Questions — CAIE (7646 questions)

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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]
CAIE P2 2015 June Q2
6 marks Moderate -0.8
  1. Given that \((x + 2)\) is a factor of $$4x^3 + ax^2 - (a + 1)x - 18,$$ find the value of the constant \(a\). [3]
  2. When \(a\) has this value, factorise \(4x^3 + ax^2 - (a + 1)x - 18\) completely. [3]
CAIE P2 2015 June Q3
6 marks Standard +0.3
It is given that \(\theta\) is an acute angle measured in degrees such that $$2\sec^2\theta + 3\tan\theta = 22.$$
  1. Find the value of \(\tan\theta\). [3]
  2. Use an appropriate formula to find the exact value of \(\tan(\theta + 135°)\). [3]
CAIE P2 2015 June Q4
7 marks Moderate -0.3
\includegraphics{figure_4} The diagram shows the curve \(y = e^x + 4e^{-2x}\) and its minimum point \(M\).
  1. Show that the \(x\)-coordinate of \(M\) is \(\ln 2\). [3]
  2. The region shaded in the diagram is enclosed by the curve and the lines \(x = 0\), \(x = \ln 2\) and \(y = 0\). Use integration to show that the area of the shaded region is \(\frac{5}{2}\). [4]
CAIE P2 2015 June Q5
12 marks Standard +0.3
  1. By sketching a suitable pair of graphs, show that the equation $$|3x| = 16 - x^4$$ has two real roots. [3]
  2. Use the iterative formula \(x_{n+1} = \sqrt[4]{16 - 3x_n}\) to find one of the real roots correct to 3 decimal places. Give the result of each iteration to 5 decimal places. [3]
  3. Hence find the coordinates of each of the points of intersection of the graphs \(y = |3x|\) and \(y = 16 - x^4\), giving your answers correct to 3 decimal places. [2]
CAIE P2 2015 June Q6
9 marks Standard +0.3
\includegraphics{figure_6} The diagram shows part of the curve with equation $$y = 4\sin^2 x + 8\sin x + 3$$ and its point of intersection \(P\) with the \(x\)-axis.
  1. Find the exact \(x\)-coordinate of \(P\). [3]
  2. Show that the equation of the curve can be written $$y = 5 + 8\sin x - 2\cos 2x,$$ and use integration to find the exact area of the shaded region enclosed by the curve and the axes. [6]
CAIE P2 2015 June Q7
10 marks Standard +0.3
  1. Find the gradient of the curve $$3\ln x + 4\ln y + 6xy = 6$$ at the point \((1, 1)\). [4]
  2. The parametric equations of a curve are $$x = \frac{10}{t} - t, \quad y = \sqrt{2t - 1}.$$ Find the gradient of the curve at the point \((-3, 3)\). [6]
CAIE P2 2003 November Q1
3 marks Moderate -0.8
Find the set of values of \(x\) satisfying the inequality \(|8 - 3x| < 2\). [3]
CAIE P2 2003 November Q2
5 marks Moderate -0.3
\includegraphics{figure_2} Two variable quantities \(x\) and \(y\) are related by the equation $$y = k(a^{-x}),$$ where \(a\) and \(k\) are constants. Four pairs of values of \(x\) and \(y\) are measured experimentally. The result of plotting \(\ln y\) against \(x\) is shown in the diagram. Use the diagram to estimate the values of \(a\) and \(k\). [5]
CAIE P2 2003 November Q3
6 marks Moderate -0.8
The polynomial \(x^4 - 6x^2 + x + a\) is denoted by \(f(x)\).
  1. It is given that \((x + 1)\) is a factor of \(f(x)\). Find the value of \(a\). [2]
  2. When \(a\) has this value, verify that \((x - 2)\) is also a factor of \(f(x)\) and hence factorise \(f(x)\) completely. [4]
CAIE P2 2003 November Q4
7 marks Moderate -0.3
  1. Express \(\cos \theta + (\sqrt{3}) \sin \theta\) in the form \(R \cos(\theta - \alpha)\), where \(R > 0\) and \(0 < \alpha < \frac{1}{2}\pi\), giving the exact value of \(\alpha\). [3]
  2. Hence show that one solution of the equation $$\cos \theta + (\sqrt{3}) \sin \theta = \sqrt{2}$$ is \(\theta = \frac{7}{12}\pi\), and find the other solution in the interval \(0 < \theta < 2\pi\). [4]