CAIE P2 (Pure Mathematics 2) 2003 November

Find the set of values of \(x\) satisfying the inequality \(|8 - 3x| < 2\). [3]

\includegraphics{figure_2} Two variable quantities \(x\) and \(y\) are related by the equation $$y = k(a^{-x}),$$ where \(a\) and \(k\) are constants. Four pairs of values of \(x\) and \(y\) are measured experimentally. The result of plotting \(\ln y\) against \(x\) is shown in the diagram. Use the diagram to estimate the values of \(a\) and \(k\). [5]

The polynomial \(x^4 - 6x^2 + x + a\) is denoted by \(f(x)\).

1. It is given that \((x + 1)\) is a factor of \(f(x)\). Find the value of \(a\). [2]
2. When \(a\) has this value, verify that \((x - 2)\) is also a factor of \(f(x)\) and hence factorise \(f(x)\) completely. [4]

1. Express \(\cos \theta + (\sqrt{3}) \sin \theta\) in the form \(R \cos(\theta - \alpha)\), where \(R > 0\) and \(0 < \alpha < \frac{1}{2}\pi\), giving the exact value of \(\alpha\). [3]
2. Hence show that one solution of the equation $$\cos \theta + (\sqrt{3}) \sin \theta = \sqrt{2}$$ is \(\theta = \frac{7}{12}\pi\), and find the other solution in the interval \(0 < \theta < 2\pi\). [4]

1. By sketching a suitable pair of graphs, for \(x < 0\), show that exactly one root of the equation \(x^2 = 2^x\) is negative. [2]
2. Verify by calculation that this root lies between \(-1.0\) and \(-0.5\). [2]
3. Use the iterative formula $$x_{n+1} = -\sqrt{(2^{x_n})}$$ to determine this root correct to 2 significant figures, showing the result of each iteration. [3]

\includegraphics{figure_6} The diagram shows the curve \(y = (4 - x)e^x\) and its maximum point \(M\). The curve cuts the \(x\)-axis at \(A\) and the \(y\)-axis at \(B\).

1. Write down the coordinates of \(A\) and \(B\). [2]
2. Find the \(x\)-coordinate of \(M\). [4]
3. The point \(P\) on the curve has \(x\)-coordinate \(p\). The tangent to the curve at \(P\) passes through the origin \(O\). Calculate the value of \(p\). [5]

1. By differentiating \(\frac{\cos x}{\sin x}\), show that if \(y = \cot x\) then \(\frac{dy}{dx} = -\cosec^2 x\). [3]
2. Hence show that \(\int_{\frac{\pi}{6}}^{\frac{\pi}{2}} \cosec^2 x \, dx = \sqrt{3}\). [2]

By using appropriate trigonometrical identities, find the exact value of

1. \(\int_{\frac{\pi}{6}}^{\frac{\pi}{2}} \cot^2 x \, dx\), [3]
2. \(\int_{\frac{\pi}{6}}^{\frac{\pi}{2}} \frac{1}{1 - \cos 2x} \, dx\). [3]