Edexcel P2 (Pure Mathematics 2) 2024 January

1. $$f ( x ) = a x ^ { 3 } + 3 x ^ { 2 } - 8 x + 2 \quad \text { where } a \text { is a constant }$$ Given that when \(\mathrm { f } ( x )\) is divided by \(( x - 2 )\) the remainder is 3 , find the value of \(a\).

1. Find the coefficient of the term in \(x ^ { 7 }\) of the binomial expansion of

$$\left( \frac { 3 } { 8 } + 4 x \right) ^ { 12 }$$ giving your answer in simplest form.

1. The circle \(C\)

• has centre \(A ( 3,5 )\)
• passes through the point \(B ( 8 , - 7 )\)
  1. Find an equation for \(C\).

The points \(M\) and \(N\) lie on \(C\) such that \(M N\) is a chord of \(C\).
Given that \(M N\)

• lies above the \(x\)-axis
• is parallel to the \(x\)-axis
• has length \(4 \sqrt { 22 }\)
• find an equation for the line passing through points \(M\) and \(N\).

1. (a) Sketch the curve with equation

$$y = a ^ { - x } + 4$$ where \(a\) is a constant and \(a > 1\)
On your sketch show

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

The table above shows corresponding values of \(x\) and \(y\) for \(y = 3 ^ { - \frac { 1 } { 2 } x } + 4\)
The values of \(y\) are given to four significant figures, as appropriate.
Using the trapezium rule with all the values of \(y\) in the table,
(b) find an approximate value for $$\int _ { - 4 } ^ { 8.5 } \left( 3 ^ { - \frac { 1 } { 2 } x } + 4 \right) d x$$ giving your answer to two significant figures.
(c) Using the answer to part (b), find an approximate value for

1. \(\int _ { - 4 } ^ { 8.5 } \left( 3 ^ { - \frac { 1 } { 2 } x } \right) \mathrm { d } x\)
2. \(\int _ { - 4 } ^ { 8.5 } \left( 3 ^ { - \frac { 1 } { 2 } x } + 4 \right) \mathrm { d } x + \int _ { - 8.5 } ^ { 4 } \left( 3 ^ { \frac { 1 } { 2 } x } + 4 \right) \mathrm { d } x\)

1. 1. Find the value of

$$\sum _ { r = 1 } ^ { \infty } 6 \times ( 0.25 ) ^ { r }$$ (3)
(ii) A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$\begin{aligned} u _ { 1 } & = 3
u _ { n + 1 } & = \frac { u _ { n } - 3 } { u _ { n } - 2 } \quad n \in \mathbb { N } \end{aligned}$$

1. Show that this sequence is periodic.
2. State the order of this sequence.
3. Hence find $$\sum _ { n = 1 } ^ { 70 } u _ { n }$$

1. (a) Given that

$$2 \log _ { 4 } ( x + 3 ) + \log _ { 4 } x = \log _ { 4 } ( 4 x + 2 ) + \frac { 1 } { 2 }$$ show that $$x ^ { 3 } + 6 x ^ { 2 } + x - 4 = 0$$ (b) Given also that - 1 is a root of the equation $$x ^ { 3 } + 6 x ^ { 2 } + x - 4 = 0$$

1. use algebra to find the other two roots of the equation.
2. Hence solve $$2 \log _ { 4 } ( x + 3 ) + \log _ { 4 } x = \log _ { 4 } ( 4 x + 2 ) + \frac { 1 } { 2 }$$

1. Wheat is grown on a farm.

• In year 1 , the farm produced 300 tonnes of wheat.
• In year 12 , the farm is predicted to produce 4000 tonnes of wheat.

Model \(A\) assumes that the amount of wheat produced on the farm will increase by the same amount each year.

1. Using model \(A\), find the amount of wheat produced on the farm in year 4. Give your answer to the nearest 10 tonnes. Model \(B\) assumes that the amount of wheat produced on the farm will increase by the same percentage each year.
2. Using model \(B\), find the amount of wheat produced on the farm in year 2. Give your answer to the nearest 10 tonnes.
3. Calculate, according to the two models, the difference between the total amounts of wheat predicted to be produced on the farm from year 1 to year 12 inclusive. Give your answer to the nearest 10 tonnes.

1. (i) Use a counter example to show that the following statement is false

$$\text { " } n ^ { 2 } + 3 n + 1 \text { is prime for all } n \in \mathbb { N } \text { " }$$ (ii) Use algebra to prove by exhaustion that for all \(n \in \mathbb { N }\) $$\text { " } n ^ { 2 } - 2 \text { is not a multiple of } 4 \text { " }$$

1. In this question you must show detailed reasoning.

Solutions relying entirely on calculator technology are not acceptable.

1. Solve, for \(0 \leqslant x < 360 ^ { \circ }\), the equation $$\sin x \tan x = 5$$ giving your answers to one decimal place.
2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{0e08d931-aa1c-48a8-8b39-47096f981950-26_643_736_721_660} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure} Figure 1 shows a sketch of part of the curve with equation $$y = A \sin \left( 2 \theta - \frac { 3 \pi } { 8 } \right) + 2$$ where \(A\) is a constant and \(\theta\) is measured in radians.
   The points \(P , Q\) and \(R\) lie on the curve and are shown in Figure 1.
   Given that the \(y\) coordinate of \(P\) is 7
   (a) state the value of \(A\),
   (b) find the exact coordinates of \(Q\),
   (c) find the value of \(\theta\) at \(R\), giving your answer to 3 significant figures.

1. In this question you must show detailed reasoning.

Solutions relying entirely on calculator technology are not acceptable. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve with equation $$y = \frac { 1 } { 2 } x ^ { 2 } + \frac { 1458 } { \sqrt { x ^ { 3 } } } - 74 \quad x > 0$$ The point \(P\) is the only stationary point on the curve.

1. Use calculus to show that the \(x\) coordinate of \(P\) is 9 The line \(l\) passes through the point \(P\) and is parallel to the \(x\)-axis.
   The region \(R\), shown shaded in Figure 2, is bounded by the curve, the line \(l\) and the line with equation \(x = 4\)
2. Use algebraic integration to find the exact area of \(R\).