Questions — CAIE P2 (709 questions)

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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]
CAIE P2 2024 November Q4
5 marks Moderate -0.8
The polynomial \(\text{p}(x)\) is defined by $$\text{p}(x) = ax^3 - ax^2 - 15x + 18,$$ where \(a\) is a constant. It is given that \((x + 2)\) is a factor of \(\text{p}(x)\).
  1. Find the value of \(a\). [2]
  2. Hence factorise \(\text{p}(x)\) completely. [3]
CAIE P2 2024 November Q4
3 marks Moderate -0.3
  1. Solve the equation \(\text{p}(\cos ec^2 \theta) = 0\) for \(-90° < \theta < 90°\). [3]
CAIE P2 2024 November Q5
17 marks Standard +0.3
It is given that \(\int_a^{a^2} \frac{10}{2x+1} dx = 7\), where \(a\) is a constant greater than \(1\).
  1. Show that \(a = \sqrt[9]{0.5e^{1.4}(2a+1) - 0.5}\). [5]
  2. Use an iterative formula, based on the equation in part (a), to find the value of \(a\) correct to \(3\) significant figures. Use an initial value of \(2\) and give the result of each iteration to \(5\) significant figures. [3]
CAIE P2 2024 November Q6
7 marks Standard +0.3
A curve has parametric equations $$x = \frac{e^{2t} - 2}{e^{2t} + 1}, \quad y = e^{3t} + 1.$$
  1. Find an expression for \(\frac{dy}{dx}\) in terms of \(t\). [4]
  2. Find the exact gradient of the curve at the point where the curve crosses the \(y\)-axis. [3]
CAIE P2 2024 November Q7
11 marks Standard +0.8
  1. Prove that \(\cos(\theta + 30°)\cos(\theta + 60°) = \frac{1}{4}\sqrt{3} - \frac{1}{2}\sin 2\theta\). [4]
  2. Solve the equation \(5\cos(2\alpha + 30°)\cos(2\alpha + 60°) = 1\) for \(0° < \alpha < 90°\). [4]
  3. Show that the exact value of \(\cos 20° \cos 50° + \cos 40° \cos 70°\) is \(\frac{1}{2}\sqrt{3}\). [3]
CAIE P2 2015 June Q1
4 marks Moderate -0.8
  1. Use logarithms to solve the equation \(2^x = 20^5\), giving the answer correct to 3 significant figures. [2]
  2. Hence determine the number of integers \(n\) satisfying $$20^{-5} < 2^n < 20^5.$$ [2]