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CAIE P2 2024 June Q1
4 marks Standard +0.3
Solve the inequality \(|5x + 7| > |2x - 3|\). [4]
CAIE P2 2024 June Q2
4 marks Standard +0.3
Use logarithms to solve the equation \(6^{2x-1} = 5e^{3x+2}\). Give your answer correct to 4 significant figures. [4]
CAIE P2 2024 June Q3
8 marks Moderate -0.3
\includegraphics{figure_3} The diagram shows the curve with equation \(y = 8e^{-x} - e^{2x}\). The curve crosses the y-axis at the point A and the x-axis at the point B. The shaded region is bounded by the curve and the two axes.
  1. Find the gradient of the curve at A. [3]
  2. Show that the x-coordinate of B is \(\ln 2\) and hence find the area of the shaded region. [5]
CAIE P2 2024 June Q4
7 marks Standard +0.3
A curve is defined by the parametric equations $$x = 4\cos^2 t, \quad y = \sqrt{3}\sin 2t,$$ for values of \(t\) such that \(0 < t < \frac{1}{2}\pi\). Find the equation of the normal to the curve at the point for which \(t = \frac{1}{6}\pi\). Give your answer in the form \(ax + by + c = 0\) where \(a\), \(b\) and \(c\) are integers. [7]
CAIE P2 2024 June Q5
8 marks Standard +0.3
The polynomial \(p(x)\) is defined by \(p(x) = 9x^3 + 18x^2 + 5x + 4\).
  1. Find the quotient when \(p(x)\) is divided by \((3x + 2)\), and show that the remainder is 6. [3]
  2. Find the value of \(\int_0^2 \frac{p(x)}{3x + 2} \, dx\), giving your answer in the form \(a + \ln b\) where \(a\) and \(b\) are integers. [5]
CAIE P2 2024 June Q6
9 marks Standard +0.3
\includegraphics{figure_6} The diagram shows the curve with equation \(y = \frac{\ln(2x + 1)}{x + 3}\). The curve has a maximum point M.
  1. Find an expression for \(\frac{dy}{dx}\). [2]
  2. Show that the x-coordinate of M satisfies the equation \(x = \frac{x + 3}{\ln(2x + 1)} - 0.5\). [2]
  3. Show by calculation that the x-coordinate of M lies between 2.5 and 3.0. [2]
  4. Use an iterative formula based on the equation in part (b) to find the x-coordinate of M correct to 4 significant figures. Give the result of each iteration to 6 significant figures. [3]
CAIE P2 2024 June Q7
10 marks Standard +0.3
  1. Prove that \(2\sin\theta\cosec 2\theta \equiv \sec\theta\). [2]
  2. Solve the equation \(\tan^2\theta + 7\sin\theta\cosec 2\theta = 8\) for \(-\pi < \theta < \pi\). [5]
  3. Find \(\int 8\sin^2\frac{1}{2}x\cosec^2 x \, dx\). [3]
CAIE P2 2023 March Q1
4 marks Standard +0.3
Find the exact value of \(\int_0^{\frac{\pi}{4}} 2 \tan^2(\frac{1}{2}x) \, dx\). [4]
CAIE P2 2023 March Q2
5 marks Standard +0.8
Solve the equation \(\tan(\theta - 60°) = 3 \cot \theta\) for \(-90° < \theta < 90°\). [5]
CAIE P2 2023 March Q3
8 marks Moderate -0.3
The polynomial \(p(x)\) is defined by $$p(x) = ax^3 - ax^2 + ax + b,$$ where \(a\) and \(b\) are constants. It is given that \((x + 2)\) is a factor of \(p(x)\), and that the remainder is 35 when \(p(x)\) is divided by \((x - 3)\).
  1. Find the values of \(a\) and \(b\). [5]
  2. Hence factorise \(p(x)\) and show that the equation \(p(x) = 0\) has exactly one real root. [3]
CAIE P2 2023 March Q4
7 marks Moderate -0.3
  1. Sketch, on the same diagram, the graphs of \(y = |2x - 1|\) and \(y = 3x - 3\). [2]
  2. Solve the inequality \(|2x - 1| < 3x - 3\). [3]
  3. Find the smallest integer \(N\) satisfying the inequality \(|2 \ln N - 1| < 3 \ln N - 3\). [2]
CAIE P2 2023 March Q5
8 marks Standard +0.3
It is given that \(\int_1^a \left(\frac{4}{1 + 2x} + \frac{3}{x}\right) dx = \ln 10\), where \(a\) is a constant greater than 1.
  1. Show that \(a = \sqrt{90(1 + 2a)^{-2}}\). [5]
  2. Use an iterative formula, based on the equation in (a), to find the value of \(a\) correct to 3 significant figures. Use an initial value of 1.7 and give the result of each iteration to 5 significant figures. [3]
CAIE P2 2023 March Q6
8 marks Standard +0.3
\includegraphics{figure_6} The diagram shows the curve with equation \(y = \frac{4e^{2x} + 9}{e^x + 2}\). The curve has a minimum point \(M\) and crosses the \(y\)-axis at the point \(P\).
  1. Find the exact value of the gradient of the curve at \(P\). [4]
  2. Find the exact coordinates of \(M\). [4]
CAIE P2 2023 March Q7
10 marks Standard +0.3
\includegraphics{figure_7} The diagram shows the curve with parametric equations $$x = k \tan t, \quad y = 3 \sin 2t - 4 \sin t,$$ for \(0 < t < \frac{1}{2}\pi\). It is given that \(k\) is a positive constant. The curve crosses the \(x\)-axis at the point \(P\).
