Edexcel P2 (Pure Mathematics 2) 2022 June

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

$$\left( 2 + \frac { 3 } { 8 } x \right) ^ { 10 }$$ Give each coefficient as an integer.

2. \begin{figure}[h] \end{figure} Figure 1 shows the graph of $$y = 1 - \log _ { 10 } ( \sin x ) \quad 0 < x < \pi$$ where \(x\) is in radians. The table below shows some values of \(x\) and \(y\) for this graph, with values of \(y\) given to 3 decimal places.

1. Complete the table above, giving values of \(y\) to 3 decimal places.
2. Use the trapezium rule with all the \(y\) values in the completed table to find, to 2 decimal places, an estimate for $$\int _ { 0.5 } ^ { 3 } \left( 1 - \log _ { 10 } ( \sin x ) \right) \mathrm { d } x$$
3. Use your answer to part (b) to find an estimate for $$\int _ { 0.5 } ^ { 3 } \left( 3 + \log _ { 10 } ( \sin x ) \right) \mathrm { d } x$$

3. (i) Show that the following statement is false: $$\text { " } ( n + 1 ) ^ { 3 } - n ^ { 3 } \text { is prime for all } n \in \mathbb { N } \text { " }$$ (ii) Given that the points \(A ( 1,0 ) , B ( 3 , - 10 )\) and \(C ( 7 , - 6 )\) lie on a circle, prove that \(A B\) is a diameter of this circle.

4. In this question you must show all stages of your working. Give your answers in fully simplified surd form. Given that \(a\) and \(b\) are positive constants, solve the simultaneous equations $$\begin{gathered} a - b = 8
\log _ { 4 } a + \log _ { 4 } b = 3 \end{gathered}$$ (6)

5. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
Solve, for \(- 180 ^ { \circ } < \theta \leqslant 180 ^ { \circ }\), the equation $$3 \tan \left( \theta + 43 ^ { \circ } \right) = 2 \cos \left( \theta + 43 ^ { \circ } \right)$$

7
6. In a geometric sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\)

• the common ratio is \(r\)
• \(u _ { 2 } + u _ { 3 } = 6\)
• \(u _ { 4 } = 8\)
  1. Show that \(r\) satisfies

$$3 r ^ { 2 } - 4 r - 4 = 0$$ Given that the geometric sequence has a sum to infinity,

find \(u _ { 1 }\)

find \(S _ { \infty }\) 7. $$f ( x ) = A x ^ { 3 } + 6 x ^ { 2 } - 4 x + B$$ where \(A\) and \(B\) are constants. Given that

• ( \(x + 2\) ) is a factor of \(\mathrm { f } ( x )\)
• \(\int _ { 3 } ^ { 5 } \mathrm { f } ( x ) \mathrm { d } x = 176\)
  find the value of \(A\) and the value of \(B\).

8. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
A curve has equation $$y = 256 x ^ { 4 } - 304 x - 35 + \frac { 27 } { x ^ { 2 } } \quad x \neq 0$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
2. Hence find the coordinates of the stationary points of the curve.

9. A scientist is using carbon-14 dating to determine the age of some wooden items. The equation for carbon-14 dating an item is given by $$N = k \lambda ^ { t }$$ where

• \(N\) grams is the amount of carbon-14 currently present in the item
• \(k\) grams was the initial amount of carbon-14 present in the item
• \(t\) is the number of years since the item was made
• \(\lambda\) is a constant, with \(0 < \lambda < 1\)
  1. Sketch the graph of \(N\) against \(t\) for \(k = 1\)

Given that it takes 5700 years for the amount of carbon-14 to reduce to half its initial value,

show that the value of the constant \(\lambda\) is 0.999878 to 6 decimal places. Given that Item \(A\)

• is known to have had 15 grams of carbon-14 present initially
• is thought to be 3250 years old
• calculate, to 3 significant figures, how much carbon-14 the equation predicts is currently in Item \(A\).

Item \(B\) is known to have initially had 25 grams of carbon-14 present, but only 18 grams now remain.

Use algebra to calculate the age of Item \(B\) to the nearest 100 years.

10. The circle \(C\) has centre \(X ( 3,5 )\) and radius \(r\) The line \(l\) has equation \(y = 2 x + k\), where \(k\) is a constant.

1. Show that \(l\) and \(C\) intersect when $$5 x ^ { 2 } + ( 4 k - 26 ) x + k ^ { 2 } - 10 k + 34 - r ^ { 2 } = 0$$ Given that \(l\) is a tangent to \(C\),
2. show that \(5 r ^ { 2 } = ( k + p ) ^ { 2 }\), where \(p\) is a constant to be found. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{db4ec300-8081-4d29-acd5-0aae789d8f95-28_636_572_902_687} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure} The line \(l\)
   • cuts the \(y\)-axis at the point \(A\)
   • touches the circle \(C\) at the point \(B\)
     as shown in Figure 2.
     Given that \(A B = 2 r\)
   • find the value of \(k\)