Edexcel P2 (Pure Mathematics 2) 2018 Specimen

1. $$\mathrm { f } ( x ) = x ^ { 4 } + x ^ { 3 } + 2 x ^ { 2 } + a x + b ,$$ where \(a\) and \(b\) are constants.
When \(\mathrm { f } ( x )\) is divided by ( \(x - 1\) ), the remainder is 7

1. Show that \(a + b = 3\) When \(\mathrm { f } ( x )\) is divided by ( \(x + 2\) ), the remainder is - 8
2. Find the value of \(a\) and the value of \(b\)
   

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2. The first term of a geometric series is 20 and the common ratio is \(\frac { 7 } { 8 }\). The sum to infinity of the series is \(S _ { \infty }\)

1. Find the value of \(S _ { \infty }\) The sum to \(N\) terms of the series is \(S _ { N }\)
2. Find, to 1 decimal place, the value of \(S _ { 12 }\)
3. Find the smallest value of \(N\), for which \(S _ { \infty } - S _ { N } < 0.5\)
   2. The first term of a geometric series is 20 and the common ratio is \(\frac { 7 } { 8 }\). The sum to infinity
   of the series is \(S _ { \infty }\)

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

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

   \(x\)	0	0.25	0.5	0.75	1
   \(y\)	1	1.251			2

2. Use the trapezium rule with all the values of \(y\) from your table to find an approximation for the value of $$\int _ { 0 } ^ { 1 } \sqrt { \left( 3 ^ { x } + x \right) } \mathrm { d } x$$ You must show clearly how you obtained your answer.
3. Explain how the trapezium rule could be used to obtain a more accurate estimate for the value of $$\int _ { 0 } ^ { 1 } \sqrt { \left( 3 ^ { x } + x \right) } d x$$
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Given \(n \in \mathbb { N }\), prove, by exhaustion, that \(n ^ { 2 } + 2\) is not divisible by 4 .
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An arithmetic series has first term \(a\) and common difference \(d\).

1. Prove that the sum of the first \(n\) terms of the series is $$\frac { 1 } { 2 } n [ 2 a + ( n - 1 ) d ]$$ A company, which is making 200 mobile phones each week, plans to increase its production. The number of mobile phones produced is to be increased by 20 each week from 200 in week 1 to 220 in week 2, to 240 in week 3 and so on, until it is producing 600 in week \(N\).
2. Find the value of \(N\) The company then plans to continue to make 600 mobile phones each week.
3. Find the total number of mobile phones that will be made in the first 52 weeks starting from and including week 1.
   
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6. (i) Find the exact value of \(x\) for which $$\log _ { 2 } ( 2 x ) = \log _ { 2 } ( 5 x + 4 ) - 3$$ (ii) Given that $$\log _ { a } y + 3 \log _ { a } 2 = 5$$ express \(y\) in terms of \(a\). Give your answer in its simplest form.
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7. \begin{figure}[h] \end{figure} The circle with equation $$x ^ { 2 } + y ^ { 2 } - 20 x - 16 y + 139 = 0$$ had centre \(C\) and radius \(r\).

1. Find the coordinates of \(C\).
2. Show that \(r = 5\) The line with equation \(x = 13\) crosses the circle at the points \(P\) and \(Q\) as shown in Figure 1 .
3. Find the \(y\) coordinate of \(P\) and the \(y\) coordinate of \(Q\). A tangent to the circle from \(O\) touches the circle at point \(X\).
4. Find, in surd form, the length \(O X\).
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8. Figure 2 Figure 2 shows a sketch of part of the curves \(C _ { 1 }\) and \(C _ { 2 }\) with equations $$\begin{array} { l l } C _ { 1 } : y = 10 x - x ^ { 2 } - 8 & x > 0
C _ { 2 } : y = x ^ { 3 } & x > 0 \end{array}$$ The curves \(C _ { 1 }\) and \(C _ { 2 }\) intersect at the points \(A\) and \(B\).

1. Verify that the point \(A\) has coordinates (1, 1)
2. Use algebra to find the coordinates of the point \(B\) The finite region \(R\) is bounded by \(C _ { 1 }\) and \(C _ { 2 }\)
3. Use calculus to find the exact area of \(R\)
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9. (i) Solve, for \(0 \leqslant \theta < \pi\), the equation $$\sin 3 \theta - \sqrt { 3 } \cos 3 \theta = 0$$ giving your answers in terms of \(\pi\)
(ii) Given that $$4 \sin ^ { 2 } x + \cos x = 4 - k , \quad 0 \leqslant k \leqslant 3$$

1. find \(\cos x\) in terms of \(k\)
2. When \(k = 3\), find the values of \(x\) in the range \(0 \leqslant x < 360 ^ { \circ }\)
   
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