Pure Mathematics 3 — Variant 1

Expand $(3 + x)(1 - 2x)^{\frac{1}{2}}$ in ascending powers of $x$, up to and including the term in $x^{2}$, simplifying the coefficients.

Solve the equation $\ln(x - 5) = 7 - \ln x$. Give your answer correct to 2 decimal places.

The equation of a curve is $y = \frac{e^{\sin x}}{\cos^{2}x}$ for $0 \leqslant x \leqslant 2\pi$. Find $\frac{\mathrm{d}y}{\mathrm{d}x}$ and hence find the $x$-coordinates of the stationary points of the curve.

By sketching a suitable pair of graphs, show that the equation $\operatorname{cosec}\frac{1}{2}x = e^{x} - 3$ has exactly one root, denoted by $\alpha$, in the interval $0 < x < \pi$.

Verify by calculation that $\alpha$ lies between $1$ and $2$.

Show that if a sequence of values in the interval $0 < x < \pi$ given by the iterative formula $x_{n+1} = \ln(\operatorname{cosec}\frac{1}{2}x_{n} + 3)$ converges, then it converges to $\alpha$.

Use this iterative formula with an initial value of $1.4$ to determine $\alpha$ correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

State the minimum number of calculated iterations needed with this initial value to determine $\alpha$ correct to 2 decimal places.

On a single Argand diagram sketch the loci given by the equations $|z - 3 + 2i| = 2$ and $|w - 3 + 2i| = |w + 3 - 4i|$ where $z$ and $w$ are complex numbers.

Hence find the least value of $|z - w|$ for points on these loci. Give your answer in an exact form.

Use the substitution $u = 1 - \sin x$ to find the exact value of $\int_{\pi}^{\frac{3}{2}\pi} \frac{\sin 2x}{\sqrt{1 - \sin x}} \, \mathrm{d}x$. Give your answer in the form $a + b\sqrt{2}$ where $a$ and $b$ are rational numbers to be determined.

The equations of two straight lines $l_{1}$ and $l_{2}$ are $l_{1}: \mathbf{r} = \mathbf{i} - 2\mathbf{j} + 3\mathbf{k} + \lambda(2\mathbf{i} - \mathbf{j} + a\mathbf{k})$ and $l_{2}: \mathbf{r} = -\mathbf{i} - \mathbf{j} - \mathbf{k} + \mu(3\mathbf{i} - 2\mathbf{j} - 2\mathbf{k})$, where $a$ is a constant. The lines $l_{1}$ and $l_{2}$ are perpendicular.

Show that $a = 4$. The lines $l_{1}$ and $l_{2}$ also intersect.

Find the position vector of the point of intersection. The point $A$ has position vector $-5\mathbf{i} + \mathbf{j} - 9\mathbf{k}$.

Show that $A$ lies on $l_{1}$. The point $B$ is the image of $A$ after a reflection in the line $l_{2}$.

Find the position vector of $B$.

Given that $2x = \tan y$, show that $\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{2}{1 + 4x^{2}}$.

Hence find the exact value of $\int_{\frac{1}{2}}^{\frac{\sqrt{3}}{2}} x\tan^{-1}(2x) \, \mathrm{d}x$.

In a field there are 300 plants of a certain species, all of which can be infected by a particular disease. At time $t$ after the first plant is infected there are $x$ infected plants. The rate of change of $x$ is proportional to the product of the number of plants infected and the number of plants that are not yet infected. The variables $x$ and $t$ are treated as continuous, and it is given that $\frac{\mathrm{d}x}{\mathrm{d}t} = 0.2$ and $x = 1$ when $t = 0$.

Show that $x$ and $t$ satisfy the differential equation $1495\frac{\mathrm{d}x}{\mathrm{d}t} = x(300 - x)$.

Using partial fractions, solve the differential equation and obtain an expression for $t$ in terms of a single logarithm involving $x$.