## MATH1901 Quizzes

Quiz 1: Numbers and sets
Question 1 Questions
Which of the following are correct ways of writing the set
$A=\left\{x\in ℝ\phantom{\rule{1em}{0ex}}|\phantom{\rule{1em}{0ex}}-3
Exactly one option must be correct)
 a) $\left(-3,\infty \right)$ b) $\left[-3,\infty \right)$ c) $\left[-3,-1\right]\cap \left[0,\infty \right)$ d) $\left(-3,-1\right)\cup \left[0,\infty \right)$ e) $\left(-3,-1\right]\cup \left[0,\infty \right)$

Choice (a) is incorrect
$\left(-3,\infty \right)$ includes the real numbers between $-1$ and $0$,
which do not belong to $A$.
Choice (b) is incorrect
$\left[-3,\infty \right)$ includes $-3$, as well as the real numbers between $-1$ and $0$, none of which belong to $A$.
Choice (c) is incorrect
$\left[-3,-1\right]\cap \left[0,\infty \right)$ is the empty set $\varnothing$. As $A$ is not empty (for example, $A$ includes $-1$), this option can’t be correct.
Choice (d) is incorrect
$-1$ is not in $\left(-3,-1\right)\cup \left[0,\infty \right)$, but $-1$ is in $A$.
Choice (e) is correct!
What is another way of writing the set
$B=\left\{x\in ℝ\phantom{\rule{1em}{0ex}}|\phantom{\rule{1em}{0ex}}|x-3|<2\right\}\phantom{\rule{1em}{0ex}}?$
Exactly one option must be correct)
 a) $\left(2,3\right]$ b) $\left[2,4\right]$ c) $\left(1,5\right)$ d) $\left[1,5\right]$ e) $\left[2,3\right)$

Choice (a) is incorrect
For example, $4$ belongs to $B$ but is not in $\left(2,3\right]$.
Choice (b) is incorrect
For example, $1.5$ belongs to $B$ but is not in $\left[2,4\right]$.
Choice (c) is correct!
$B$ is the set of all points whose distance from 3 on the number line is less than 2.
The solution to $|x-3|<2$ is $1.
Choice (d) is incorrect
Neither $1$ nor $5$ belong to $B$, but both $1$ and $5$ belong to $\left[1,5\right]$.
Choice (e) is incorrect
For example, $4$ belongs to $B$ but is not in $\left[2,3\right)$.
If $A=\left\{7,8,9,10\right\}$ and $B=\left\{5,6,7,8\right\}$ then $\left(A\setminus B\right)\cup \left(B\setminus A\right)$ is Exactly one option must be correct)
 a) $\left\{5,6,7,8,9,10\right\}$ b) $\left\{5,6,9,10\right\}$ c) $\varnothing$, the empty set. d) $\left\{7,8\right\}$ e) None of the above.

Choice (a) is incorrect
Choice (b) is correct!
$A\setminus B=\left\{9,10\right\}$ and $B\setminus A=\left\{5,6\right\}$ so $\left(A\setminus B\right)\cup \left(B\setminus A\right)=\left\{5,6,9,10\right\}$.
Choice (c) is incorrect
Choice (d) is incorrect
Choice (e) is incorrect
The set $\left\{0,1,±\sqrt{-1},\pi ,12\right\}$ is a subset of Exactly one option must be correct)
 a) $ℕ$ b) $ℤ$ c) $ℚ$ d) $ℝ$ e) $ℂ$

Choice (a) is incorrect
The number $\pi$ is not a natural number.
Choice (b) is incorrect
The number $\pi$ is not an integer.
Choice (c) is incorrect
The number $\pi$ is not a rational number.
Choice (d) is incorrect
$\sqrt{-1}$ is not real.
Choice (e) is correct!
Since $±\sqrt{-1}$ denotes the two imaginary numbers $i$ and $-i$, the given set cannot be in any of the sets $ℕ,ℤ,ℚ\phantom{\rule{1em}{0ex}}or\phantom{\rule{1em}{0ex}}ℝ$.
Hence the right answer must be $ℂ$ which contains all imaginary numbers.
Which of the following alternatives is the best response to ‘Solve ${x}^{2}-3x+4=0$ over $ℂ$’. Exactly one option must be correct)
 a) There are no real solutions. b) $x=1,4$ c) $x=\frac{3±\sqrt{7}}{2}$ d) $x=\frac{3±i\sqrt{7}}{2}$ e) None of the above is correct.

Choice (a) is incorrect
As the question asks us to solve the equation over $ℂ$ (that is, to find all solutions belonging to the set of complex numbers), this is not the best response.
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
Using the quadratic formula, $x=\frac{3±\sqrt{9-16}}{2}=\frac{3±\sqrt{-7}}{2}$.
Choice (e) is incorrect
If $z=9+3i$ and $w=2-i$ then $z+w$ equals Exactly one option must be correct)
 a) $9-i$ b) $11+2i$ c) $6+3i$ d) $8$ e) None of the above

Choice (a) is incorrect
Choice (b) is correct!
$z+w=\left(9+3i\right)+\left(2-i\right)=\left(9+2\right)+\left(3-1\right)i=11+2i$.
Choice (c) is incorrect
Choice (d) is incorrect
Choice (e) is incorrect
If $w=2-i$ then $\overline{w}$ equals Exactly one option must be correct)
 a) $2-i$ b) $2$ c) $2+i$ d) $-2+i$ e) None of the above

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
$\overline{w}=\overline{2-i}=2+i$.
Choice (d) is incorrect
Choice (e) is incorrect
If $p=9+3i$ and $q=2-i$ then $p\overline{q}$ equals
Exactly one option must be correct)
 a) $15+15i$ b) $21+15i$ c) $18+3i$ d) $1-i$ e) None of the above

Choice (a) is correct!
$p\overline{q}=\left(9+3i\right)\overline{\left(2-i\right)}$
$=\left(9+3i\right)\left(2+i\right)=\left(18-3\right)+\left(6+9\right)i$
$=15+15i$.
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Choice (e) is incorrect
If $z=9+3i$ and $w=2-i$ then $\frac{z}{w}$ equals Exactly one option must be correct)
 a) $15+15i$ b) $6+3i$ c) $12+15i$ d) $3-3i$ e) None of the above

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Choice (e) is correct!
$\frac{z}{w}=\frac{9+3i}{2-i}=\frac{9+3i}{2-i}×\frac{2+i}{2+i}=\frac{15+15i}{5}=3+3i$.
The shaded region in the graph
corresponds to which set of complex numbers? Exactly one option must be correct)
 a) $\left\{z\in ℂ:|z-\left(i+1\right)|<2\right\}$ b) $\left\{z\in ℂ:|z|-|1+i|<2\right\}$ c) $\left\{z\in ℂ:\text{Re}\left(z+\left(i+1\right)\right)<2\right\}$ d) $\left\{z\in ℂ:|z-2|<|i+1-2|\right\}$ e) None of the above.

Choice (a) is correct!
Choice (b) is incorrect
This set corresponds to the interior of a circle, centre the origin, radius $2+\sqrt{2}$.
Choice (c) is incorrect
This set corresponds to the open half plane containing all complex numbers $z=x+iy$ with $x<1$.
Choice (d) is incorrect
This set corresponds to the interior of a circle, centre $2$, radius $\sqrt{2}$.
Choice (e) is incorrect