CAIE P2 (Pure Mathematics 2) 2016 November

1. It is given that \(x\) satisfies the equation \(3^{2x} = 5(3^x) + 14\). Find the value of \(3^x\) and, using logarithms, find the value of \(x\) correct to 3 significant figures. [4]
2. Hence state the values of \(x\) satisfying the equation \(3^{2|x|} = 5(3^{|x|}) + 14\). [1]

\includegraphics{figure_2} The variables \(x\) and \(y\) satisfy the equation \(y = Ae^{px}\), where \(A\) and \(p\) are constants. The graph of \(\ln y\) against \(x\) is a straight line passing through the points \((5, 3.17)\) and \((10, 4.77)\), as shown in the diagram. Find the values of \(A\) and \(p\) correct to 2 decimal places. [5]

A curve has equation \(y = 2\sin 2x - 5\cos 2x + 6\) and is defined for \(0 \leq x \leq \pi\). Find the \(x\)-coordinates of the stationary points of the curve, giving your answers correct to 3 significant figures. [6]

It is given that the positive constant \(a\) is such that $$\int_{-a}^a (4e^{2x} + 5) dx = 100.$$

1. Show that \(a = \frac{1}{4}\ln(50 + e^{-2a} - 5a)\). [4]
2. Use the iterative formula \(a_{n+1} = \frac{1}{4}\ln(50 + e^{-2a_n} - 5a_n)\) to find \(a\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places. [3]

1. Show that \(\frac{\cos 2x + 9\cos x + 5}{\cos x + 4} \equiv 2\cos x + 1\). [3]
2. Hence find the exact value of \(\int_{-\pi}^{\pi} \frac{\cos 4x + 9\cos 2x + 5}{\cos 2x + 4} dx\). [4]

The equation of a curve is \(3x^2 + 4xy + y^2 = 24\). Find the equation of the normal to the curve at the point \((1, 3)\), giving your answer in the form \(ax + by + c = 0\) where \(a\), \(b\) and \(c\) are integers. [8]

The polynomial \(p(x)\) is defined by $$p(x) = ax^3 + 3x^2 + bx + 12,$$ where \(a\) and \(b\) are constants. It is given that \((x + 3)\) is a factor of \(p(x)\). It is also given that the remainder is 18 when \(p(x)\) is divided by \((x + 2)\).

1. Find the values of \(a\) and \(b\). [5]
2. When \(a\) and \(b\) have these values,
   1. show that the equation \(p(x) = 0\) has exactly one real root, [4]
   2. solve the equation \(p(\sec y) = 0\) for \(-180° < y < 180°\). [3]