Chooose all options giving polar exponential form of the complex number
$z=-1+\sqrt{3}i$. (Zero or more
options can be correct)

For example, choice (a) should be False.

For example, choice (b) should be False.

For example, choice (c) should be True.

For example, choice (d) should be True.

For example, choice (e) should be False.

*There is at least one mistake.*

For example, choice (a) should be False.

Plot $z$ in the
complex plane to determine which quadrant it lies in. Its modulus is found using the formula
$|a+ib|=\sqrt{{a}^{2}+{b}^{2}}$. Then find its
argument $\theta $ using,
for example, $tan\theta =\sqrt{3}\u2215\left(-1\right)$.
However you must remember to adjust your answer to take account of the quadrant!

*There is at least one mistake.*

For example, choice (b) should be False.

Plot
$z$ in the
complex plane to determine which quadrant it lies in. Its modulus is found using the formula
$|a+ib|=\sqrt{{a}^{2}+{b}^{2}}$. Then find its
argument $\theta $ using,
for example, $tan\theta =\sqrt{3}\u2215\left(-1\right)$.
However you must remember to adjust your answer to take account of the quadrant!

*There is at least one mistake.*

For example, choice (c) should be True.

The modulus
of $z$ is
$2$ and its principal
argument is $2\pi \u22153$,
($z$ is in the second
quadrant).

*There is at least one mistake.*

For example, choice (d) should be True.

The
modulus of $z$ is
$2$ and its principal
argument is $2\pi \u22153$,
($z$
is in the second quadrant). However an argument is also
$2\pi \u22153-2\pi =-4\pi i\u22153$. This leads to the polar
exponential form $2{e}^{-4\pi i\u22153}$.
There is no compulsion to use the principal argument when writing a complex number in polar
form of any kind.

*There is at least one mistake.*

For example, choice (e) should be False.

Plot $z$ in the
complex plane to determine which quadrant it lies in. Its modulus is found using the formula
$|a+ib|=\sqrt{{a}^{2}+{b}^{2}}$. Then find its
argument $\theta $ using,
for example, $tan\theta =\sqrt{3}\u2215\left(-1\right)$.
However you must remember to adjust your answer to take account of the quadrant!

*Correct!*

*False*Plot $z$ in the complex plane to determine which quadrant it lies in. Its modulus is found using the formula $|a+ib|=\sqrt{{a}^{2}+{b}^{2}}$. Then find its argument $\theta $ using, for example, $tan\theta =\sqrt{3}\u2215\left(-1\right)$. However you must remember to adjust your answer to take account of the quadrant!*False*Plot $z$ in the complex plane to determine which quadrant it lies in. Its modulus is found using the formula $|a+ib|=\sqrt{{a}^{2}+{b}^{2}}$. Then find its argument $\theta $ using, for example, $tan\theta =\sqrt{3}\u2215\left(-1\right)$. However you must remember to adjust your answer to take account of the quadrant!*True*The modulus of $z$ is $2$ and its principal argument is $2\pi \u22153$, ($z$ is in the second quadrant).*True*The modulus of $z$ is $2$ and its principal argument is $2\pi \u22153$, ($z$ is in the second quadrant). However an argument is also $2\pi \u22153-2\pi =-4\pi i\u22153$. This leads to the polar exponential form $2{e}^{-4\pi i\u22153}$. There is no compulsion to use the principal argument when writing a complex number in polar form of any kind.*False*Plot $z$ in the complex plane to determine which quadrant it lies in. Its modulus is found using the formula $|a+ib|=\sqrt{{a}^{2}+{b}^{2}}$. Then find its argument $\theta $ using, for example, $tan\theta =\sqrt{3}\u2215\left(-1\right)$. However you must remember to adjust your answer to take account of the quadrant!

Find all solutions of the equation ${e}^{z}=-1+\sqrt{3}i$.
Exactly one option must be correct)

*Choice (a) is incorrect*

Don’t forget that when you equate polar forms of two equal complex numbers, you
have to remember that arguments are determined only up to an integer multiple of
$2\pi $.

*Choice (b) is incorrect*

Recall the
definition: when $z=x+iy$
then ${e}^{z}={e}^{x}\left(cosy+isiny\right)={e}^{x}\phantom{\rule{0.3em}{0ex}}{e}^{iy}$.

*Choice (c) is incorrect*

Recall the
definition: when $z=x+iy$
then ${e}^{z}={e}^{x}\left(cosy+isiny\right)={e}^{x}\phantom{\rule{0.3em}{0ex}}{e}^{iy}$.

*Choice (d) is incorrect*

Recall the
definition: when $z=x+iy$
then ${e}^{z}={e}^{x}\left(cosy+isiny\right)={e}^{x}\phantom{\rule{0.3em}{0ex}}{e}^{iy}$.

*Choice (e) is correct!*

In fact, the set of solutions of
${e}^{z}=-1+\sqrt{3}i$ is the set
$\left\{z\in \u2102\mid z=ln2+i\left(2\pi \u22153+2k\pi \right),k\in \mathbb{Z}\right\}.$ Thus there
are infinitely many solutions, each having the same real part and with imaginary parts
spaced $2\pi $
apart.

The notation $f:A\to B$
means Exactly one option must be correct)

*Choice (a) is incorrect*

The range of $f$
is a subset of $B$ and is not
necessarily equal to $B$.

