Mathematics Extension 2

Using the points defined in part (d), find $|\overrightarrow{CB}|$.

Using the substitution $t=\tan\frac{x}{2}$, find
$\int\frac{dx}{1+\cos x-\sin x}.$

Let $\ell_1$ be the line with equation
$\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}-1\\7\end{pmatrix}+\lambda\begin{pmatrix}3\\2\end{pmatrix},\quad \lambda\in\mathbb R.$

The line $\ell_2$ passes through the point $A(-6,5)$ and is parallel to $\ell_1$. Find the equation of the line in the form .

Hence, find the exact value of $(-\sqrt3+i)^{10}$, giving your answer in the form $x+iy$.

Write the complex number $-\sqrt3+i$ in exponential form.

Evaluate $\displaystyle\int \sin^3(2x)\cos(2x)\,dx$.

If $z=-\sqrt3+i$ and $w=1+i$, find $\operatorname{Arg}(zw)$ in terms of $\pi$.

If $z=-\sqrt3+i$ and $w=1+i$, write $\frac zw$ in the form $x+iy$, where $x$ and $y$ are real numbers.

Find $\frac{-1+i\sqrt3}{1+i}$ in Cartesian form.

Express $\dfrac{3-i}{2+i}$ in the form $x+iy$, where $x$ and $y$ are real numbers.

By letting $(x+iy)^2=7+24i$, where $x,y$ are real, find $\sqrt{7+24i}$ in Cartesian form.

A triangle is formed in three-dimensional space with vertices $A(1,-1,2)$, $B(0,2,-1)$ and $C(2,1,1)$. Find the size of $\angle ABC$, giving your answer to the nearest degree.

Express $-1+i\sqrt3$ and $1+i$ in modulus/argument form.

Find the vector equation for the sphere $x^2+4x+y^2-6y+z^2=12$.

Hence find the exact value of $\tan\frac{5\pi}{12}$.

Using De Moivre’s Theorem, find the 3 cube roots of $-4\sqrt2+4i\sqrt2$. Leave your answers in modulus/argument form.

Hence solve the equation $z^2-3iz-(4+6i)=0$.

Point $A$ has position vector $8\mathbf i+13\mathbf j-2\mathbf k$, point $B$ has position vector $10\mathbf i+14\mathbf j-4\mathbf k$, and point $C$ has position vector $9\mathbf i+9\mathbf j+6\mathbf k$. Find $\overrightarrow{AB}$.

Find $A$ and $B$ such that
$\frac{15}{x^2-2x-15}=\frac A{x+3}+\frac B{x-5}.$

Hence, or otherwise, find
$\int\frac{x^2-2x}{x^2-2x-15}\,dx.$

Evaluate
$\int_0^{\pi/4}\sec^4x\tan^2x\,dx.$

Use proof by contrapositive to prove that: If $a^2-2a+7$ is even, then $a$ is odd.

Find $A$, $B$ and $C$ so that
$\frac{A}{x-1}+\frac{Bx+C}{x^2+4}=\frac{-7x-18}{(x-1)(x^2+4)}.$

Find $\displaystyle\int\frac{1}{2+\cos x}\,dx$.