CAIE P1 (Pure Mathematics 1) 2010 June

1. Show that the equation $$3(2\sin x - \cos x) = 2(\sin x - 3\cos x)$$ can be written in the form \(\tan x = -\frac{4}{5}\). [2]
2. Solve the equation \(3(2\sin x - \cos x) = 2(\sin x - 3\cos x)\), for \(0° \leq x \leq 360°\). [2]

\includegraphics{figure_2} The diagram shows part of the curve \(y = \frac{a}{x}\), where \(a\) is a positive constant. Given that the volume obtained when the shaded region is rotated through \(360°\) about the \(x\)-axis is \(24\pi\), find the value of \(a\). [4]

The functions f and g are defined for \(x \in \mathbb{R}\) by $$f : x \mapsto 4x - 2x^2,$$ $$g : x \mapsto 5x + 3.$$

1. Find the range of f. [2]
2. Find the value of the constant \(k\) for which the equation \(gf(x) = k\) has equal roots. [3]

\includegraphics{figure_4} In the diagram, \(A\) is the point \((-1, 3)\) and \(B\) is the point \((3, 1)\). The line \(L_1\) passes through \(A\) and is parallel to \(OB\). The line \(L_2\) passes through \(B\) and is perpendicular to \(AB\). The lines \(L_1\) and \(L_2\) meet at \(C\). Find the coordinates of \(C\). [6]

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

1. Find the value of \(p\) for which \(\overrightarrow{OA}\) is perpendicular to \(\overrightarrow{OB}\). [2]
2. Find the values of \(p\) for which the magnitude of \(\overrightarrow{AB}\) is 7. [4]

1. Find the first 3 terms in the expansion of \((1 + ax)^4\) in ascending powers of \(x\). [2]
2. Given that there is no term in \(x\) in the expansion of \((1 - 2x)(1 + ax)^5\), find the value of the constant \(a\). [2]
3. For this value of \(a\), find the coefficient of \(x^2\) in the expansion of \((1 - 2x)(1 + ax)^5\). [3]

1. Find the sum of all the multiples of 5 between 100 and 300 inclusive. [3]
2. A geometric progression has a common ratio of \(-\frac{2}{3}\) and the sum of the first 3 terms is 35. Find
   1. the first term of the progression, [3]
   2. the sum to infinity. [2]

A solid rectangular block has a square base of side \(x\) cm. The height of the block is \(h\) cm and the total surface area of the block is \(96\) cm\(^2\).

1. Express \(h\) in terms of \(x\) and show that the volume, \(V\) cm\(^3\), of the block is given by $$V = 24x - \frac{1}{2}x^3.$$ [3]

Given that \(x\) can vary,

1. find the stationary value of \(V\), [3]
2. determine whether this stationary value is a maximum or a minimum. [2]

\includegraphics{figure_9} The diagram shows the curve \(y = (x - 2)^2\) and the line \(y + 2x = 7\), which intersect at points \(A\) and \(B\). Find the area of the shaded region. [8]

The equation of a curve is \(y = \frac{1}{6}(2x - 3)^3 - 4x\).

1. Find \(\frac{dy}{dx}\). [3]
2. Find the equation of the tangent to the curve at the point where the curve intersects the \(y\)-axis. [3]
3. Find the set of values of \(x\) for which \(\frac{1}{6}(2x - 3)^3 - 4x\) is an increasing function of \(x\). [3]

The function \(f : x \mapsto 4 - 3\sin x\) is defined for the domain \(0 \leq x < 2\pi\).

1. Solve the equation \(f(x) = 2\). [3]
2. Sketch the graph of \(y = f(x)\). [2]
3. Find the set of values of \(k\) for which the equation \(f(x) = k\) has no solution. [2]

The function \(g : x \mapsto 4 - 3\sin x\) is defined for the domain \(\frac{1}{2}\pi \leq x \leq A\).

1. State the largest value of \(A\) for which \(g\) has an inverse. [1]
2. For this value of \(A\), find the value of \(g^{-1}(3)\). [2]