Edexcel P2 (Pure Mathematics 2) 2021 June

1. Adina is saving money to buy a new computer. She saves \(\pounds 5\) in week \(1 , \pounds 5.25\) in week 2 , \(\pounds 5.50\) in week 3 and so on until she has enough money, in total, to buy the computer.

She decides to model her savings using either an arithmetic series or a geometric series.
Using the information given,

1. 1. state with a reason whether an arithmetic series or a geometric series should be used,
   2. write down an expression, in terms of \(n\), for the amount, in pounds ( \(\pounds\) ), saved in week \(n\). Given that the computer Adina wants to buy costs \(\pounds 350\)
2. find the number of weeks it will take for Adina to save enough money to buy the computer.
   

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2. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = 4 ^ { x }\) A copy of Figure 1, labelled Diagram 1, is shown on the next page.

1. On Diagram 1, sketch the curve with equation
   1. \(y = 2 ^ { x }\)
   2. \(y = 4 ^ { x } - 6\) Label clearly the coordinates of any points of intersection with the coordinate axes. The curve with equation \(y = 2 ^ { x }\) meets the curve with equation \(y = 4 ^ { x } - 6\) at the point \(P\).
2. Using algebra, find the exact coordinates of \(P\).
   \includegraphics[max width=\textwidth, alt={}]{515f245f-9c5b-4263-ab2c-0a4f96f3bff0-05_1009_1490_264_219}
   \section*{Diagram 1}

3. (i) Prove that for all single digit prime numbers, \(p\), $$p ^ { 3 } + p \text { is a multiple of } 10$$ (ii) Show, using algebra, that for \(n \in \mathbb { N }\) $$( n + 1 ) ^ { 3 } - n ^ { 3 } \text { is not a multiple of } 3$$

1. (a) Find, in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), the binomial expansion of

$$\left( 2 + \frac { x } { 8 } \right) ^ { 13 }$$ fully simplifying each coefficient.
(b) Use the answer to part (a) to find an approximation for \(2.0125 ^ { 13 }\) Give your answer to 3 decimal places. Without calculating \(2.0125 { } ^ { 13 }\)
(c) state, with a reason, whether the answer to part (b) is an overestimate or an underestimate.

5. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the graph of the curves \(C _ { 1 }\) and \(C _ { 2 }\)
The curves intersect when \(x = 2.5\) and when \(x = 4\) A table of values for some points on the curve \(C _ { 1 }\) is shown below, with \(y\) values given to 3 decimal places as appropriate. Using the trapezium rule with all the values of \(y\) in the table,

1. find, to 2 decimal places, an estimate for the area bounded by the curve \(C _ { 1 }\), the line with equation \(x = 2.5\), the \(x\)-axis and the line with equation \(x = 4\) The curve \(C _ { 2 }\) has equation $$y = x ^ { \frac { 3 } { 2 } } - 3 x + 9 \quad x > 0$$
2. Find \(\int \left( x ^ { \frac { 3 } { 2 } } - 3 x + 9 \right) \mathrm { d } x\) The region \(R\), shown shaded in Figure 2, is bounded by the curves \(C _ { 1 }\) and \(C _ { 2 }\)
3. Use the answers to part (a) and part (b) to find, to one decimal place, an estimate for the area of the region \(R\).
   (3)

1. A circle has equation

$$x ^ { 2 } - 6 x + y ^ { 2 } + 8 y + k = 0$$ where \(k\) is a positive constant. Given that the \(x\)-axis is a tangent to this circle,

1. find the value of \(k\). The circle meets the coordinate axes at the points \(R , S\) and \(T\).
2. Find the exact area of the triangle \(R S T\).
   \includegraphics[max width=\textwidth, alt={}, center]{515f245f-9c5b-4263-ab2c-0a4f96f3bff0-21_2647_1840_118_111}

7. (a) Given that $$3 \log _ { 3 } ( 2 x - 1 ) = 2 + \log _ { 3 } ( 14 x - 25 )$$ show that $$2 x ^ { 3 } - 3 x ^ { 2 } - 30 x + 56 = 0$$ (b) Show that - 4 is a root of this cubic equation.
(c) Hence, using algebra and showing each step of your working, solve $$3 \log _ { 3 } ( 2 x - 1 ) = 2 + \log _ { 3 } ( 14 x - 25 )$$

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

1. Solve, for \(0 < \theta < 360 ^ { \circ }\), the equation $$3 \sin \left( \theta + 30 ^ { \circ } \right) = 7 \cos \left( \theta + 30 ^ { \circ } \right)$$ giving your answers to one decimal place.
2. (a) Show that the equation $$3 \sin ^ { 3 } x = 5 \sin x - 7 \sin x \cos x$$ can be written in the form $$\sin x \left( a \cos ^ { 2 } x + b \cos x + c \right) = 0$$ where \(a , b\) and \(c\) are constants to be found.
   (b) Hence solve for \(- \frac { \pi } { 2 } \leqslant x \leqslant \frac { \pi } { 2 }\) the equation $$3 \sin ^ { 3 } x = 5 \sin x - 7 \sin x \cos x$$ \includegraphics[max width=\textwidth, alt={}, center]{515f245f-9c5b-4263-ab2c-0a4f96f3bff0-27_2644_1840_118_111}

9. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of a square based, open top box.
The height of the box is \(h \mathrm {~cm}\), and the base edges each have length \(l \mathrm {~cm}\).
Given that the volume of the box is \(250000 \mathrm {~cm} ^ { 3 }\)

1. show that the external surface area, \(S \mathrm {~cm} ^ { 2 }\), of the box is given by $$S = \frac { 250000 } { h } + 2000 \sqrt { h }$$
2. Use algebraic differentiation to show that \(S\) has a stationary point when \(h = 250 ^ { k }\) where \(k\) is a rational constant to be found.
3. Justify by further differentiation that this value of \(h\) gives the minimum external surface area of the box.
   \includegraphics[max width=\textwidth, alt={}]{515f245f-9c5b-4263-ab2c-0a4f96f3bff0-32_2647_1838_118_116}