CAIE P1 (Pure Mathematics 1) 2012 June

1. Prove the identity \(\tan^2 \theta - \sin^2 \theta = \tan^2 \theta \sin^2 \theta\). [3]
2. Use this result to explain why \(\tan \theta > \sin \theta\) for \(0° < \theta < 90°\). [1]

Relative to an origin \(O\), the position vectors of the points \(A\), \(B\) and \(C\) are given by $$\overrightarrow{OA} = \begin{pmatrix} 2 \\ -1 \\ 4 \end{pmatrix}, \quad \overrightarrow{OB} = \begin{pmatrix} 4 \\ 2 \\ -2 \end{pmatrix} \quad \text{and} \quad \overrightarrow{OC} = \begin{pmatrix} 1 \\ 3 \\ p \end{pmatrix}.$$ Find

1. the unit vector in the direction of \(\overrightarrow{AB}\), [3]
2. the value of the constant \(p\) for which angle \(BOC = 90°\). [2]

The first three terms in the expansion of \((1 - 2x)^2(1 + ax)^6\), in ascending powers of \(x\), are \(1 - x + bx^2\). Find the values of the constants \(a\) and \(b\). [6]

1. Solve the equation \(\sin 2x + 3 \cos 2x = 0\) for \(0° \leqslant x \leqslant 360°\). [5]
2. How many solutions has the equation \(\sin 2x + 3 \cos 2x = 0\) for \(0° \leqslant x \leqslant 1080°\)? [1]

\includegraphics{figure_5} The diagram shows part of the curve \(x = \frac{8}{y^2} - 2\), crossing the \(y\)-axis at the point \(A\). The point \(B (6, 1)\) lies on the curve. The shaded region is bounded by the curve, the \(y\)-axis and the line \(y = 1\). Find the exact volume obtained when this shaded region is rotated through \(360°\) about the \(y\)-axis. [6]

The first term of an arithmetic progression is 12 and the sum of the first 9 terms is 135.

1. Find the common difference of the progression. [2]

The first term, the ninth term and the \(n\)th term of this arithmetic progression are the first term, the second term and the third term respectively of a geometric progression.

1. Find the common ratio of the geometric progression and the value of \(n\). [5]

The curve \(y = \frac{10}{2x + 1} - 2\) intersects the \(x\)-axis at \(A\). The tangent to the curve at \(A\) intersects the \(y\)-axis at \(C\).

1. Show that the equation of \(AC\) is \(5y + 4x = 8\). [5]
2. Find the distance \(AC\). [2]

\includegraphics{figure_8} In the diagram, \(AB\) is an arc of a circle with centre \(O\) and radius \(r\). The line \(XB\) is a tangent to the circle at \(B\) and \(A\) is the mid-point of \(OX\).

1. Show that angle \(AOB = \frac{1}{3}\pi\) radians. [2]

Express each of the following in terms of \(r\), \(\pi\) and \(\sqrt{3}\):

1. the perimeter of the shaded region, [3]
2. the area of the shaded region. [2]

A curve is such that \(\frac{d^2y}{dx^2} = -4x\). The curve has a maximum point at \((2, 12)\).

1. Find the equation of the curve. [6]

A point \(P\) moves along the curve in such a way that the \(x\)-coordinate is increasing at 0.05 units per second.

1. Find the rate at which the \(y\)-coordinate is changing when \(x = 3\), stating whether the \(y\)-coordinate is increasing or decreasing. [2]

The equation of a line is \(2y + x = k\), where \(k\) is a constant, and the equation of a curve is \(xy = 6\).

1. In the case where \(k = 8\), the line intersects the curve at the points \(A\) and \(B\). Find the equation of the perpendicular bisector of the line \(AB\). [6]
2. Find the set of values of \(k\) for which the line \(2y + x = k\) intersects the curve \(xy = 6\) at two distinct points. [3]

The function \(f\) is such that \(f(x) = 8 - (x - 2)^2\), for \(x \in \mathbb{R}\).

1. Find the coordinates and the nature of the stationary point on the curve \(y = f(x)\). [3]

The function \(g\) is such that \(g(x) = 8 - (x - 2)^2\), for \(k \leqslant x \leqslant 4\), where \(k\) is a constant.

1. State the smallest value of \(k\) for which \(g\) has an inverse. [1]

For this value of \(k\),

1. find an expression for \(g^{-1}(x)\), [3]
2. sketch, on the same diagram, the graphs of \(y = g(x)\) and \(y = g^{-1}(x)\). [3]