*Choice (b) is correct!*

The notation $f:A\to B$
means that the function is well-defined on all elements of
$A$ (its input values come
from $A$). Its output
values all lie in $B$.
However $B$
need not be equal to the range. The range (the set of values which
$f$ actually maps to)
is a subset of $B$ and
may be equal to $B$.

*Choice (c) is incorrect*

The set
$A$ is equal to the
domain of $f$.

*Choice (d) is incorrect*

What is the largest possible domain and the corresponding range of the following
function?

$$f\left(x\right)=ln\left(x+2\right)$$

Exactly one option must be correct)
*Choice (a) is incorrect*

*Choice (b) is incorrect*

*Choice (c) is incorrect*

*Choice (d) is incorrect*

*Choice (e) is correct!*

What is the largest possible domain and the corresponding range of the
function

$$f\left(x\right)=sin\left({e}^{x}\right)\phantom{\rule{1em}{0ex}}?$$

Exactly one option must be correct)
*Choice (a) is incorrect*

*Choice (b) is correct!*

*Choice (c) is incorrect*

*Choice (d) is incorrect*

*Choice (e) is incorrect*

If $f\left(x\right)={x}^{2}$ and
$g\left(x\right)=x+1$ then the composite
function $\left(f\circ g\right)\left(x\right)$
is equal to Exactly one option must be correct)

*Choice (a) is incorrect*

*Choice (b) is incorrect*

*Choice (c) is incorrect*

*Choice (d) is correct!*

*Choice (e) is incorrect*

If $f\left(x\right)=\sqrt{x}$,
$g\left(x\right)=x+1$ and
$h\left(x\right)={e}^{x}$ then
$\left(\right.f\circ g\circ h\left)\right.\left(x\right)$ is given by Exactly one
option must be correct)

*Choice (a) is incorrect*

*Choice (b) is correct!*

*Choice (c) is incorrect*

*Choice (d) is incorrect*

*Choice (e) is incorrect*

When $f\left(x\right)=\frac{-4+3x}{-3+2x}$, what is
$f\left(f\left(x\right)\right)$? Exactly one option
must be correct)

*Choice (a) is incorrect*

*Choice (b) is incorrect*

*Choice (c) is incorrect*

*Choice (d) is incorrect*

*Choice (e) is correct!*

The formulas given below are formulas for various functions of the complex variable
$z$.
Which two functions have the same natural domain and corresponding range?
$f\left(z\right)={e}^{z},\phantom{\rule{2em}{0ex}}g\left(z\right)={e}^{\left|z\right|},\phantom{\rule{2em}{0ex}}h\left(z\right)={e}^{{z}^{2}},\phantom{\rule{2em}{0ex}}k\left(z\right)={z}^{2}$ (Zero or more options
can be correct)

For example, choice (a) should be False.

For example, choice (b) should be False.

For example, choice (c) should be True.

For example, choice (d) should be False.

For example, choice (e) should be False.

*There is at least one mistake.*

For example, choice (a) should be False.

$f$ and
$g$
have the same natural domain but different ranges.

*There is at least one mistake.*

For example, choice (b) should be False.

$f$ and
$k$
have the same natural domain but different ranges.

*There is at least one mistake.*

For example, choice (c) should be True.

Both
$f$ and
$h$ have
domain $\u2102$
and range $\u2102\setminus \left\{0\right\}$.

*There is at least one mistake.*

For example, choice (d) should be False.

$h$ and
$k$
have the same natural domain but different ranges.

*There is at least one mistake.*

For example, choice (e) should be False.

$g$ and
$k$ have
the same natural domain but different ranges.

*Correct!*

*False*$f$ and $g$ have the same natural domain but different ranges.*False*$f$ and $k$ have the same natural domain but different ranges.*True*Both $f$ and $h$ have domain $\u2102$ and range $\u2102\setminus \left\{0\right\}$.*False*$h$ and $k$ have the same natural domain but different ranges.*False*$g$ and $k$ have the same natural domain but different ranges.

Find the natural domain and corresponding range of the function
$f$ of the real
variable $x$ given
by the formula $f\left(x\right)=ln\left(x-{x}^{2}\right)$.
Exactly one option must be correct)

*Choice (a) is correct!*

We need
$x-{x}^{2}>0$,and this occurs
when $0<x<1$, giving the
domain $\left(0,1\right)$. The
values of $x-{x}^{2}$ reach
a maximum of $\frac{1}{4}$
when $x=\frac{1}{2}$, and
as $x$ gets
close to $0$
and to $1$,
$x-{x}^{2}$ gets closer and
closer to $0$. Therefore
$f\left(x\right)$ has vertical
asymptotes to $-\infty $
at $x=0$ and
$x=1$, and has a
maximum value of $ln\frac{1}{4}=-ln4$
when $x=\frac{1}{2}$. Hence the
range is $\left(-\infty ,-ln4\right]$.

*Choice (b) is incorrect*

Remember
that for the natural logarithm to be defined, it must be used with positive real inputs, so we
need $x-{x}^{2}>0$.

*Choice (c) is incorrect*

The natural domain is correct, but the corresponding range is incorrect.
Draw a rough sketch of the graph to help find the range.

*Choice (d) is incorrect*

Remember
that for the natural logarithm to be defined, it must be used with positive real inputs, so we
need $x-{x}^{2}>0$.

*Choice (e) is incorrect*

The natural domain is correct, but the corresponding range is incorrect. Draw a
rough sketch of the graph to help find the range.