CAIE P2 (Pure Mathematics 2) 2023 March

Find the exact value of \(\int_0^{\frac{\pi}{4}} 2 \tan^2(\frac{1}{2}x) \, dx\). [4]

Solve the equation \(\tan(\theta - 60°) = 3 \cot \theta\) for \(-90° < \theta < 90°\). [5]

The polynomial \(p(x)\) is defined by $$p(x) = ax^3 - ax^2 + ax + b,$$ where \(a\) and \(b\) are constants. It is given that \((x + 2)\) is a factor of \(p(x)\), and that the remainder is 35 when \(p(x)\) is divided by \((x - 3)\).

1. Find the values of \(a\) and \(b\). [5]
2. Hence factorise \(p(x)\) and show that the equation \(p(x) = 0\) has exactly one real root. [3]

1. Sketch, on the same diagram, the graphs of \(y = |2x - 1|\) and \(y = 3x - 3\). [2]
2. Solve the inequality \(|2x - 1| < 3x - 3\). [3]
3. Find the smallest integer \(N\) satisfying the inequality \(|2 \ln N - 1| < 3 \ln N - 3\). [2]

It is given that \(\int_1^a \left(\frac{4}{1 + 2x} + \frac{3}{x}\right) dx = \ln 10\), where \(a\) is a constant greater than 1.

1. Show that \(a = \sqrt{90(1 + 2a)^{-2}}\). [5]
2. Use an iterative formula, based on the equation in (a), to find the value of \(a\) correct to 3 significant figures. Use an initial value of 1.7 and give the result of each iteration to 5 significant figures. [3]

\includegraphics{figure_6} The diagram shows the curve with equation \(y = \frac{4e^{2x} + 9}{e^x + 2}\). The curve has a minimum point \(M\) and crosses the \(y\)-axis at the point \(P\).

1. Find the exact value of the gradient of the curve at \(P\). [4]
2. Find the exact coordinates of \(M\). [4]

\includegraphics{figure_7} The diagram shows the curve with parametric equations $$x = k \tan t, \quad y = 3 \sin 2t - 4 \sin t,$$ for \(0 < t < \frac{1}{2}\pi\). It is given that \(k\) is a positive constant. The curve crosses the \(x\)-axis at the point \(P\).

1. Find the value of \(\cos t\) at \(P\), giving your answer as an exact fraction. [3]
2. Express \(\frac{dy}{dx}\) in terms of \(k\) and \(\cos t\). [4]
3. Given that the normal to the curve at \(P\) has gradient \(\frac{9}{10}\), find the value of \(k\), giving your answer as an exact fraction. [3]