Edexcel P2 (Pure Mathematics 2) 2021 January

1. $$f ( x ) = x ^ { 4 } + a x ^ { 3 } - 3 x ^ { 2 } + b x + 5$$ where \(a\) and \(b\) are constants.
When \(\mathrm { f } ( x )\) is divided by ( \(x + 1\) ), the remainder is 4

1. Show that \(a + b = - 1\) When \(\mathrm { f } ( x )\) is divided by ( \(x - 2\) ), the remainder is - 23
2. Find the value of \(a\) and the value of \(b\).

2. A curve has equation $$y = x ^ { 3 } - x ^ { 2 } - 16 x + 2$$

1. Using calculus, find the \(x\) coordinates of the stationary points of the curve.
2. Justify, by further calculus, the nature of all of the stationary points of the curve.

3. (i) Solve $$7 ^ { x + 2 } = 3$$ giving your answer in the form \(x = \log _ { 7 } a\) where \(a\) is a rational number in its simplest form.
(ii) Using the laws of logarithms, solve $$1 + \log _ { 2 } y + \log _ { 2 } ( y + 4 ) = \log _ { 2 } ( 5 - y )$$

4. (a) Find the first three terms, in ascending powers of \(x\), of the binomial expansion of $$( 2 + p x ) ^ { 6 }$$ where \(p\) is a constant. Give each term in simplest form. Given that in the expansion of $$\left( 3 - \frac { 1 } { 2 } x \right) ( 2 + p x ) ^ { 6 }$$ the coefficient of \(x ^ { 2 }\) is \(- \frac { 3 } { 4 }\)
(b) find the possible values of \(p\).
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5. (i) Use algebra to prove that for all \(x \geqslant 0\) $$3 x + 1 \geqslant 2 \sqrt { 3 x }$$ (ii) Show that the following statement is not true.
"The sum of three consecutive prime numbers is always a multiple of 5 "

1. (a) Show that the equation

$$\frac { 3 \sin \theta \cos \theta } { 2 \sin \theta - 1 } = 5 \tan \theta \quad \sin \theta \neq \frac { 1 } { 2 }$$ can be written in the form $$3 \sin ^ { 3 } \theta + 10 \sin ^ { 2 } \theta - 8 \sin \theta = 0$$ (b) Hence solve, for \(- \frac { \pi } { 4 } < x < \frac { \pi } { 4 }\) $$\frac { 3 \sin 2 x \cos 2 x } { 2 \sin 2 x - 1 } = 5 \tan 2 x$$ giving your answers to 3 decimal places where appropriate.

7. Figure 1 Solar panels are installed on the roof of a building. The power, \(P\), produced on a particular day, in kW , can be modelled by the equation $$P = 0.95 + 2 ^ { t - 12 } + 2 ^ { 12 - t } - ( t - 12 ) ^ { 2 } \quad 8.5 \leqslant t \leqslant 15.2$$ where \(t\) is the time in hours after midnight. The graph of \(P\) against \(t\) is shown in Figure 1. A table of values of \(t\) and \(P\) is shown below, with the values of \(P\) given to 4 significant figures where appropriate.

1. Use the given equation to complete the table, giving the values of \(P\) to 4 significant figures where appropriate. The amount of energy, in kWh , produced between 10:00 and 12:00 can be found by calculating the area of region \(R\), shown shaded in Figure 1.
2. Use the trapezium rule, with all the values of \(P\) in the completed table, to find an estimate for the amount of energy produced between 10:00 and 12:00. Give your answer to 2 decimal places.
   7.
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8. A sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by $$\begin{gathered} a _ { n + 1 } = 2 \left( a _ { n } + 3 \right) ^ { 2 } - 7
a _ { 1 } = p - 3 \end{gathered}$$ where \(p\) is a constant.

1. Find an expression for \(a _ { 2 }\) in terms of \(p\), giving your answer in simplest form. Given that \(\sum _ { n = 1 } ^ { 3 } a _ { n } = p + 15\)
2. find the possible values of \(a _ { 2 }\)
   

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   VIAV SIHI NI JYHAM ION OO

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9. A circle \(C\) has equation $$( x - k ) ^ { 2 } + ( y - 2 k ) ^ { 2 } = k + 7$$ where \(k\) is a positive constant.

1. Write down, in terms of \(k\),
   1. the coordinates of the centre of \(C\),
   2. the radius of \(C\). Given that the point \(P ( 2,3 )\) lies on \(C\)
2. 1. show that \(5 k ^ { 2 } - 17 k + 6 = 0\)
   2. hence find the possible values of \(k\). The tangent to the circle at \(P\) intersects the \(x\)-axis at point \(T\).
      Given that \(k < 2\)
3. calculate the exact area of triangle \(O P T\).

10. In this question you must show detailed reasoning. Owen wants to train for 12 weeks in preparation for running a marathon. During the 12-week period he will run every Sunday and every Wednesday.

• On Sunday in week 1 he will run 15 km
• On Sunday in week 12 he will run 37 km

He considers two different 12-week training plans. In training plan \(A\), he will increase the distance he runs each Sunday by the same amount.

1. Calculate the distance he will run on Sunday in week 5 under training plan \(A\). In training plan \(B\), he will increase the distance he runs each Sunday by the same percentage.
2. Calculate the distance he will run on Sunday in week 5 under training plan \(B\). Give your answer in km to one decimal place. Owen will also run a fixed distance, \(x \mathrm {~km}\), each Wednesday over the 12-week period. Given that
   • \(x\) is an integer
   • the total distance that Owen will run on Sundays and Wednesdays over the 12 weeks will not exceed 360 km
   • 1. find the maximum value of \(x\), if he uses training plan \(A\),
     2. find the maximum value of \(x\), if he uses training plan \(B\).
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