Edexcel PURE 2024 October

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

A table of values of \(x\) and \(y\) for \(y = \mathrm { f } ( x )\) is shown below. Using the trapezium rule with all the values of \(y\) in the given table,

1. find an estimate for $$\int _ { 0.5 } ^ { 5.5 } \mathrm { f } ( x ) \mathrm { d } x$$ giving your answer to one decimal place.
2. Using your answer to part (a) and making your method clear, estimate $$\int _ { 0.5 } ^ { 5.5 } ( \mathrm { f } ( x ) + 4 x ) \mathrm { d } x$$

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

$$\begin{gathered} u _ { 1 } = 7 \\ u _ { n + 1 } = ( - 1 ) ^ { n } u _ { n } + k \end{gathered}$$ where \(k\) is a constant.

1. Show that \(u _ { 5 } = 7\) Given that \(\sum _ { r = 1 } ^ { 4 } u _ { r } = 30\)
2. find the value of \(k\).
3. Hence find the value of \(\sum _ { r = 1 } ^ { 150 } u _ { r }\)

3. $$f ( x ) = 2 x ^ { 3 } - x ^ { 2 } + A x + B$$ where \(A\) and \(B\) are integers.
Given that when \(\mathrm { f } ( x )\) is divided by \(( x + 3 )\) the remainder is 55

1. show that $$3 A - B = - 118$$ Given also that \(( 2 x - 5 )\) is a factor of \(\mathrm { f } ( x )\),
2. find the value of \(A\) and the value of \(B\).
3. Hence find the quotient when \(\mathrm { f } ( x )\) is divided by ( \(x - 7\) )

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

Solutions relying entirely on calculator technology are not acceptable.
The curve \(C\) has equation $$y = 4 x ^ { \frac { 1 } { 2 } } + 9 x ^ { - \frac { 1 } { 2 } } + 3 \quad x > 0$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) giving each term in simplest form.
2. Hence find the \(x\) coordinate of the stationary point of \(C\).
3. 1. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) giving each term in simplest form.
   2. Hence determine the nature of the stationary point of \(C\), giving a reason for your answer.
4. State the range of values of \(x\) for which \(y\) is decreasing.

1. (a) Find, in terms of \(a\), the first 3 terms, in ascending powers of \(x\), of the binomial expansion of

$$( 2 + a x ) ^ { 6 }$$ where \(a\) is a non-zero constant. Give each term in simplest form. $$f ( x ) = \left( 3 + \frac { 1 } { x } \right) ^ { 2 } ( 2 + a x ) ^ { 6 }$$ Given that the constant term in the expansion of \(\mathrm { f } ( x )\) is 576
(b) find the value of \(a\).

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

Using the laws of logarithms, solve $$\log _ { 4 } ( 12 - 2 x ) = 2 + 2 \log _ { 4 } ( x + 1 )$$

1. Jem pays money into a savings scheme, \(A\), over a period of 300 months.

Jem pays \(\pounds 20\) into scheme \(A\) in month \(1 , \pounds 20.50\) in month \(2 , \pounds 21\) in month 3 and so on, so that the amounts Jem pays each month form an arithmetic sequence.

1. Show that Jem pays \(\pounds 69.50\) into scheme \(A\) in month 100
2. Find the total amount that Jem pays into scheme \(A\) over the period of 300 months. Kim pays money into a different savings scheme, \(B\), over the same period of 300 months. In a model, the amounts Kim pays into scheme \(B\) increase by the same percentage each month, so that the amounts Kim pays each month form a geometric sequence. Given that Kim pays
   • \(\pounds 20\) into scheme \(B\) in month 1
   • \(\pounds 250\) into scheme \(B\) in month 300
   • use the model to calculate, to the nearest \(\pounds 10\), the difference between the total amount paid into scheme \(A\) and the total amount paid into scheme \(B\) over the period of 300 months.

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

Solutions relying entirely on calculator technology are not acceptable. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve \(C _ { 1 }\) with equation $$y = x ^ { 2 } + 3 \quad x > 0$$ and part of the curve \(C _ { 2 }\) with equation $$y = 13 - \frac { 9 } { x ^ { 2 } } \quad x > 0$$ The curves \(C _ { 1 }\) and \(C _ { 2 }\) intersect at the points \(P\) and \(Q\) as shown in Figure 1 .

1. Use algebra to find the \(x\) coordinate of \(P\) and the \(x\) coordinate of \(Q\). 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 \(R\).

1. In this question you must show detailed reasoning.

Solutions relying entirely on calculator technology are not acceptable.

1. Show that the equation $$2 \tan \theta = 3 \cos \theta$$ can be written as $$3 \sin ^ { 2 } \theta + 2 \sin \theta - 3 = 0$$
2. Hence solve, for \(- \pi < x < \pi\), the equation $$2 \tan \left( 2 x + \frac { \pi } { 3 } \right) = 3 \cos \left( 2 x + \frac { \pi } { 3 } \right)$$ giving your answers to 3 significant figures.

1. The circle \(C\) has equation

$$x ^ { 2 } + y ^ { 2 } + 4 x - 30 y + 209 = 0$$

1. Find
   1. the coordinates of the centre of \(C\),
   2. the exact value of the radius of \(C\). The line \(L\) has equation \(y = m x + 1\), where \(m\) is a constant.
      Given that \(L\) is the tangent to \(C\) at the point \(P\),
2. show that $$2 m ^ { 2 } - 7 m - 22 = 0$$
3. Hence find the possible pairs of coordinates of \(P\).

1. (i) Prove by counter example that the statement
   "If \(n\) is a prime number then \(3 ^ { n } + 2\) is also a prime number." is false.
   (ii) Use proof by exhaustion to prove that if \(m\) is an integer that is not divisible by 3 , then

$$m ^ { 2 } - 1$$ is divisible by 3