (a) Prove that for all positive values of \(x\) and \(y\), $$\frac { x + y } { 2 } \geqslant \sqrt { x y }$$ (b) Prove by counter-example that this inequality does not hold when \(x\) and \(y\) are both negative.
(1)
\includegraphics[max width=\textwidth, alt={}, center]{124ee19f-8a49-42df-9f4b-5a1cc2139be9-29_61_54_2608_1852}

10. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

1. Solve, for \(- \frac { \pi } { 2 } < x < \frac { \pi } { 2 }\) $$\tan ^ { 2 } \left( 2 x + \frac { \pi } { 4 } \right) = 3$$
2. Solve, for \(0 < \theta < 360 ^ { \circ }\) $$( 2 \sin \theta - \cos \theta ) ^ { 2 } = 1$$ giving your answers, as appropriate, to one decimal place.