Questions — Edexcel P2 (158 questions)

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Edexcel P2 2022 June Q3
7 marks Moderate -0.8
  1. Show that the following statement is false: "\((n + 1)^3 - n^3\) is prime for all \(n \in \mathbb{N}\)" [2]
  2. Given that the points \(A(1, 0)\), \(B(3, -10)\) and \(C(7, -6)\) lie on a circle, prove that \(AB\) is a diameter of this circle. [5]
Edexcel P2 2022 June Q4
6 marks Standard +0.3
In this question you must show all stages of your working. Give your answers in fully simplified surd form. Given that \(a\) and \(b\) are positive constants, solve the simultaneous equations \begin{align} a - b &= 8
\log_5 a + \log_5 b &= 3 \end{align} [6]
Edexcel P2 2022 June Q5
6 marks Standard +0.3
In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. Solve, for \(-180° < \theta \leq 180°\), the equation $$3\tan(\theta + 43°) = 2\cos(\theta + 43°)$$ [6]
Edexcel P2 2022 June Q6
8 marks Moderate -0.3
In a geometric sequence \(u_1, u_2, u_3, \ldots\)
  • the common ratio is \(r\)
  • \(u_2 + u_3 = 6\)
  • \(u_4 = 8\)
  1. Show that \(r\) satisfies $$3r^2 - 4r - 4 = 0$$ [3]
Given that the geometric sequence has a sum to infinity,
  1. find \(u_1\) [3]
  2. find \(S_∞\) [2]
Edexcel P2 2022 June Q7
7 marks Standard +0.3
$$f(x) = Ax^3 + 6x^2 - 4x + B$$ where \(A\) and \(B\) are constants. Given that
  • \((x + 2)\) is a factor of \(f(x)\)
  • \(\int_{-3}^{5} f(x)dx = 176\)
Find the value of \(A\) and the value of \(B\). [7]
Edexcel P2 2022 June Q8
8 marks Standard +0.3
In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. A curve has equation $$y = 256x^4 - 304x - 35 + \frac{27}{x^2} \quad x \neq 0$$
  1. Find \(\frac{dy}{dx}\) [3]
  2. Hence find the coordinates of the stationary points of the curve. [5]
Edexcel P2 2022 June Q9
9 marks Moderate -0.8
A scientist is using carbon-14 dating to determine the age of some wooden items. The equation for carbon-14 dating an item is given by $$N = k\lambda^t$$ where
  • \(N\) grams is the amount of carbon-14 currently present in the item
  • \(k\) grams was the initial amount of carbon-14 present in the item
  • \(t\) is the number of years since the item was made
  • \(\lambda\) is a constant, with \(0 < \lambda < 1\)
  1. Sketch the graph of \(N\) against \(t\) for \(k = 1\) [2]
Given that it takes 5700 years for the amount of carbon-14 to reduce to half its initial value,
  1. show that the value of the constant \(\lambda\) is 0.999878 to 6 decimal places. [2]
Given that Item A
  • is known to have had 15 grams of carbon-14 present initially
  • is thought to be 3250 years old
  1. calculate, to 3 significant figures, how much carbon-14 the equation predicts is currently in Item A. [2]
Item B is known to have initially had 25 grams of carbon-14 present, but only 18 grams now remain.
  1. Use algebra to calculate the age of Item B to the nearest 100 years. [3]
Edexcel P2 2022 June Q10
12 marks Standard +0.3
The circle \(C\) has centre \(X(3, 5)\) and radius \(r\) The line \(l\) has equation \(y = 2x + k\), where \(k\) is a constant.
  1. Show that \(l\) and \(C\) intersect when $$5x^2 + (4k - 26)x + k^2 - 10k + 34 - r^2 = 0$$ [3]
Given that \(l\) is a tangent to \(C\),
  1. show that \(5r^2 = (k + p)^2\), where \(p\) is a constant to be found. [3]
\includegraphics{figure_2} The line \(l\)
  • cuts the \(y\)-axis at the point \(A\)
  • touches the circle \(C\) at the point \(B\)
as shown in Figure 2. Given that \(AB = 2r\)
  1. find the value of \(k\) [6]