CAIE P2 (Pure Mathematics 2) 2024 June

Solve the inequality \(|5x + 7| > |2x - 3|\). [4]

Use logarithms to solve the equation \(6^{2x-1} = 5e^{3x+2}\). Give your answer correct to 4 significant figures. [4]

\includegraphics{figure_3} The diagram shows the curve with equation \(y = 8e^{-x} - e^{2x}\). The curve crosses the y-axis at the point A and the x-axis at the point B. The shaded region is bounded by the curve and the two axes.

1. Find the gradient of the curve at A. [3]
2. Show that the x-coordinate of B is \(\ln 2\) and hence find the area of the shaded region. [5]

A curve is defined by the parametric equations $$x = 4\cos^2 t, \quad y = \sqrt{3}\sin 2t,$$ for values of \(t\) such that \(0 < t < \frac{1}{2}\pi\). Find the equation of the normal to the curve at the point for which \(t = \frac{1}{6}\pi\). Give your answer in the form \(ax + by + c = 0\) where \(a\), \(b\) and \(c\) are integers. [7]

The polynomial \(p(x)\) is defined by \(p(x) = 9x^3 + 18x^2 + 5x + 4\).

1. Find the quotient when \(p(x)\) is divided by \((3x + 2)\), and show that the remainder is 6. [3]
2. Find the value of \(\int_0^2 \frac{p(x)}{3x + 2} \, dx\), giving your answer in the form \(a + \ln b\) where \(a\) and \(b\) are integers. [5]

\includegraphics{figure_6} The diagram shows the curve with equation \(y = \frac{\ln(2x + 1)}{x + 3}\). The curve has a maximum point M.

1. Find an expression for \(\frac{dy}{dx}\). [2]
2. Show that the x-coordinate of M satisfies the equation \(x = \frac{x + 3}{\ln(2x + 1)} - 0.5\). [2]
3. Show by calculation that the x-coordinate of M lies between 2.5 and 3.0. [2]
4. Use an iterative formula based on the equation in part (b) to find the x-coordinate of M correct to 4 significant figures. Give the result of each iteration to 6 significant figures. [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]