CAIE P3 (Pure Mathematics 3) 2017 November

Find the quotient and remainder when \(x^4\) is divided by \(x^2 + 2x - 1\). [3]

Two variable quantities \(x\) and \(y\) are believed to satisfy an equation of the form \(y = C(a^x)\), where \(C\) and \(a\) are constants. An experiment produced four pairs of values of \(x\) and \(y\). The table below gives the corresponding values of \(x\) and \(\ln y\). By plotting \(\ln y\) against \(x\) for these four pairs of values and drawing a suitable straight line, estimate the values of \(C\) and \(a\). Give your answers correct to 2 significant figures. [5] \includegraphics{figure_2}

The equation \(x^3 = 3x + 7\) has one real root, denoted by \(\alpha\).

1. Show by calculation that \(\alpha\) lies between 2 and 3. [2]

Two iterative formulae, \(A\) and \(B\), derived from this equation are as follows: $$x_{n+1} = (3x_n + 7)^{\frac{1}{3}}, \quad (A)$$ $$x_{n+1} = \frac{x_n^3 - 7}{3}. \quad (B)$$ Each formula is used with initial value \(x_1 = 2.5\).

1. Show that one of these formulae produces a sequence which fails to converge, and use the other formula to calculate \(\alpha\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places. [4]

1. Prove the identity \(\tan(45° + x) + \tan(45° - x) = 2 \sec 2x\). [4]
2. Hence sketch the graph of \(y = \tan(45° + x) + \tan(45° - x)\) for \(0° \leqslant x \leqslant 90°\). [3]

The equation of a curve is \(2x^4 + xy^3 + y^4 = 10\).

1. Show that \(\frac{dy}{dx} = -\frac{8x^3 + y^3}{3xy^2 + 4y^3}\). [4]
2. Hence show that there are two points on the curve at which the tangent is parallel to the \(x\)-axis and find the coordinates of these points. [4]

The variables \(x\) and \(y\) satisfy the differential equation $$\frac{dy}{dx} = 4 \cos^2 y \tan x,$$ for \(0 \leqslant x < \frac{1}{2}\pi\), and \(x = 0\) when \(y = \frac{1}{4}\pi\). Solve this differential equation and find the value of \(x\) when \(y = \frac{1}{8}\pi\). [8]

1. The complex number \(u\) is given by \(u = 8 - 15\text{i}\). Showing all necessary working, find the two square roots of \(u\). Give answers in the form \(a + ib\), where the numbers \(a\) and \(b\) are real and exact. [5]
2. On an Argand diagram, shade the region whose points represent complex numbers satisfying both the inequalities \(|z - 2 - \text{i}| \leqslant 2\) and \(0 \leqslant \arg(z - \text{i}) \leqslant \frac{1}{4}\pi\). [4]

Let \(\text{f}(x) = \frac{4x^2 + 9x - 8}{(x + 2)(2x - 1)}\).

1. Express \(\text{f}(x)\) in the form \(A + \frac{B}{x + 2} + \frac{C}{2x - 1}\). [4]
2. Hence show that \(\int_1^4 \text{f}(x) \, dx = 6 + \frac{1}{2} \ln\left(\frac{16}{7}\right)\). [5]

\includegraphics{figure_9} The diagram shows the curve \(y = (1 + x^2)\text{e}^{-\frac{3x}{4}}\) for \(x \geqslant 0\). The shaded region \(R\) is enclosed by the curve, the \(x\)-axis and the lines \(x = 0\) and \(x = 2\).

1. Find the exact values of the \(x\)-coordinates of the stationary points of the curve. [4]
2. Show that the exact value of the area of \(R\) is \(18 - \frac{42}{\text{e}}\). [5]

The equations of two lines \(l\) and \(m\) are \(\mathbf{r} = 3\mathbf{i} - \mathbf{j} - 2\mathbf{k} + \lambda(-\mathbf{i} + \mathbf{j} + 4\mathbf{k})\) and \(\mathbf{r} = 4\mathbf{i} + 4\mathbf{j} - 3\mathbf{k} + \mu(2\mathbf{i} + \mathbf{j} - 2\mathbf{k})\) respectively.

1. Show that the lines do not intersect. [3]
2. Calculate the acute angle between the directions of the lines. [3]
3. Find the equation of the plane which passes through the point \((3, -2, -1)\) and which is parallel to both \(l\) and \(m\). Give your answer in the form \(ax + by + cz = d\). [5]