CAIE P1 (Pure Mathematics 1) 2011 June

Find \(\int \left(x^3 + \frac{1}{x^3}\right) \mathrm{d}x\). [3]

1. Find the terms in \(x^2\) and \(x^3\) in the expansion of \(\left(1 - \frac{3}{2}x\right)^6\). [3]
2. Given that there is no term in \(x^3\) in the expansion of \((k + 2x)\left(1 - \frac{3}{2}x\right)^6\), find the value of the constant \(k\). [2]

The equation \(x^2 + px + q = 0\), where \(p\) and \(q\) are constants, has roots \(-3\) and \(5\).

1. Find the values of \(p\) and \(q\). [2]
2. Using these values of \(p\) and \(q\), find the value of the constant \(r\) for which the equation \(x^2 + px + q + r = 0\) has equal roots. [3]

A curve has equation \(y = \frac{4}{3x - 4}\) and \(P(2, 2)\) is a point on the curve.

1. Find the equation of the tangent to the curve at \(P\). [4]
2. Find the angle that this tangent makes with the \(x\)-axis. [2]

1. Prove the identity \(\frac{\cos \theta}{\tan \theta(1 - \sin \theta)} \equiv 1 + \frac{1}{\sin \theta}\). [3]
2. Hence solve the equation \(\frac{\cos \theta}{\tan \theta(1 - \sin \theta)} = 4\), for \(0° \leq \theta \leq 360°\). [3]

The function \(f\) is defined by \(f : x \mapsto \frac{x + 3}{2x - 1}\), \(x \in \mathbb{R}\), \(x \neq \frac{1}{2}\).

1. Show that \(f f(x) = x\). [3]
2. Hence, or otherwise, obtain an expression for \(f^{-1}(x)\). [2]

The line \(L_1\) passes through the points \(A(2, 5)\) and \(B(10, 9)\). The line \(L_2\) is parallel to \(L_1\) and passes through the origin. The point \(C\) lies on \(L_2\) such that \(AC\) is perpendicular to \(L_2\). Find

1. the coordinates of \(C\), [5]
2. the distance \(AC\). [2]

Relative to the origin \(O\), the position vectors of the points \(A\), \(B\) and \(C\) are given by $$\overrightarrow{OA} = \begin{pmatrix} 2 \\ 3 \\ 5 \end{pmatrix}, \quad \overrightarrow{OB} = \begin{pmatrix} 4 \\ 2 \\ 3 \end{pmatrix} \quad \text{and} \quad \overrightarrow{OC} = \begin{pmatrix} 10 \\ 0 \\ 6 \end{pmatrix}.$$

1. Find angle \(ABC\). [6]

The point \(D\) is such that \(ABCD\) is a parallelogram.

1. Find the position vector of \(D\). [2]

The function \(f\) is such that \(f(x) = 3 - 4\cos^k x\), for \(0 \leq x \leq \pi\), where \(k\) is a constant.

1. In the case where \(k = 2\),
   1. find the range of \(f\), [2]
   2. find the exact solutions of the equation \(f(x) = 1\). [3]
2. In the case where \(k = 1\),
   1. sketch the graph of \(y = f(x)\), [2]
   2. state, with a reason, whether \(f\) has an inverse. [1]

1. A circle is divided into 6 sectors in such a way that the angles of the sectors are in arithmetic progression. The angle of the largest sector is 4 times the angle of the smallest sector. Given that the radius of the circle is 5 cm, find the perimeter of the smallest sector. [6]
2. The first, second and third terms of a geometric progression are \(2k + 3\), \(k + 6\) and \(k\), respectively. Given that all the terms of the geometric progression are positive, calculate
   1. the value of the constant \(k\), [3]
   2. the sum to infinity of the progression. [2]

\includegraphics{figure_11} The diagram shows part of the curve \(y = 4\sqrt{x} - x\). The curve has a maximum point at \(M\) and meets the \(x\)-axis at \(O\) and \(A\).

1. Find the coordinates of \(A\) and \(M\). [5]
2. Find the volume obtained when the shaded region is rotated through \(360°\) about the \(x\)-axis, giving your answer in terms of \(\pi\). [6]