Multiple Choice

1. Which of the following numbers is an irrational number?

   $2.\overline{1}$

   $0.85$

   $\sqrt{64}$

   $\sqrt{32}$

2. Which statement about rational and irrational numbers is true?

   The product of two irrationals is always irrational.

   The sum of two irrationals is always irrational.

   The sum of a rational and an irrational is always irrational.

   The product of a non-zero rational and an irrational is always rational.

3. Simplify $\displaystyle \frac{(2x^3y^2)^3 \cdot (3x^2y^4)}{12x^5y^7}$.

   $2x^6y^2$

   $6x^6y^3$

   $2x^6y^3$

   $2x^5y^3$

4. Assume $a,b\neq0$. Simplify $\displaystyle \left(\frac{a^5b^3}{ab^2}\right)^3\cdot (a^2b)^0$.

   $a^{15}b^6$

   $a^{12}b^2$

   $a^{12}b^3$

   $a^{12}b^0$

5. Earth has mass $5.97\times10^{24}$ kg, Sun has mass $1.99\times10^{30}$ kg. How many times more massive is the Sun?

   $3.33\times10^6$

   $3.33\times10^{-5}$

   $0.333\times10^6$

   $3.33\times10^5$

6. A virus is $2.5\times10^{-7}$ m; a bacterium is $4.0\times10^{-6}$ m. What is the combined length of 5 viruses and 2 bacteria (scientific notation)?

   $9.25\times10^{-6}$

   $9.25\times10^{-5}$

   $5.25\times10^{-6}$

   $9.25\times10^{-7}$

7. Solve $3(2x-5)+7=4x-2(x+1)$.

   $x=-2$

   $x=\tfrac{3}{2}$

   $x=\tfrac{2}{3}$

   $x=3$

8. Solve $\displaystyle \frac{2x}{3}-\frac{x-1}{2}=5$.

   $x=15$

   $x=33$

   $x=24$

   $x=27$

9. Which scenario does \emph{not} represent a function?

   Voltage $\mapsto$ power (one power for each voltage)

   $A\mapsto s$ via $s=\sqrt{A}$ (area to side length)

   Temperature value $F\mapsto$ time(s) $t$ recorded

   Minutes reading $\mapsto$ pages (constant rate)

10. Function $F$ has $(-3,11),(-1,7),(0,5),(2,1)$. Function $H$ is the line through $(-2,9),(2,-7)$. Compare intercepts.

    H has greater $x$- and $y$-intercepts.

    H has greater $x$-intercept, F has greater $y$-intercept.

    F has greater $x$-intercept, H has greater $y$-intercept.

    F has greater $x$- and $y$-intercepts.

11. A plant grows linearly: day $5$ height $18$ cm; day $12$ height $32$ cm. Find height on day $2$.

    $12$ cm

    $8$ cm

    $10$ cm

    $14$ cm

12. Line $f(x)=mx+b$ has slope $-3$ and passes through $(2,-5)$. Which statement is true?

    The $y$-intercept is $-1$.

    The $y$-intercept is $-5$.

    The $y$-intercept is $2$.

    The $y$-intercept is $1$.

13. Solve $3(x-4)-2x>5x+8$.

    $x<-5$

    $x>-20$

    $x<5$

    $x>-5$

14. Solve $5-2(3x+1)\ge 4x-13$.

    $x\le 2$

    $x\ge \tfrac{8}{5}$

    $x\le \tfrac{8}{5}$

    $x\le \tfrac{5}{8}$

15. A laptop was \$900 after a $25\%$ discount from the original price, then increased by $20\%$. Final price vs original?

    The final price is $10\%$ higher than the original price.

    The final price is the same as the original price.

    The final price is $10\%$ lower than the original price.

    The final price is $5\%$ lower than the original price.

