CAIE P3 (Pure Mathematics 3) 2024 June

Solve the equation \(8^{3-6x} = 4 \times 5^{-2x}\). Give your answer correct to 3 decimal places. [4]

Find the exact coordinates of the stationary point of the curve \(y = e^{2x} \sin 2x\) for \(0 \leqslant x < \frac{1}{2}\pi\). [5]

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]

\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]

Express \(\frac{6x^2 - 2x + 2}{(x - 1)(2x + 1)}\) in partial fractions. [5]

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]

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]

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]

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]

\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]

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]

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]