Questions P3 (1243 questions)

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CAIE P3 2024 June Q3
5 marks Standard +0.3
The square roots of \(24 - 7i\) can be expressed in the Cartesian form \(x + iy\), where \(x\) and \(y\) are real and exact. By first forming a quartic equation in \(x\) or \(y\), find the square roots of \(24 - 7i\) in exact Cartesian form. [5]
CAIE P3 2024 June Q4
4 marks Moderate -0.5
\includegraphics{figure_4} The variables \(x\) and \(y\) satisfy the equation \(ky = e^{cx}\), where \(k\) and \(c\) are constants. The graph of \(\ln y\) against \(x\) is a straight line passing through the points \((2.80, 0.372)\) and \((5.10, 2.21)\), as shown in the diagram. Find the values of \(k\) and \(c\). Give each value correct to 2 significant figures. [4]
CAIE P3 2024 June Q5
5 marks Moderate -0.8
Express \(\frac{6x^2 - 2x + 2}{(x - 1)(2x + 1)}\) in partial fractions. [5]
CAIE P3 2024 June Q6
7 marks Standard +0.3
  1. On an Argand diagram shade the region whose points represent complex numbers \(z\) which satisfy both the inequalities \(|z - 4 - 3i| \leqslant 2\) and \(\arg(z - 2 - i) \geqslant \frac{1}{4}\pi\). [5]
  2. Calculate the greatest value of \(\arg z\) for points in this region. [2]
CAIE P3 2024 June Q7
3 marks Easy -1.2
Let \(f(x) = 8x^3 + 54x^2 - 17x - 21\).
  1. Show that \(x + 7\) is a factor of \(f(x)\). [1]
  2. Find the quotient when \(f(x)\) is divided by \(x + 7\). [2]
CAIE P3 2024 June Q7
3 marks Standard +0.3
  1. Hence solve the equation $$8 \cos^3 \theta + 54 \cos^2 \theta - 17 \cos \theta - 21 = 0,$$ for \(0° \leqslant \theta \leqslant 360°\). [3]
CAIE P3 2024 June Q8
8 marks Challenging +1.2
  1. Express \(3 \cos 2x - \sqrt{3} \sin 2x\) in the form \(R \cos(2x + \alpha)\), where \(R > 0\) and \(0 < \alpha < \frac{1}{2}\pi\). Give the exact values of \(R\) and \(\alpha\). [3]
  2. Hence find the exact value of \(\int_0^{\frac{1}{2}\pi} \frac{3}{(3 \cos 2x - \sqrt{3} \sin 2x)^2} \, dx\), simplifying your answer. [5]
CAIE P3 2024 June Q9
11 marks Challenging +1.2
\includegraphics{figure_9} A container in the shape of a cuboid has a square base of side \(x\) and a height of \((10 - x)\). It is given that \(x\) varies with time, \(t\), where \(t > 0\). The container decreases in volume at a rate which is inversely proportional to \(t\). When \(t = \frac{1}{10}\), \(x = \frac{1}{2}\) and the rate of decrease of \(x\) is \(\frac{20}{37}\).
  1. Show that \(x\) and \(t\) satisfy the differential equation $$\frac{dx}{dt} = \frac{-1}{2t(20x - 3x^2)}$$ [5]
  2. Solve the differential equation, obtaining an expression for \(t\) in terms of \(x\). [6]
CAIE P3 2024 June Q10
11 marks Standard +0.3
The equations of two straight lines are $$\mathbf{r} = \mathbf{i} + \mathbf{j} + 2a\mathbf{k} + \lambda(3\mathbf{i} + 4\mathbf{j} + a\mathbf{k}) \quad \text{and} \quad \mathbf{r} = -3\mathbf{i} - \mathbf{j} + 4\mathbf{k} + \mu(-\mathbf{i} + 2\mathbf{j} + 2\mathbf{k}),$$ where \(a\) is a constant.
  1. Given that the acute angle between the directions of these lines is \(\frac{1}{4}\pi\), find the possible values of \(a\). [6]
  2. Given instead that the lines intersect, find the value of \(a\) and the position vector of the point of intersection. [5]
CAIE P3 2024 June Q11
9 marks Standard +0.8
Use the substitution \(2x = \tan \theta\) to find the exact value of $$\int_0^{\frac{1}{2}} \frac{12}{(1 + 4x^2)^2} \, dx .$$ Give your answer in the form \(a + b\pi\), where \(a\) and \(b\) are rational numbers. [9]
CAIE P3 2021 March Q1
3 marks Moderate -0.3
Solve the equation \(\ln(x^3 - 3) = 3 \ln x - \ln 3\). Give your answer correct to 3 significant figures. [3]
CAIE P3 2021 March Q2
5 marks Moderate -0.8
The polynomial \(ax^3 + 5x^2 - 4x + b\), where \(a\) and \(b\) are constants, is denoted by p\((x)\). It is given that \((x + 2)\) is a factor of p\((x)\) and that when p\((x)\) is divided by \((x + 1)\) the remainder is 2. Find the values of \(a\) and \(b\). [5]
CAIE P3 2021 March Q3
6 marks Standard +0.3
By first expressing the equation \(\tan(x + 45°) = 2 \cot x + 1\) as a quadratic equation in \(\tan x\), solve the equation for \(0° < x < 180°\). [6]
CAIE P3 2021 March Q4
7 marks Standard +0.3
The variables \(x\) and \(y\) satisfy the differential equation $$(1 - \cos x)\frac{dy}{dx} = y \sin x.$$ It is given that \(y = 4\) when \(x = \pi\).
