Edexcel P2 (Pure Mathematics 2) 2020 October

1. (a) Find the first 4 terms, in ascending powers of \(x\), of the binomial expansion of

$$\left( 2 - \frac { x } { 4 } \right) ^ { 10 }$$ giving each term in its simplest form.
(b) Hence find the constant term in the series expansion of $$\left( 3 - \frac { 1 } { x } \right) ^ { 2 } \left( 2 - \frac { x } { 4 } \right) ^ { 10 }$$

2. $$y = \frac { 2 ^ { x } } { \sqrt { \left( 5 x ^ { 2 } + 3 \right) } }$$

1. Complete the table below,giving the values of \(y\) to 3 decimal places.

   \(x\)	- 0.25	0	0.25	0.5	0.75
   \(y\)	0.462		0.653		0.698

2. Use the trapezium rule,with all the values of \(y\) from the completed table,to find an approximate value for
   . $$\int _ { - 0.25 } ^ { 0.75 } \frac { 2 ^ { x } } { \sqrt { \left( 5 x ^ { 2 } + 3 \right) } } \mathrm { d } x$$

3. $$f ( x ) = a x ^ { 3 } - x ^ { 2 } + b x + 4$$ where \(a\) and \(b\) are constants. When \(\mathrm { f } ( x )\) is divided by ( \(x + 4\) ), the remainder is - 108

1. Use the remainder theorem to show that $$16 a + b = 24$$ Given also that ( \(2 x - 1\) ) is a factor of \(\mathrm { f } ( x )\),
2. find the value of \(a\) and the value of \(b\).
3. Find \(\mathrm { f } ^ { \prime } ( x )\).
4. Hence find the exact coordinates of the stationary points of the curve with equation \(y = \mathrm { f } ( x )\).

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4. The points \(P\) and \(Q\) have coordinates \(( - 11,6 )\) and \(( - 3,12 )\) respectively. Given that \(P Q\) is a diameter of the circle \(C\),

1. 1. find the coordinates of the centre of \(C\),
   2. find the radius of \(C\).
2. Hence find an equation of \(C\).
3. Find an equation of the tangent to \(C\) at the point \(Q\) giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers to be found.
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5. Ben is saving for the deposit for a house over a period of 60 months. Ben saves \(\pounds 100\) in the first month and in each subsequent month, he saves \(\pounds 5\) more than the previous month, so that he saves \(\pounds 105\) in the second month, \(\pounds 110\) in the third month, and so on, forming an arithmetic sequence.

1. Find the amount Ben saves in the 40th month.
2. Find the total amount Ben saves over the 60 -month period. Lina is also saving for a deposit for a house.
   Lina saves \(\pounds 600\) in the first month and in each subsequent month, she saves \(\pounds 10\) less than the previous month, so that she saves \(\pounds 590\) in the second month, \(\pounds 580\) in the third month, and so on, forming an arithmetic sequence. Given that, after \(n\) months, Lina will have saved exactly \(\pounds 18200\) for her deposit,
3. form an equation in \(n\) and show that it can be written as $$n ^ { 2 } - 121 n + 3640 = 0$$
4. Solve the equation in part (c).
5. State, with a reason, which of the solutions to the equation in part (c) is not a sensible value for \(n\).

6. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curves \(C _ { 1 }\) and \(C _ { 2 }\) with equations $$\begin{array} { l l } C _ { 1 } : y = x ^ { 3 } - 6 x + 9 & x \geqslant 0
C _ { 2 } : y = - 2 x ^ { 2 } + 7 x - 1 & x \geqslant 0 \end{array}$$ The curves \(C _ { 1 }\) and \(C _ { 2 }\) intersect at the points \(A\) and \(B\) as shown in Figure 1 .
The point \(A\) has coordinates (1,4). Using algebra and showing all steps of your working,

1. find the coordinates of the point \(B\). The finite region \(R\), shown shaded in Figure 1, is bounded by \(C _ { 1 }\) and \(C _ { 2 }\)
2. Use algebraic integration to find the exact area of \(R\).

7. (i) Show that $$\tan \theta + \frac { 1 } { \tan \theta } \equiv \frac { 1 } { \sin \theta \cos \theta } \quad \theta \neq \frac { \mathrm { n } \pi } { 2 } \quad n \in \mathbb { Z }$$ (ii) Solve, for \(0 \leqslant x < 90 ^ { \circ }\), the equation $$3 \cos ^ { 2 } \left( 2 x + 10 ^ { \circ } \right) = 1$$ giving your answers in degrees to one decimal place.
(Solutions based entirely on graphical or numerical methods are not acceptable.)

8. A geometric series has first term \(a\) and common ratio \(r\).

1. Prove that the sum of the first \(n\) terms of this series is given by $$S _ { n } = \frac { a \left( 1 - r ^ { n } \right) } { 1 - r }$$ The second term of a geometric series is - 320 and the fifth term is \(\frac { 512 } { 25 }\)
2. Find the value of the common ratio.
3. Hence find the sum of the first 13 terms of the series, giving your answer to 2 decimal places.

9. (i) Find the exact value of \(x\) for which $$\log _ { 3 } ( x + 5 ) - 4 = \log _ { 3 } ( 2 x - 1 )$$ (ii) Given that $$3 ^ { y + 3 } \times 2 ^ { 1 - 2 y } = 108$$

1. show that $$0.75 ^ { y } = 2$$
2. Hence find the value of \(y\), giving your answer to 3 decimal places.

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