  1. Find the value of \(\cos t\) at \(P\), giving your answer as an exact fraction. [3]
  2. Express \(\frac{dy}{dx}\) in terms of \(k\) and \(\cos t\). [4]
  3. Given that the normal to the curve at \(P\) has gradient \(\frac{9}{10}\), find the value of \(k\), giving your answer as an exact fraction. [3]
CAIE P2 2024 March Q1
4 marks Moderate -0.5
Use logarithms to solve the equation \(3^{4t+3} = 5^{2t+7}\). Give your answer correct to 3 significant figures. [4]
CAIE P2 2024 March Q2
4 marks Moderate -0.3
  1. Sketch the graph of \(y = |3x - 7|\), stating the coordinates of the points where the graph meets the axes. [2]
  2. Hence find the set of values of the constant \(k\) for which the equation \(|3x - 7| = k(x - 4)\) has exactly two real roots. [2]
CAIE P2 2024 March Q3
7 marks Moderate -0.3
The polynomial \(\mathrm{p}(x)\) is defined by $$\mathrm{p}(x) = 6x^3 + ax^2 + 3x - 10,$$ where \(a\) is a constant. It is given that \((2x - 1)\) is a factor of \(\mathrm{p}(x)\).
  1. Find the value of \(a\) and hence factorise \(\mathrm{p}(x)\) completely. [5]
  2. Solve the equation \(\mathrm{p}(\cos\theta) = 0\) for \(-90° < \theta < 90°\). [2]
CAIE P2 2024 March Q4
7 marks Moderate -0.3
\includegraphics{figure_4} The diagram shows the curve with equation \(y = \sqrt{1 + e^{0.5x}}\). The shaded region is bounded by the curve and the straight lines \(x = 0\), \(x = 6\) and \(y = 0\).
  1. Use the trapezium rule with three intervals to find an approximation to the area of the shaded region. Give your answer correct to 3 significant figures. [3]
  2. The shaded region is rotated completely about the \(x\)-axis. Find the exact volume of the solid produced. [4]
CAIE P2 2024 March Q5
12 marks Standard +0.3
\includegraphics{figure_5} The diagram shows part of the curve with equation \(y = \frac{x^3}{x + 2}\). At the point \(P\), the gradient of the curve is 6.
  1. Show that the \(x\)-coordinate of \(P\) satisfies the equation \(x = \sqrt[3]{12x + 12}\). [4]
  2. Show by calculation that the \(x\)-coordinate of \(P\) lies between 3.8 and 4.0. [2]
  3. Use an iterative formula, based on the equation in part (a), to find the \(x\)-coordinate of \(P\) correct to 3 significant figures. Show the result of each iteration to 5 significant figures. [3]
CAIE P2 2024 March Q6
9 marks Standard +0.3
\includegraphics{figure_6} The diagram shows the curve with parametric equations $$x = 1 + \sqrt{t}, \quad y = (\ln t + 2)(\ln t - 3),$$ for \(0 < t < 25\). The curve crosses the \(x\)-axis at the points \(A\) and \(B\) and has a minimum point \(M\).
  1. Show that \(\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{4\ln t - 2}{\sqrt{t}}\). [4]
  2. Find the exact gradient of the curve at \(B\). [2]
  3. Find the exact coordinates of \(M\). [3]
CAIE P2 2024 March Q7
10 marks Standard +0.8
  1. Prove that $$\sin 2\theta (a \cot\theta + b \tan\theta) \equiv a + b + (a - b) \cos 2\theta,$$ where \(a\) and \(b\) are constants. [4]
  2. Find the exact value of \(\int_{\frac{\pi}{12}}^{\frac{\pi}{6}} \sin 2\theta (5 \cot\theta + 3 \tan\theta) \mathrm{d}\theta\). [3]
  3. Solve the equation \(\sin^2\alpha\left(2\cot\frac{1}{2}\alpha + 7\tan\frac{1}{2}\alpha\right) = 11\) for \(-\pi < \alpha < \pi\). [3]
CAIE P2 2024 November Q1
5 marks Moderate -0.3
The variables \(x\) and \(y\) satisfy the equation \(a^{2y} = e^{3x+k}\), where \(a\) and \(k\) are constants. The graph of \(y\) against \(x\) is a straight line.
  1. Use logarithms to show that the gradient of the straight line is \(\frac{3}{2\ln a}\). [1]
  2. Given that the straight line passes through the points \((0.4, 0.95)\) and \((3.3, 3.80)\), find the values of \(a\) and \(k\). [4]
CAIE P2 2024 November Q2
4 marks Standard +0.3
Solve the inequality \(|x - 7| > 4x + 3\). [4]
CAIE P2 2024 November Q3
3 marks Moderate -0.3
The function \(\text{f}\) is defined by \(\text{f}(x) = \tan^2\left(\frac{1}{2}x\right)\) for \(0 \leqslant x < \pi\).
  1. Find the exact value of \(\text{f}'\left(\frac{\pi}{3}\right)\). [3]
CAIE P2 2024 November Q3
4 marks Moderate -0.3
  1. Find the exact value of \(\int_0^{\frac{\pi}{4}} \left(\text{f}(x) + \sin x\right) dx\). [4]