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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 PURE S1 S2 S3 S4 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
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CAIE P2 2018 November Q1
5 marks Moderate -0.3
  1. Solve the equation \(|9x - 2| = |3x + 2|\). [3]
  2. Hence, using logarithms, solve the equation \(|3^{x+2} - 2| = |3^{x+1} + 2|\), giving your answer correct to 3 significant figures. [2]
CAIE P2 2018 November Q2
5 marks Moderate -0.5
Show that \(\int_1^7 \frac{6}{2x + 1} \, dx = \ln 125\). [5]
CAIE P2 2018 November Q3
5 marks Standard +0.3
Solve the equation \(\sec^2 \theta = 3 \cosec \theta\) for \(0° < \theta < 180°\). [5]
CAIE P2 2018 November Q4
11 marks Moderate -0.3
\includegraphics{figure_4} The diagram shows the curve with equation $$y = x^4 + 2x^3 + 2x^2 - 12x - 32.$$ The curve crosses the \(x\)-axis at points with coordinates \((\alpha, 0)\) and \((\beta, 0)\).
  1. Use the factor theorem to show that \((x + 2)\) is a factor of $$x^4 + 2x^3 + 2x^2 - 12x - 32.$$ [2]
  2. Show that \(\beta\) satisfies an equation of the form \(x = \sqrt[3]{p + qx}\), and state the values of \(p\) and \(q\). [3]
  3. Use an iterative formula based on the equation in part (ii) to find the value of \(\beta\) correct to 4 significant figures. Give the result of each iteration to 6 significant figures. [3]
CAIE P2 2018 November Q5
9 marks Standard +0.3
A curve has parametric equations $$x = t + \ln(t + 1), \quad y = 3te^{2t}.$$
  1. Find the equation of the tangent to the curve at the origin. [5]
  2. Find the coordinates of the stationary point, giving each coordinate correct to 2 decimal places. [4]
CAIE P2 2018 November Q6
8 marks Standard +0.3
\includegraphics{figure_6} The diagram shows the curve with equation \(y = \sqrt{1 + 3\cos^2(\frac{1}{2}x)}\) for \(0 \leqslant x \leqslant \pi\). The region \(R\) is bounded by the curve, the axes and the line \(x = \pi\).
  1. Use the trapezium rule with two intervals to find an approximation to the area of \(R\), giving your answer correct to 3 significant figures. [3]
  2. The region \(R\) is rotated completely about the \(x\)-axis. Without using a calculator, find the exact volume of the solid produced. [5]
CAIE P2 2018 November Q7
10 marks Standard +0.3
\includegraphics{figure_7} The diagram shows the curve with equation \(y = \sin 2x + 3\cos 2x\) for \(0 \leqslant x \leqslant \pi\). At the points \(P\) and \(Q\) on the curve, the gradient of the curve is 3.
  1. Find an expression for \(\frac{dy}{dx}\). [2]
  2. By first expressing \(\frac{dy}{dx}\) in the form \(R\cos(2x + \alpha)\), where \(R > 0\) and \(0 < \alpha < \frac{1}{2}\pi\), find the \(x\)-coordinates of \(P\) and \(Q\), giving your answers correct to 4 significant figures. [8]
Edexcel P2 2022 June Q1
4 marks Easy -1.2
Find the first four terms, in ascending powers of \(x\), of the binomial expansion of $$\left(2 + \frac{3}{8}x\right)^{10}$$ Give each coefficient as an integer. [4]
Edexcel P2 2022 June Q2
8 marks Moderate -0.3
\includegraphics{figure_1} Figure 1 shows the graph of $$y = 1 - \log_{10}(\sin x) \quad 0 < x < \pi$$ where \(x\) is in radians. The table below shows some values of \(x\) and \(y\) for this graph, with values of \(y\) given to 3 decimal places.
\(x\)0.511.522.53
\(y\)1.3191.0011.2231.850
  1. Complete the table above, giving values of \(y\) to 3 decimal places. [2]
  2. Use the trapezium rule with all the \(y\) values in the completed table to find, to 2 decimal places, an estimate for $$\int_{0.5}^{3} \left(1 - \log_{10}(\sin x)\right) dx$$ [3]
  3. Use your answer to part (b) to find an estimate for $$\int_{0.5}^{3} \left(3 + \log_{10}(\sin x)\right) dx$$ [3]
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]