Questions — CAIE P3 (1110 questions)

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CAIE P3 2018 November Q1
4 marks Standard +0.8
Find the set of values of \(x\) satisfying the inequality \(2|2x - a| < |x + 3a|\), where \(a\) is a positive constant. [4]
CAIE P3 2018 November Q2
4 marks Moderate -0.3
Showing all necessary working, solve the equation \(\frac{2e^x + e^{-x}}{e^x - e^{-x}} = 4\), giving your answer correct to 2 decimal places. [4]
CAIE P3 2018 November Q3
7 marks Standard +0.3
  1. By sketching a suitable pair of graphs, show that the equation \(x^3 = 3 - x\) has exactly one real root. [2]
  2. Show that if a sequence of real values given by the iterative formula $$x_{n+1} = \frac{2x_n^3 + 3}{3x_n^2 + 1}$$ converges, then it converges to the root of the equation in part (i). [2]
  3. Use this iterative formula to determine the root correct to 3 decimal places. Give the result of each iteration to 5 decimal places. [3]
CAIE P3 2018 November Q4
7 marks Standard +0.3
The parametric equations of a curve are $$x = 2\sin\theta + \sin 2\theta, \quad y = 2\cos\theta + \cos 2\theta,$$ where \(0 < \theta < \pi\).
  1. Obtain an expression for \(\frac{dy}{dx}\) in terms of \(\theta\). [3]
  2. Hence find the exact coordinates of the point on the curve at which the tangent is parallel to the \(y\)-axis. [4]
CAIE P3 2018 November Q5
7 marks Standard +0.3
The coordinates \((x, y)\) of a general point on a curve satisfy the differential equation $$x\frac{dy}{dx} = (2 - x^2)y.$$ The curve passes through the point \((1, 1)\). Find the equation of the curve, obtaining an expression for \(y\) in terms of \(x\). [7]
CAIE P3 2018 November Q6
8 marks Standard +0.8
  1. Show that the equation \((\sqrt{2})\cos ec x + \cot x = \sqrt{3}\) can be expressed in the form \(R\sin(x - \alpha) = \sqrt{2}\), where \(R > 0\) and \(0° < \alpha < 90°\). [4]
  2. Hence solve the equation \((\sqrt{2})\cos ec x + \cot x = \sqrt{3}\), for \(0° < x < 180°\). [4]
CAIE P3 2018 November Q7
9 marks Standard +0.3
\includegraphics{figure_7} The diagram shows the curve \(y = 5\sin^2 x \cos^3 x\) for \(0 \leqslant x \leqslant \frac{1}{2}\pi\), and its maximum point \(M\). The shaded region \(R\) is bounded by the curve and the \(x\)-axis.
  1. Find the \(x\)-coordinate of \(M\), giving your answer correct to 3 decimal places. [5]
  2. Using the substitution \(u = \sin x\) and showing all necessary working, find the exact area of \(R\). [4]
CAIE P3 2018 November Q8
9 marks Standard +0.3
  1. Showing all necessary working, express the complex number \(\frac{2 + 3i}{1 - 2i}\) in the form \(re^{i\theta}\), where \(r > 0\) and \(-\pi < \theta \leqslant \pi\). Give the values of \(r\) and \(\theta\) correct to 3 significant figures. [5]
  2. On an Argand diagram sketch the locus of points representing complex numbers \(z\) satisfying the equation \(|z - 3 + 2i| = 1\). Find the least value of \(|z|\) for points on this locus, giving your answer in an exact form. [4]
CAIE P3 2018 November Q9
10 marks Standard +0.3
Let \(f(x) = \frac{6x^2 + 8x + 9}{(2 - x)(3 + 2x)^2}\).
  1. Express \(f(x)\) in partial fractions. [5]
  2. Hence, showing all necessary working, show that \(\int_{-1}^0 f(x) dx = 1 + \frac{1}{2}\ln\left(\frac{4}{3}\right)\). [5]
CAIE P3 2018 November Q10
10 marks Standard +0.3
The planes \(m\) and \(n\) have equations \(3x + y - 2z = 10\) and \(x - 2y + 2z = 5\) respectively. The line \(l\) has equation \(\mathbf{r} = 4\mathbf{i} + 2\mathbf{j} + \mathbf{k} + \lambda(\mathbf{i} + \mathbf{j} + 2\mathbf{k})\).
  1. Show that \(l\) is parallel to \(m\). [3]
  2. Calculate the acute angle between the planes \(m\) and \(n\). [3]
  3. A point \(P\) lies on the line \(l\). The perpendicular distance of \(P\) from the plane \(n\) is equal to 2. Find the position vectors of the two possible positions of \(P\). [4]