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asked 19 hours ago in Mathematics by ♦Pieter Bronze Status (6,963 points) | 7 views

Sketch the function $f(x)=\left\{\begin{array}{cc}|x| & \text { if }|x| \leq 1 \\ 1 & \text { if }|x| \geq 1\end{array}\right.$ and write down its domain and range.

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asked 19 hours ago in Mathematics by ♦Pieter Bronze Status (6,963 points) | 6 views

Sketch the function $f(x)=-\frac{1}{2} \sqrt{x+1}$ and write down its domain and range.

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asked 19 hours ago in Mathematics by ♦Pieter Bronze Status (6,963 points) | 44 views

Evaluate $\lim _{x \rightarrow 1}\left(\frac{y-4 \sqrt{y}+3}{y^{2}-1}\right)$

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asked 19 hours ago in Mathematics by ♦Pieter Bronze Status (6,963 points) | 7 views

List the Compound Angle Identities in trigonometry

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asked 19 hours ago in Mathematics by ♦Pieter Bronze Status (6,963 points) | 7 views

List the Double Angle Identities in trigonometry

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asked 19 hours ago in Mathematics by ♦Pieter Bronze Status (6,963 points) | 14 views

There are 14 people at a party. Every pair of people shakes hands exactly once. How many handshakes occur?

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asked 19 hours ago in Mathematics by ♦Pieter Bronze Status (6,963 points) | 6 views

Find the $83^{\text {rd }}$ term in the sequence, $1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,6,6, \ldots$

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asked 19 hours ago in Mathematics by ♦Pieter Bronze Status (6,963 points) | 5 views

Which fraction is the smallest?

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asked 19 hours ago in Mathematics by ♦Pieter Bronze Status (6,963 points) | 9 views

Find $726 \times 32+726 \times 68$.

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asked 19 hours ago in Mathematics by ♦Pieter Bronze Status (6,963 points) | 6 views

My friend is 4 years older than me. The sum of our ages is 24 . How old am I?

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asked 20 hours ago in Mathematics by ♦Pieter Bronze Status (6,963 points) | 6 views

What is 30 percent of 20 percent of 50 percent of 7000 ?

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asked 20 hours ago in Mathematics by ♦Pieter Bronze Status (6,963 points) | 7 views

Which of the following numbers is closest to $0,000246 \times 7982413 ?$

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asked 20 hours ago in Mathematics by ♦Pieter Bronze Status (6,963 points) | 7 views

Some students in a Grade five class line up. Thabo is $10^{t h}$ from the front and $12^{t h}$ from the back. How many students are in the class?

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asked 20 hours ago in Mathematics by ♦Pieter Bronze Status (6,963 points) | 6 views

Find the value of $0,2019-0,02019$.

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asked 20 hours ago in Mathematics by ♦Pieter Bronze Status (6,963 points) | 6 views

Find the value of $1+2+3+4+5+6+7+8+9+10$.

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asked 20 hours ago in Mathematics by ♦Pieter Bronze Status (6,963 points) | 6 views

Nkosi played soccer with some friends. They started at 5:20 pm and finished at 6:05 pm. How long was their game?

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asked 20 hours ago in Mathematics by ♦Pieter Bronze Status (6,963 points) | 23 views

Mr Smith has a sweet shop. The graph below shows his sales for each day of a week, from Monday to Sunday. On which day were sales highest?

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1 answer

asked 20 hours ago in Mathematics by ♦Pieter Bronze Status (6,963 points) | 6 views

Find the value of $20 \times 19$.

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asked 20 hours ago in Mathematics by ♦Pieter Bronze Status (6,963 points) | 6 views

Let $f(z)$ be the principal branch of $\sqrt[3]{z}$.

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asked 20 hours ago in Mathematics by ♦Pieter Bronze Status (6,963 points) | 6 views

Find $i^{i}$ and its principal value.

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asked 20 hours ago in Mathematics by ♦Pieter Bronze Status (6,963 points) | 6 views

Compute $\cos \left(\frac{\pi}{3}+i\right)$

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asked 20 hours ago in Mathematics by ♦Pieter Bronze Status (6,963 points) | 6 views

Calculate $\sin \left(\frac{\pi}{4}+i\right)$.

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asked 20 hours ago in Mathematics by ♦Pieter Bronze Status (6,963 points) | 7 views

Find all the complex roots of the equation $\cos z=3$.

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asked 20 hours ago in Mathematics by ♦Pieter Bronze Status (6,963 points) | 7 views

Show that $$ \tan \left(z_{1}+z_{2}\right)=\frac{\tan z_{1}+\tan z_{2}}{1-\left(\tan z_{1}\right)\left(\tan z_{2}\right)} $$

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asked 20 hours ago in Mathematics by ♦Pieter Bronze Status (6,963 points) | 7 views

Show that $$ |\cos (z)|^{2}=(\cos x)^{2}+(\sinh y)^{2} $$ for all $z \in \mathbb{C}$, where $x=\operatorname{Re}(z)$ and $y=\operatorname{Im}(z)$

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asked 20 hours ago in Mathematics by ♦Pieter Bronze Status (6,963 points) | 6 views

Show that $$ |\sin z|^{2}=(\sin x)^{2}+(\sinh y)^{2} $$ for all complex numbers $z=x+y i$.