16. Population $+10\%$ in 2022, then $-10\%$ in 2023. End of 2023 population $=9900$. Find population at start of 2022.

    $9800$

    $10000$

    $10100$

    $9900$

17. Boys:Girls $=3:5$. After $12$ boys leave and $4$ girls join, ratio becomes $1:2$. Initial total?

    $140$

    $168$

    $28$

    $224$

18. Ratios: French:Italian $=3:4$, Italian:German $=2:5$. If German exceeds French by $42$, how many Italian?

    $42$

    $18$

    $60$

    $24$

19. Boys:Girls $=3:4$, Girls:Teachers $=8:1$. If there are $140$ students, how many teachers?

    $80$

    $140$

    $60$

    $10$

20. A bag has red:blue $=5:3$. After adding $10$ red, ratio becomes $15:7$. Total marbles before addition?

    $35$

    $56$

    $7$

    $21$

21. $U=\{1,2,\dots,50\}$; $P=$ multiples of $3$; $Q=$ multiples of $5$. How many in $U$ are neither multiples of $3$ nor $5$?

    $3$

    $27$

    $16$

    $23$

22. $U=\{1\le x\le 10\}$; $P=\{1,2,3,4,5\}$, $Q=\{4,5,6,7,8\}$, $R=\{2,3,5,7,9\}$. Find $(P\cap Q)'\cup R$ (complement relative to $U$).

    $\{2,3,5,7,9\}$

    $\{1,2,3,5,6,7,8,9,10\}$

    $\{4,5\}$

    $\{1,2,3,6,7,8,9,10\}$

23. For what $x$ is $x^2=9x$ true?

    $x=-9$ or $x=9$

    $x=0$ or $x=9$

    $x=3$

    $x=9$

24. Auditorium rows: $12,15,18,\dots$ seats. Expression for $n$th row?

    $n+11$

    $3n+9$

    $3n+12$

    $3n$

25. Sequence $2,9,22,41,\dots$. Find the $7$th term.

    $100$

    $134$

    $119$

    $125$

26. $a_n=2a_{n-1}-a_{n-2}$ with $a_1=3,a_2=5$. Find $a_5$.

    $9$

    $11$

    $13$

    $10$

27. Sequence $\tfrac12,\tfrac14,\tfrac18,\tfrac1{16},\dots$. Which term equals $\tfrac1{512}$?

    $12$th

    $9$th

    $10$th

    $8$th

28. Sequence $3,8,15,24,\dots$. Next term?

    $36$

    $35$

    $30$

    $33$

29. Given the sequence $a_n=\dfrac{n^2+n}{2}$. What is the sum of the first $4$ terms?

    $20$

    $10$

    $25$

    $15$

30. Evaluate $\left(1\frac{1}{2}-\frac{3}{4}\right)\div\left(\frac{5}{6}+\frac{1}{3}\right)\times 2$.

    $\dfrac{3}{2}$

    $\dfrac{7}{9}$

    $\dfrac{18}{7}$

    $\dfrac{9}{7}$

31. Compute $\displaystyle \frac{2}{3} \div \left(\frac{1}{2}+\frac{1}{3}\right)\times\left(\frac{3}{4}-\frac{1}{5}\right)+\frac{1}{4}$.

    $\dfrac{11}{25}$

    $\dfrac{5}{6}$

    $\dfrac{3}{4}$

    $\dfrac{69}{100}$

32. Simplify $\left(\frac12\right)^2 + 3\frac{1}{3} \div \left(1\frac{1}{4}-\frac12\right)$.

    $\dfrac{40}{9}$

    $\dfrac{169}{12}$

    $\dfrac{169}{36}$

    $\dfrac{1}{4}$

33. Simplify $\displaystyle \frac{8^3\cdot(2^{-2})^4}{4^{-5}}$.

    $2^{11}$

    $2^1$

    $2^{27}$

    $2^{-9}$

34. Simplify $(\tfrac{1}{5})^{-3}\cdot(5^2)^{-1}\cdot 5^0$.

    $5^3$

    $1$

    $5^{-1}$

    $5$

35. If $9^x\cdot (3^2)^5=\dfrac{1}{27^{-2}}$, find $x$.

    $x=2$

    $x=-2$

    $x=3$

    $x=-8$