CAIE P1 (Pure Mathematics 1) 2015 June

Given that \(\theta\) is an obtuse angle measured in radians and that \(\sin \theta = k\), find, in terms of \(k\), an expression for

1. \(\cos \theta\), [1]
2. \(\tan \theta\), [2]
3. \(\sin(\theta + \pi)\). [1]

\includegraphics{figure_2} The diagram shows the curve \(y = 2x^2\) and the points \(X(-2, 0)\) and \(P(p, 0)\). The point \(Q\) lies on the curve and \(PQ\) is parallel to the \(y\)-axis.

1. Express the area, \(A\), of triangle \(XPQ\) in terms of \(p\). [2]

The point \(P\) moves along the \(x\)-axis at a constant rate of 0.02 units per second and \(Q\) moves along the curve so that \(PQ\) remains parallel to the \(y\)-axis.

1. Find the rate at which \(A\) is increasing when \(p = 2\). [3]

1. Find the first three terms, in ascending powers of \(x\), in the expansion of
   1. \((1 - x)^6\), [2]
   2. \((1 + 2x)^6\). [2]
2. Hence find the coefficient of \(x^2\) in the expansion of \([(1 - x)(1 + 2x)]^6\). [3]

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

1. Find the cosine of angle \(AOB\). [3]

The position vector of \(C\) is given by \(\overrightarrow{OC} = \begin{pmatrix} k \\ -2k \\ 2k - 3 \end{pmatrix}\).

1. Given that \(AB\) and \(OC\) have the same length, find the possible values of \(k\). [4]

A piece of wire of length 24 cm is bent to form the perimeter of a sector of a circle of radius \(r\) cm.

1. Show that the area of the sector, \(A\) cm\(^2\), is given by \(A = 12r - r^2\). [3]
2. Express \(A\) in the form \(a - (r - b)^2\), where \(a\) and \(b\) are constants. [2]
3. Given that \(r\) can vary, state the greatest value of \(A\) and find the corresponding angle of the sector. [2]

The line with gradient \(-2\) passing through the point \(P(3t, 2t)\) intersects the \(x\)-axis at \(A\) and the \(y\)-axis at \(B\).

1. Find the area of triangle \(AOB\) in terms of \(t\). [3]

The line through \(P\) perpendicular to \(AB\) intersects the \(x\)-axis at \(C\).

1. Show that the mid-point of \(PC\) lies on the line \(y = x\). [4]

1. The third and fourth terms of a geometric progression are \(\frac{1}{4}\) and \(\frac{2}{9}\) respectively. Find the sum to infinity of the progression. [4]
2. A circle is divided into 5 sectors in such a way that the angles of the sectors are in arithmetic progression. Given that the angle of the largest sector is 4 times the angle of the smallest sector, find the angle of the largest sector. [4]

The function \(\text{f} : x \mapsto 5 + 3\cos(\frac{1}{3}x)\) is defined for \(0 \leqslant x \leqslant 2\pi\).

1. Solve the equation \(\text{f}(x) = 7\), giving your answer correct to 2 decimal places. [3]
2. Sketch the graph of \(y = \text{f}(x)\). [2]
3. Explain why \(\text{f}\) has an inverse. [1]
4. Obtain an expression for \(\text{f}^{-1}(x)\). [3]

The equation of a curve is \(y = x^3 + px^2\), where \(p\) is a positive constant.

1. Show that the origin is a stationary point on the curve and find the coordinates of the other stationary point in terms of \(p\). [4]
2. Find the nature of each of the stationary points. [3]

Another curve has equation \(y = x^3 + px^2 + px\).

1. Find the set of values of \(p\) for which this curve has no stationary points. [3]

\includegraphics{figure_10} The diagram shows part of the curve \(y = \frac{8}{\sqrt{(3x + 4)}}\). The curve intersects the \(y\)-axis at \(A(0, 4)\). The normal to the curve at \(A\) intersects the line \(x = 4\) at the point \(B\).

1. Find the coordinates of \(B\). [5]
2. Show, with all necessary working, that the areas of the regions marked \(P\) and \(Q\) are equal. [6]