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asked 20 hours ago in Mathematics by ♦Pieter Bronze Status (6,963 points) | 7 views

Sketch the following sets in the complex plane $\mathbb{C}$ and determine whether they are open, closed, or neither; bounded; connected. Briefly state your reason.

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asked 20 hours ago in Mathematics by ♦Pieter Bronze Status (6,963 points) | 6 views

Let $$ T(z)=\frac{z}{z+1} $$ Find the inverse image of the disk $|z|<1 / 2$ under $T$ and sketch it.

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1 answer

asked 20 hours ago in Mathematics by ♦Pieter Bronze Status (6,963 points) | 8 views

Suppose $p(z)$ is a polynomial with real coefficients. Prove that

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asked 21 hours ago in Mathematics by ♦Pieter Bronze Status (6,963 points) | 6 views

Suppose that $f(z)=x^{2}-y^{2}-2 y+i(2 x-2 x y)$, where $z=x+i y$.

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asked 21 hours ago in Mathematics by ♦Pieter Bronze Status (6,963 points) | 80 views

Write the following functions $f(z)$ in the forms $f(z)=u(x, y)+i v(x, y)$ under Cartesian coordinates with $u(x, y)=\operatorname{Re}(f(z))$ and $v(x, y)=\operatorname{Im}(f(z))$ :

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asked 21 hours ago in Mathematics by ♦Pieter Bronze Status (6,963 points) | 6 views

Use complex numbers to prove the Law of Cosine: Let $\triangle A B C$ be a triangle with

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1 answer

asked 21 hours ago in Mathematics by ♦Pieter Bronze Status (6,963 points) | 8 views

Establish the identity and then use it to derive Lagrange's trigonometric identity

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1 answer

asked 21 hours ago in Mathematics by ♦Pieter Bronze Status (6,963 points) | 5 views

Use exponential form to compute

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asked 21 hours ago in Mathematics by ♦Pieter Bronze Status (6,963 points) | 5 views

Find the four roots of the polynomial $z^{4}+16$ and use these to factor $z^{4}+16$ into two quadratic polynomials with real coefficients.

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asked 21 hours ago in Mathematics by ♦Pieter Bronze Status (6,963 points) | 6 views

Find all the complex roots of the equations:

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asked 21 hours ago in Mathematics by ♦Pieter Bronze Status (6,963 points) | 6 views

Find the principal argument and exponential form of

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asked 21 hours ago in Mathematics by ♦Pieter Bronze Status (6,963 points) | 7 views

Express the following in the form $x+i y$, with $x, y \in \mathbb{R}$ :

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asked 21 hours ago in Mathematics by ♦Pieter Bronze Status (6,963 points) | 7 views

Show that $$ |\log (z)| \leq|\ln | z||+\pi $$ for all $z \neq 0$

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asked 21 hours ago in Mathematics by ♦Pieter Bronze Status (6,963 points) | 6 views

Show that $$ \frac{R^{4}-R}{R^{2}+R+1} \leq\left|\frac{z^{4}+i z}{z^{2}+z+1}\right| \leq \frac{R^{4}+R}{(R-1)^{2}} $$ for all $z$ satisfying $|z|=R>1$.

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asked 21 hours ago in Mathematics by ♦Pieter Bronze Status (6,963 points) | 6 views

Sketch the curves in the complex plane given by

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asked 21 hours ago in Mathematics by ♦Pieter Bronze Status (6,963 points) | 6 views

Verify that $\sqrt{2}|z| \geq|\operatorname{Re} z|+|\operatorname{Im} z|$.

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1 answer

asked 21 hours ago in Mathematics by ♦Pieter Bronze Status (6,963 points) | 6 views

Show that $$ \left|z_{1}-z_{2}\right|^{2}+\left|z_{1}+z_{2}\right|^{2}=2\left(\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}\right) $$ for all $z_{1}, z_{2} \in \mathbb{C}$.

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asked 21 hours ago in Mathematics by ♦Pieter Bronze Status (6,963 points) | 6 views

The set $\mathbb{Q}$ adjoin $\sqrt{2}$ is defined by $\mathbb{Q}(\sqrt{2})=\{p+q \sqrt{2}: p, q \in \mathbb{Q}\}$.

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asked 21 hours ago in Mathematics by ♦Pieter Bronze Status (6,963 points) | 6 views

Suppose that $z_{1}$ and $z_{2}$ are complex numbers, with $z_{1} z_{2}$ real and non-zero. Show that there exists a real number $r$ such that $z_{1}=r \bar{z}_{2}$.

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asked 21 hours ago in Mathematics by ♦Pieter Bronze Status (6,963 points) | 5 views

Find all complex solutions of the following equations:

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asked 21 hours ago in Mathematics by ♦Pieter Bronze Status (6,963 points) | 5 views

Graph the following regions in the complex plane:

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asked 21 hours ago in Mathematics by ♦Pieter Bronze Status (6,963 points) | 5 views

Use the binomial theorem to expand

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asked 21 hours ago in Mathematics by ♦Pieter Bronze Status (6,963 points) | 6 views

Show that $f\left(z_{1} z_{2}\right)=f\left(z_{1}\right) f\left(z_{2}\right)$ for all $z_{1}, z_{2} \in \mathbb{C}$.

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asked 21 hours ago in Mathematics by ♦Pieter Bronze Status (6,963 points) | 5 views

Compute (a) $\frac{2+i}{2-i}$;

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