Edexcel P2 (Pure Mathematics 2) 2023 June

1. The continuous curve \(C\) has equation \(y = \mathrm { f } ( x )\).

A table of values of \(x\) and \(y\) for \(y = \mathrm { f } ( x )\) is shown below. Use the trapezium rule with all the values of \(y\) in the table to find an approximation for $$\int _ { 4 } ^ { 5 } f ( x ) d x$$ giving your answer to 3 decimal places.

1. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.

$$f ( x ) = 4 x ^ { 3 } - 8 x ^ { 2 } + 5 x + a$$ where \(a\) is a constant.
Given that ( \(2 x - 3\) ) is a factor of \(\mathrm { f } ( x )\),

1. use the factor theorem to show that \(a = - 3\)
2. Hence show that the equation \(\mathrm { f } ( x ) = 0\) has only one real root.

1. A circle \(C\) has centre \(( 2,5 )\)

Given that the point \(P ( 8 , - 3 )\) lies on \(C\)

1. 1. find the radius of \(C\)
   2. find an equation for \(C\)
2. Find the equation of the tangent to \(C\) at \(P\) giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers to be found.

1. The binomial expansion, in ascending powers of \(x\), of

$$( 3 + p x ) ^ { 5 }$$ where \(p\) is a constant, can be written in the form $$A + B x + C x ^ { 2 } + D x ^ { 3 } \ldots$$ where \(A\), \(B\), \(C\) and \(D\) are constants.

1. Find the value of \(A\) Given that
   • \(B = 18 D\)
   • \(p < 0\)
   • find
     1. the value of \(p\)
     2. the value of \(C\)

1. Use the laws of logarithms to solve

$$\log _ { 2 } ( 16 x ) + \log _ { 2 } ( x + 1 ) = 3 + \log _ { 2 } ( x + 6 )$$

1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

A software developer released an app to download.
The numbers of downloads of the app each month, in thousands, for the first three months after the app was released were $$2 k - 15 \quad k \quad k + 4$$ where \(k\) is a constant.
Given that the numbers of downloads each month are modelled as a geometric series,

1. show that \(k ^ { 2 } - 7 k - 60 = 0\)
2. predict the number of downloads in the 4th month. The total number of all downloads of the app is predicted to exceed 3 million for the first time in the \(N\) th month.
3. Calculate the value of \(N\) according to the model.

1. The height of a river above a fixed point on the riverbed was monitored over a 7-day period.

The height of the river, \(H\) metres, \(t\) days after monitoring began, was given by $$H = \frac { \sqrt { t } } { 20 } \left( 20 + 6 t - t ^ { 2 } \right) + 17 \quad 0 \leqslant t \leqslant 7$$ Given that \(H\) has a stationary value at \(t = \alpha\)

1. use calculus to show that \(\alpha\) satisfies the equation $$5 \alpha ^ { 2 } - 18 \alpha - 20 = 0$$
2. Hence find the value of \(\alpha\), giving your answer to 3 decimal places.
3. Use further calculus to prove that \(H\) is a maximum at this value of \(\alpha\).

1. (i) A student writes the following statement:
   "When \(a\) and \(b\) are consecutive prime numbers, \(a ^ { 2 } + b ^ { 2 }\) is never a multiple of 10 "
   Prove by counter example that this statement is not true.
   (ii) Given that \(x\) and \(y\) are even integers greater than 0 and less than 6 , prove by exhaustion, that

$$1 < x ^ { 2 } - \frac { x y } { 4 } < 15$$

1. In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

1. Show that $$3 \cos \theta ( \tan \theta \sin \theta + 3 ) = 11 - 5 \cos \theta$$ may be written as $$3 \cos ^ { 2 } \theta - 14 \cos \theta + 8 = 0$$
2. Hence solve, for \(0 < x < 360 ^ { \circ }\) $$3 \cos 2 x ( \tan 2 x \sin 2 x + 3 ) = 11 - 5 \cos 2 x$$ giving your answers to one decimal place.

1. The curve \(C\) has equation

$$y = \frac { ( x - k ) ^ { 2 } } { \sqrt { x } } \quad x > 0$$ where \(k\) is a positive constant.

1. Show that $$\int _ { 1 } ^ { 16 } \frac { ( x - k ) ^ { 2 } } { \sqrt { x } } \mathrm {~d} x = a k ^ { 2 } + b k + \frac { 2046 } { 5 }$$ where \(a\) and \(b\) are integers to be found. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{0e3b364c-151b-471d-acb6-01afb018fb75-26_645_670_904_699} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure} Figure 1 shows a sketch of the curve \(C\) and the line \(l\).
   Given that \(l\) intersects \(C\) at the point \(A ( 1,9 )\) and at the point \(B ( 16 , q )\) where \(q\) is a constant,
2. show that \(k = 4\) The region \(R\), shown shaded in Figure 1, is bounded by \(C\) and \(l\)
   Using the answers to parts (a) and (b),
3. find the area of region \(R\)

1. A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by

$$\begin{aligned} u _ { n + 1 } & = b - a u _ { n }
u _ { 1 } & = 3 \end{aligned}$$ where \(a\) and \(b\) are constants.

1. Find, in terms of \(a\) and \(b\),
   1. \(u _ { 2 }\)
   2. \(u _ { 3 }\) Given
      • \(\sum _ { n = 1 } ^ { 3 } u _ { n } = 153\)
2. \(b = a + 9\)
3. show that
$$a ^ { 2 } - 5 a - 66 = 0$$
4. Hence find the larger possible value of \(u _ { 2 }\)