Pure Mathematics 3 — Variant 2

Solve the equation $\ln(e^{2x} + 3) = 2x + \ln 3$, giving your answer correct to 3 decimal places.

Solve the equation $3\cos 2\theta = 3\cos\theta + 2$, for $0^\circ \leqslant \theta \leqslant 360^\circ$.

The polynomial $ax^{3} + x^{2} + bx + 3$ is denoted by $\mathrm{p}(x)$. It is given that $\mathrm{p}(x)$ is divisible by $(2x - 1)$ and that when $\mathrm{p}(x)$ is divided by $(x + 2)$ the remainder is 5. Find the values of $a$ and $b$.

The equation of a curve is $y = \cos^{3}x\sqrt{\sin x}$. It is given that the curve has one stationary point in the interval $0 < x < \frac{1}{2}\pi$. Find the $x$-coordinate of this stationary point, giving your answer correct to 3 significant figures.

By sketching a suitable pair of graphs, show that the equation $\ln x = 3x - x^{2}$ has one real root.

Verify by calculation that the root lies between 2 and 2.8.

Use the iterative formula $x_{n+1} = \sqrt{3x_{n} - \ln x_{n}}$ to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

The variables $x$ and $y$ satisfy the differential equation $\frac{\mathrm{d}y}{\mathrm{d}x} = xe^{y-x}$, and $y = 0$ when $x = 0$.

Solve the differential equation, obtaining an expression for $y$ in terms of $x$.

Find the value of $y$ when $x = 1$, giving your answer in the form $a - \ln b$, where $a$ and $b$ are integers.

The equation of a curve is $x^{3} + 3x^{2}y - y^{3} = 3$.

Show that $\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{x^{2} + 2xy}{y^{2} - x^{2}}$.

Find the coordinates of the points on the curve where the tangent is parallel to the $x$-axis.

Let $\mathrm{f}(x) = \frac{x^{2} + 9x}{(3x - 1)(x^{2} + 3)}$.

Express $\mathrm{f}(x)$ in partial fractions.

Hence find $\int_{1}^{3} \mathrm{f}(x) \, \mathrm{d}x$, giving your answer in a simplified exact form.

The lines $l$ and $m$ have vector equations $\mathbf{r} = -\mathbf{i} + 3\mathbf{j} + 4\mathbf{k} + \lambda(2\mathbf{i} - \mathbf{j} - \mathbf{k})$ and $\mathbf{r} = 5\mathbf{i} + 4\mathbf{j} + 3\mathbf{k} + \mu(a\mathbf{i} + b\mathbf{j} + \mathbf{k})$ respectively, where $a$ and $b$ are constants.

Given that $l$ and $m$ intersect, show that $2b - a = 4$.

Given also that $l$ and $m$ are perpendicular, find the values of $a$ and $b$.

When $a$ and $b$ have these values, find the position vector of the point of intersection of $l$ and $m$.

The complex number $-1 + \sqrt{7}i$ is denoted by $u$. It is given that $u$ is a root of the equation $2x^{3} + 3x^{2} + 14x + k = 0$, where $k$ is a real constant.

Find the value of $k$.

Find the other two roots of the equation.

On an Argand diagram, sketch the locus of points representing complex numbers $z$ satisfying the equation $|z - u| = 2$.

Determine the greatest value of $\arg z$ for points on this locus, giving your answer in radians.