Edexcel P2 (Pure Mathematics 2) 2024 June

1. (a) Find the first four terms, in ascending powers of \(x\), of the binomial expansion of

$$\left( 1 - \frac { 1 } { 6 } x \right) ^ { 9 }$$ giving each term in simplest form.
(b) Hence find the coefficient of \(x ^ { 3 }\) in the expansion of $$( 10 x + 3 ) \left( 1 - \frac { 1 } { 6 } x \right) ^ { 9 }$$ giving the answer in simplest form.

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

In an arithmetic series,

• the sixth term is 2
• the sum of the first ten terms is - 80

For this series,

1. find the value of the first term and the value of the common difference.
2. Hence find the smallest value of \(n\) for which $$S _ { n } > 8000$$

1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
   1. Using the laws of logarithms, solve
   $$2 \log _ { 2 } ( 2 - x ) = 4 + \log _ { 2 } ( x + 10 )$$
2. Find the value of $$\log _ { \sqrt { a } } a ^ { 6 }$$ where \(a\) is a positive constant greater than 1

4. $$f ( x ) = ( x - 2 ) \left( 2 x ^ { 2 } + 5 x + k \right) + 21$$ where \(k\) is a constant.

1. State the remainder when \(\mathrm { f } ( x )\) is divided by ( \(x - 2\) ) Given that ( \(2 x - 1\) ) is a factor of \(\mathrm { f } ( x )\)
2. show that \(k = 11\)
3. Hence
   1. fully factorise \(\mathrm { f } ( x )\),
   2. find the number of real solutions of the equation $$\mathrm { f } ( x ) = 0$$ giving a reason for your answer.

1. In this question you must show detailed reasoning.
   1. Given that \(x\) and \(y\) are positive numbers such that
   $$( x - y ) ^ { 3 } > x ^ { 3 } - y ^ { 3 }$$ prove that $$y > x$$
2. Using a counter example, show that the result in part (a) is not true for all real numbers.

1. (a) Sketch the curve with equation

$$y = a ^ { x } + 4$$ where \(a\) is a positive constant greater than 1
On your sketch, show

• the coordinates of the point of intersection of the curve with the \(y\)-axis
• the equation of the asymptote of the curve

The table shows corresponding values of \(x\) and \(y\) for $$y = 2 ^ { x } - 2 x$$ with the values of \(y\) given to 4 decimal places as appropriate.
Using the trapezium rule with all the values of \(y\) in the given table,
(b) obtain an estimate for \(\int _ { 2 } ^ { 3.5 } \left( 2 ^ { x } - 2 x \right) \mathrm { d } x\), giving your answer to 2 decimal places.
(c) Using your answer to part (b) and making your method clear, estimate

1. \(\int _ { 2 } ^ { 3.5 } \left( 2 ^ { x } + 2 x \right) \mathrm { d } x\)
2. \(\int _ { 2 } ^ { 3.5 } \left( 2 ^ { x + 1 } - 4 x \right) \mathrm { d } x\)

1. The circle \(C _ { 1 }\) has equation

$$x ^ { 2 } + y ^ { 2 } + 8 x - 10 y = 29$$

1. 1. Find the coordinates of the centre of \(C _ { 1 }\)
   2. Find the exact value of the radius of \(C _ { 1 }\) In part (b) you must show detailed reasoning.
      The circle \(C _ { 2 }\) has equation $$( x - 5 ) ^ { 2 } + ( y + 8 ) ^ { 2 } = 52$$
2. Prove that the circles \(C _ { 1 }\) and \(C _ { 2 }\) neither touch nor intersect.

1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
   1. Solve, for \(0 < x \leqslant \pi\), the equation
   $$5 \sin x \tan x + 13 = \cos x$$ giving your answer in radians to 3 significant figures.
2. The temperature inside a greenhouse is monitored on one particular day. The temperature, \(H ^ { \circ } \mathrm { C }\), inside the greenhouse, \(t\) hours after midnight, is modelled by the equation $$H = 10 + 12 \sin ( k t + 18 ) ^ { \circ } \quad 0 \leqslant t < 24$$ where \(k\) is a constant.
   Use the equation of the model to answer parts (a) to (c).
   Given that
   • the temperature inside the greenhouse was \(20 ^ { \circ } \mathrm { C }\) at 6 am
   • \(0 < k < 20\)
     (a) find all possible values for \(k\), giving each answer to 2 decimal places.
   Given further that \(0 < k < 10\)
   (b) find the maximum temperature inside the greenhouse,
   (c) find the time of day at which this maximum temperature occurs. Give your answer to the nearest minute.

9. \begin{figure}[h] \end{figure} Figure 1 is a sketch of the curve \(C\) with equation $$y = 2 x ^ { \frac { 3 } { 2 } } ( 4 - x ) \quad x \geqslant 0$$ The point \(P\) is the stationary point of \(C\).

1. Find, using calculus, the \(x\) coordinate of \(P\). The region \(R _ { 1 }\), shown shaded in Figure 1, is bounded by \(C\) and the \(x\)-axis.
   The region \(R _ { 2 }\), also shown shaded in Figure 1, is bounded by \(C\), the \(x\)-axis and the line with equation \(x = k\), where \(k\) is a constant. Given that the area of \(R _ { 1 }\) is equal to the area of \(R _ { 2 }\)
2. find, using calculus, the exact value of \(k\).

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

The number of dormice and the number of voles on an island are being monitored.
Initially there are 2000 dormice on the island.
A model predicts that the number of dormice will increase by \(3 \%\) each year, so that the numbers of dormice on the island at the end of each year form a geometric sequence.

1. Find, according to the model, the number of dormice on the island 6 years after monitoring began. Give your answer to 3 significant figures. The number of voles on the island is being monitored over the same period of time.
   Given that
   • 4 years after monitoring began there were 3690 voles on the island
   • 7 years after monitoring began there were 3470 voles on the island
   • the number of voles on the island at the end of each year is modelled as a geometric sequence
   • find the equation of this model in the form
   $$N = a b ^ { t }$$ where \(N\) is the number of voles, \(t\) years after monitoring began and \(a\) and \(b\) are constants. Give the value of \(a\) and the value of \(b\) to 2 significant figures. When \(t = T\), the number of dormice on the island is equal to the number of voles on the island.
2. Find, according to the models, the value of \(T\), giving your answer to one decimal place.