Set 1: Math
First set: Math
5 problems: 30min.
Focus! Block out distractions! Your time will NOT be taken into account when measuring your $iq^\ell$. You can take all of it or less, once you think you are done, you can’t edit anymore. So, finishing early gives you nothing.
No. 1
Simplify
\[\frac{\frac{a}{b}}{\frac{b}{a}} = \frac{\frac{b}{a}}{\frac{a}{b}}\]$a,b \in \mathbb{R}$ but can’t be zero.
No. 2
If $a^2 = b^2$ for $a,b \in \mathbb{R}$, does it mean that $a=b$ always?
Yes? No? Explain.
No. 3
If $a^4 = b^4$ does it always mean $a^2 = b^2$ for real $a,b$?
How about if $a^6 = b^6$, does it mean that $a = b$?
No. 4
The law of sines is written
\[\frac{\sin(a)}{A} = \frac{\sin(b)}{B} = \frac{\sin(c)}{C}\]What’s the smallest $A$ can be? how about $a$? are these two quantities directly or inversely related?
No. 5
Now the pythagorean theorem states
\[\sin^2(a) + \sin^2(\pi/2 - a) = r^2\]What’s the domain of $r$?
That is, what could the values for $r$ be?