  1. Solve the differential equation, obtaining an expression for \(y\) in terms of \(x\). [6]
  2. Sketch the graph of \(y\) against \(x\) for \(0 < x < 2\pi\). [1]
CAIE P3 2021 March Q5
8 marks Moderate -0.3
  1. Express \(\sqrt{7} \sin x + 2 \cos x\) in the form \(R \sin(x + \alpha)\), where \(R > 0\) and \(0° < \alpha < 90°\). State the exact value of \(R\) and give \(\alpha\) correct to 2 decimal places. [3]
  2. Hence solve the equation \(\sqrt{7} \sin 2\theta + 2 \cos 2\theta = 1\), for \(0° < \theta < 180°\). [5]
CAIE P3 2021 March Q6
7 marks Standard +0.3
Let \(\text{f}(x) = \frac{5a}{(2x - a)(3a - x)}\), where \(a\) is a positive constant.
  1. Express f\((x)\) in partial fractions. [3]
  2. Hence show that \(\int_a^{2a} \text{f}(x) \, dx = \ln 6\). [4]
CAIE P3 2021 March Q7
8 marks Standard +0.3
Two lines have equations \(\mathbf{r} = \begin{pmatrix} 1 \\ 3 \\ 2 \end{pmatrix} + s \begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix}\) and \(\mathbf{r} = \begin{pmatrix} 2 \\ 1 \\ 4 \end{pmatrix} + t \begin{pmatrix} 1 \\ -1 \\ 4 \end{pmatrix}\).
  1. Show that the lines are skew. [5]
  2. Find the acute angle between the directions of the two lines. [3]
CAIE P3 2021 March Q8
9 marks Standard +0.3
The complex numbers \(u\) and \(v\) are defined by \(u = -4 + 2\text{i}\) and \(v = 3 + \text{i}\).
  1. Find \(\frac{u}{v}\) in the form \(x + \text{i}y\), where \(x\) and \(y\) are real. [3]
  2. Hence express \(\frac{u}{v}\) in the form \(re^{\text{i}\theta}\), where \(r\) and \(\theta\) are exact. [2]
In an Argand diagram, with origin \(O\), the points \(A\), \(B\) and \(C\) represent the complex numbers \(u\), \(v\) and \(2u + v\) respectively.
  1. State fully the geometrical relationship between \(OA\) and \(BC\). [2]
  2. Prove that angle \(AOB = \frac{3}{4}\pi\). [2]
CAIE P3 2021 March Q9
11 marks Standard +0.3
Let \(\text{f}(x) = \frac{e^{2x} + 1}{e^{2x} - 1}\), for \(x > 0\).
  1. The equation \(x = \text{f}(x)\) has one root, denoted by \(a\). Verify by calculation that \(a\) lies between 1 and 1.5. [2]
  2. Use an iterative formula based on the equation in part (a) to determine \(a\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places. [3]
  3. Find f\('(x)\). Hence find the exact value of \(x\) for which f\('(x) = -8\). [6]
CAIE P3 2021 March Q10
11 marks Standard +0.8
\includegraphics{figure_10} The diagram shows the curve \(y = \sin 2x \cos^2 x\) for \(0 \leqslant x \leqslant \frac{1}{2}\pi\), and its maximum point \(M\).
  1. Using the substitution \(u = \sin x\), find the exact area of the region bounded by the curve and the \(x\)-axis. [5]
  2. Find the exact \(x\)-coordinate of \(M\). [6]
CAIE P3 2024 November Q1
4 marks Moderate -0.8
Expand \((9 - 3x)^{\frac{1}{2}}\) in ascending powers of \(x\), up to and including the term in \(x^2\), simplifying the coefficients. [4]
CAIE P3 2024 November Q2
3 marks Standard +0.8
  1. By sketching a suitable pair of graphs, show that the equation \(\cot 2x = \sec x\) has exactly one root in the interval \(0 < x < \frac{1}{2}\pi\). [2]
  2. Show that if a sequence of real values given by the iterative formula $$x_{n+1} = \frac{1}{2}\tan^{-1}(\cos x_n)$$ converges, then it converges to the root in part (a). [1]
CAIE P3 2024 November Q3
5 marks Standard +0.3
The square roots of \(6 - 8i\) can be expressed in the Cartesian form \(x + iy\), where \(x\) and \(y\) are real and exact. By first forming a quartic equation in \(x\) or \(y\), find the square roots of \(6 - 8i\) in exact Cartesian form. [5]
CAIE P3 2024 November Q4
3 marks Moderate -0.3
Solve the equation \(5^x = 5^{x+2} - 10\). Give your answer correct to 3 decimal places. [3]
CAIE P3 2024 November Q5
4 marks Moderate -0.8
  1. The complex number \(u\) is given by $$u = \frac{(\cos \frac{1}{4}\pi + i \sin \frac{1}{4}\pi)^4}{\cos \frac{1}{2}\pi - i \sin \frac{1}{2}\pi}$$ Find the exact value of \(\arg u\). [2]
  2. The complex numbers \(u\) and \(u^*\) are plotted on an Argand diagram. Describe the single geometrical transformation that maps \(u\) onto \(u^*\) and state the exact value of \(\arg u^*\). [2]