Write down $d$ in the form $k \ln x$, where $k\in \mathrm{\mathbb{Q}}$.

Show that $2k^2-k+0.12=0$.

Hence, find $\text{E}\left(X\right)$.

Solve f '(x) = f "(x).

A continuous random variable $X$ has the probability density function

$f\left(x\right)=\left\{\begin{array}{cc}\frac{2}{\left(b-a\right)\left(c-a\right)}\left(x-a\right),& a\le x\le c\\ \frac{2}{\left(b-a\right)\left(b-c\right)}\left(b-x\right),& c<x\le b\\ 0,& \text{otherwise}\end{array}\right.$.

The following diagram shows the graph of $y=f\left(x\right)$ for $a\le x\le b$.

Given that $c\ge \frac{a+b}{2}$, find an expression for the median of $X$ in terms of $a, b$ and $c$.

Verify that $x=\text{tan}\theta$ and $x=-\text{cot}\theta$ satisfy the equation $x^2+\left(2\text{cot}2\theta \right)x-1=0$.

Given that $\tan \theta >0$, find $\tan \theta$.

The triangle ABC is shown in the following diagram. Given that , find the range of possible values for AB.

Let $y=\frac{1}{\cos x}$, for . The graph of $y$between $x=\theta$ and $x=\frac{\pi }{4}$ is rotated 360° about the $x$-axis. Find the volume of the solid formed.

Show that the discriminant of $f^{\prime }\left(x\right)$ is $k^2-16$.

The number $351$ is a triangular number. Determine which one it is.

Deduce from part (d)(i) that the complex roots of the equation $\left(z-r\right)\left(z^2-2az+a^2+b^2\right)=0$ can be expressed as $a\pm \text{i}\sqrt{g\text{'}\left(a\right)}$.

The sum of the first $n$ terms of the series is $-3 \ln x$.

Find the value of $n$.

Show that $p=\frac{2}{3}$.

The sum of the first $n$ terms of the series is $\ln \left(\frac{1}{x^3}\right)$.

Find the value of $n$.

The third term in the expansion is the mean of the second term and the fourth term in the expansion.

Find the possible values of $x$.

Given that $\left(h\circ g\right)\left(a\right)=\frac{\mathrm{\pi }}{4}$, find the value of $a$.

Give your answer in the form $p+\frac{q}{2}\sqrt{r}$, where $p, q, r\in {\mathrm{\mathbb{Z}}}^{+}$.

Find the value of $k$, giving a reason for your answer.

Write down an expression for the product of the roots, in terms of $k$.

Hence or otherwise, determine the values of $k$ such that the equation has one positive and one negative real root.

Let $f\left(x\right)=-x^2+4x+5$ and $g\left(x\right)=-f\left(x\right)+k$.

Find the values of $k$ so that $g\left(x\right)=0$ has no real roots.

Using the results from parts (b) and (c) find the exact value of $\text{tan}\frac{\pi }{24}-\text{cot}\frac{\pi }{24}$.

Give your answer in the form $a+b\sqrt{3}$ where $a$, $b\in \mathbb{Z}$.

Hence, find the area of the region enclosed by the graphs of $f$ and $g$.

Find the area of the region enclosed by the graph of $f$ and the line $L$.

Find the set of values of $k$ that satisfy the inequality .

The quadratic equation $\left(k-1\right)x^2+2x+\left(2k-3\right)=0$, where $k\in \mathrm{ℝ}$, has real distinct roots.

Find the range of possible values for $k$.

Hence find the exact value of ${\text{cot}}^2\frac{3\mathrm{\pi }}{8}$.

Given that $f$ is an increasing function, find all possible values of $k$.

Find the coordinates of $\text{B}$.

Find the possible values for $p$.

By substituting $Y=\frac{dy}{dt}$, show that $Y=B{\text{e}}^{2t}$ where $B$ is a constant.

Let the two values found in part (c)(ii) be ${\lambda }_1$ and ${\lambda }_2$.

Verify that $y=F{\text{e}}^{{\lambda }_{1}t}+G{\text{e}}^{{\lambda }_{2}t}$ is a solution to the differential equation in (c)(i),where $G$ is a constant.

The line $y=kx-5$ is a tangent to the curve of $f$. Find the values of $k$.

Hence, or otherwise, show that the exact value of $\text{tan}\frac{\pi }{12}=2-\sqrt{3}$.

Show that $\cos \theta =\frac{3}{4}$.

The quadratic equation $x^2-2kx+(k-1)=0$ has roots $\alpha$ and $\beta$ such that ${\alpha }^2+{\beta }^2=4$. Without solving the equation, find the possible values of the real number $k$.

Show that $\text{cot}2\theta =\frac{1-\text{ta}{\text{n}}^2\theta }{2\text{tan}\theta }$.

Find the value of $p$ and of $q$.

Consider the case when $p=4$. The roots of the equation can be expressed in the form $x=\frac{a\pm \sqrt{13}}{6}$, where $a\in \mathrm{\mathbb{Z}}$. Find the value of $a$.

By solving the differential equation $\frac{dy}{dt}=y$, show that $y=A{\text{e}}^t$ where $A$ is a constant.

Show that $\frac{dx}{dt}-x=-A{\text{e}}^t$.

Solve the differential equation in part (a)(ii) to find $x$ as a function of $t$.

By differentiating $\frac{dy}{dt}=-x+y$ with respect to $t$, show that $\frac{d^{2}y}{d{t}^2}=2\frac{{\displaystyle dy}}{{\displaystyle dt}}$.

Hence find $y$ as a function of $t$.

Hence show that $x=-\frac{B}{2}{\text{e}}^{2t}+C$, where $C$ is a constant.

Show that $\frac{d^{2}y}{d{t}^2}-2\frac{dy}{dt}-3y=0$.

Find the two values for $\lambda$ that satisfy $\frac{d^{2}y}{d{t}^2}-2\frac{dy}{dt}-3y=0$.