CAIE P1 (Pure Mathematics 1) 2011 November

The coefficient of \(x^2\) in the expansion of \(\left(k + \frac{1}{x}\right)^5\) is 30. Find the value of the constant \(k\). [3]

The first and second terms of a progression are 4 and 8 respectively. Find the sum of the first 10 terms given that the progression is

1. an arithmetic progression, [2]
2. a geometric progression. [2]

\includegraphics{figure_3} The diagram shows the curve \(y = 2x^5 + 3x^3\) and the line \(y = 2x\) intersecting at points \(A\), \(O\) and \(B\).

1. Show that the \(x\)-coordinates of \(A\) and \(B\) satisfy the equation \(2x^4 + 3x^2 - 2 = 0\). [2]
2. Solve the equation \(2x^4 + 3x^2 - 2 = 0\) and hence find the coordinates of \(A\) and \(B\), giving your answers in an exact form. [3]

\includegraphics{figure_4} In the diagram, \(ABCD\) is a parallelogram with \(AB = BD = DC = 10\) cm and angle \(ABD = 0.8\) radians. \(APD\) and \(BQC\) are arcs of circles with centres \(B\) and \(D\) respectively.

1. Find the area of the parallelogram \(ABCD\). [2]
2. Find the area of the complete figure \(ABQCDP\). [2]
3. Find the perimeter of the complete figure \(ABQCDP\). [2]

1. Given that $$3\sin^2 x - 8\cos x - 7 = 0,$$ show that, for real values of \(x\), $$\cos x = -\frac{2}{3}.$$ [3]
2. Hence solve the equation $$3\sin^2(\theta + 70°) - 8\cos(\theta + 70°) - 7 = 0$$ for \(0° \leqslant \theta \leqslant 180°\). [4]

Relative to an origin \(O\), the position vectors of points \(A\) and \(B\) are \(\mathbf{3i} + 4\mathbf{j} - \mathbf{k}\) and \(5\mathbf{i} - 2\mathbf{j} - 3\mathbf{k}\) respectively.

1. Use a scalar product to find angle \(BOA\). [4]

The point \(C\) is the mid-point of \(AB\). The point \(D\) is such that \(\overrightarrow{OD} = 2\overrightarrow{OB}\).

1. Find \(\overrightarrow{DC}\). [4]

1. A straight line passes through the point \((2, 0)\) and has gradient \(m\). Write down the equation of the line. [1]
2. Find the two values of \(m\) for which the line is a tangent to the curve \(y = x^2 - 4x + 5\). For each value of \(m\), find the coordinates of the point where the line touches the curve. [6]
3. Express \(x^2 - 4x + 5\) in the form \((x + a)^2 + b\) and hence, or otherwise, write down the coordinates of the minimum point on the curve. [2]

A curve \(y = \mathrm{f}(x)\) has a stationary point at \(P(3, -10)\). It is given that \(\mathrm{f}'(x) = 2x^2 + kx - 12\), where \(k\) is a constant.

1. Show that \(k = -2\) and hence find the \(x\)-coordinate of the other stationary point, \(Q\). [4]
2. Find \(\mathrm{f}''(x)\) and determine the nature of each of the stationary points \(P\) and \(Q\). [2]
3. Find \(\mathrm{f}(x)\). [4]

Functions \(\mathrm{f}\) and \(\mathrm{g}\) are defined by \begin{align} \mathrm{f} : x \mapsto 2x + 3 \quad &\text{for } x \leqslant 0,
\mathrm{g} : x \mapsto x^2 - 6x \quad &\text{for } x \leqslant 3. \end{align}

1. Express \(\mathrm{f}^{-1}(x)\) in terms of \(x\) and solve the equation \(\mathrm{f}(x) = \mathrm{f}^{-1}(x)\). [3]
2. On the same diagram sketch the graphs of \(y = \mathrm{f}(x)\) and \(y = \mathrm{f}^{-1}(x)\), showing the coordinates of their point of intersection and the relationship between the graphs. [3]
3. Find the set of values of \(x\) which satisfy \(\mathrm{gf}(x) \leqslant 16\). [5]

\includegraphics{figure_10} The diagram shows the line \(y = x + 1\) and the curve \(y = \sqrt{(x + 1)}\), meeting at \((-1, 0)\) and \((0, 1)\).

1. Find the area of the shaded region. [5]
2. Find the volume obtained when the shaded region is rotated through \(360°\) about the \(y\)-axis. [7]