CAIE P1 (Pure Mathematics 1) 2012 June

\includegraphics{figure_1} The diagram shows the region enclosed by the curve \(y = \frac{6}{2x - 3}\), the \(x\)-axis and the lines \(x = 2\) and \(x = 3\). Find, in terms of \(\pi\), the volume obtained when this region is rotated through \(360°\) about the \(x\)-axis. [4]

The equation of a curve is \(y = 4\sqrt{x} + \frac{2}{\sqrt{x}}\).

1. Obtain an expression for \(\frac{dy}{dx}\). [3]
2. A point is moving along the curve in such a way that the \(x\)-coordinate is increasing at a constant rate of \(0.12\) units per second. Find the rate of change of the \(y\)-coordinate when \(x = 4\). [2]

The coefficient of \(x^3\) in the expansion of \((a + x)^5 + (2 - x)^6\) is \(90\). Find the value of the positive constant \(a\). [5]

The point \(A\) has coordinates \((-1, -5)\) and the point \(B\) has coordinates \((7, 1)\). The perpendicular bisector of \(AB\) meets the \(x\)-axis at \(C\) and the \(y\)-axis at \(D\). Calculate the length of \(CD\). [6]

1. Prove the identity \(\tan x + \frac{1}{\tan x} = \frac{1}{\sin x \cos x}\). [2]
2. Solve the equation \(\frac{2}{\sin x \cos x} = 1 + 3 \tan x\), for \(0° \leqslant x \leqslant 180°\). [4]

\includegraphics{figure_6} The diagram shows a metal plate made by removing a segment from a circle with centre \(O\) and radius \(8\) cm. The line \(AB\) is a chord of the circle and angle \(AOB = 2.4\) radians. Find

1. the length of \(AB\), [2]
2. the perimeter of the plate, [3]
3. the area of the plate. [3]

1. In an arithmetic progression, the sum of the first \(n\) terms, denoted by \(S_n\), is given by $$S_n = n^2 + 8n.$$ Find the first term and the common difference. [3]
2. In a geometric progression, the second term is \(9\) less than the first term. The sum of the second and third terms is \(30\). Given that all the terms of the progression are positive, find the first term. [5]

1. Find the angle between the vectors \(\mathbf{3i} - \mathbf{4k}\) and \(\mathbf{2i} + \mathbf{3j} - \mathbf{6k}\). [4]

The vector \(\overrightarrow{OA}\) has a magnitude of \(15\) units and is in the same direction as the vector \(\mathbf{3i} - \mathbf{4k}\). The vector \(\overrightarrow{OB}\) has a magnitude of \(14\) units and is in the same direction as the vector \(\mathbf{2i} + \mathbf{3j} - \mathbf{6k}\).

1. Express \(\overrightarrow{OA}\) and \(\overrightarrow{OB}\) in terms of \(\mathbf{i}\), \(\mathbf{j}\) and \(\mathbf{k}\). [3]
2. Find the unit vector in the direction of \(\overrightarrow{AB}\). [3]

\includegraphics{figure_9} The diagram shows part of the curve \(y = -x^2 + 8x - 10\) which passes through the points \(A\) and \(B\). The curve has a maximum point at \(A\) and the gradient of the line \(BA\) is \(2\).

1. Find the coordinates of \(A\) and \(B\). [7]
2. Find \(\int y \, dx\) and hence evaluate the area of the shaded region. [4]

Functions \(f\) and \(g\) are defined by $$f : x \mapsto 2x + 5 \quad \text{for } x \in \mathbb{R},$$ $$g : x \mapsto \frac{8}{x - 3} \quad \text{for } x \in \mathbb{R}, x \neq 3.$$

1. Obtain expressions, in terms of \(x\), for \(f^{-1}(x)\) and \(g^{-1}(x)\), stating the value of \(x\) for which \(g^{-1}(x)\) is not defined. [4]
2. Sketch the graphs of \(y = f(x)\) and \(y = f^{-1}(x)\) on the same diagram, making clear the relationship between the two graphs. [3]
3. Given that the equation \(fg(x) = 5 - kx\), where \(k\) is a constant, has no solutions, find the set of possible values of \(k\). [5]