Practice Test 67 - Algebra 2

20 questions. Answer them in any order, then submit to see your score and every explanation.

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1. A store models the number of reward points a customer earns as $R(d)=2d$, where $d$ is the number of dollars spent. In reality, a customer can spend any amount from $0$ to $150$ dollars (including cents). Should the graph of $R$ versus $d$ be shown as discrete points or as a continuous curve, and why?
2. Solve the system using the inverse of the coefficient matrix. First write $AX=B$, then compute $X=A^{-1}B$. $\begin{cases} 2x+y=5 \\ 3x+4y=7 \end{cases}$ Which ordered pair $(x,y)$ is the solution?
3. A phone trade-in promotion gives you $400 today, but if you wait, the offer decreases by $15\%$ each week. If you wait 6 weeks, the total value of receiving the offer each week (hypothetically adding all weekly offers) is modeled by the finite geometric series $400 + 400(0.85) + 400(0.85)^2 + \cdots + 400(0.85)^5.$ What is the exact sum in geometric-series form?
4. Use polynomial long division (or inspection) to rewrite the rational expression $\frac{2x^3+x^2-5x+3}{x-2}$ in the form $q(x)+\frac{r(x)}{x-2}$, where $\deg(r)<\deg(x-2)$.
5. A population of bacteria triples every 4 hours. If the initial population is $P_0$, which expression is equivalent to $3^{t/4}$ and reveals the approximate growth factor per hour?
6. A car's fuel efficiency function is $E(s) = -0.02s^2 + 1.6s + 10$ where $E$ is miles per gallon and $s$ is speed in mph. The car operates between 20 mph and 80 mph. Which analysis correctly identifies the optimal operating conditions and efficiency behavior?
7. A charity sells tickets for a fundraiser. Adult tickets cost $12 each and student tickets cost $8 each. Let $R$ be total revenue (dollars), $a$ be the number of adult tickets, and $s$ be the number of student tickets. Write an equation representing this relationship, and choose correct axes labels for a graph of $R$ versus $a$ when $s$ is fixed at $20$.
8. Compute the product of a complex number and its conjugate: $(4 + 3i)(4 - 3i)$. (Recall $(a+bi)(a-bi)=a^2+b^2$, which is real and equals $|a+bi|^2$.)
9. What is $f^{-1}(x)$ if $f(x)=\dfrac{x+1}{x-1}$ with domain restriction $x\ne 1$?​​​
10. Consider the complex numbers $a = 3 + 4i$ and $b = 5 - 12i$. When computing $a - b$ geometrically on the complex plane, the result vector starts at the origin and ends at which point?
11. Write an expression for $f^{-1}(x)$ given $f(x)=x^3+5$. (Use the swap-and-solve process: write $y=f(x)$, swap $x$ and $y$, then solve for $y$.)
12. Find the center and radius of the circle $x^2 + y^2 + 10x - 4y + 13 = 0$.
13. Factor $x^2+25$ over the complex numbers and state how many zeros it has counting multiplicity. (A degree-2 polynomial has exactly 2 complex zeros counting multiplicity; this is a special case preview of the Fundamental Theorem of Algebra: every polynomial of degree $n$ has exactly $n$ complex zeros counting multiplicity.)
14. Solve the system by both substitution and elimination (they should give the same result). What is the solution? $\begin{cases} x - y = 2\\ 2x + y = 10 \end{cases}$
15. Graph $p(x)=x^5-5x^3+4x$ by finding its zeros (from factoring) and using end behavior to make a rough sketch.​
16. A gym charges a monthly membership fee plus a one-time signup fee. The total cost after $m$ months is modeled by $C(m)=35m+20$. The gym only allows memberships for up to 24 months, and $m$ must represent a whole number of months. What is an appropriate realistic domain for $C(m)$ in this context?
17. What is $t$ in the equation $5\cdot 2^{3t}=70$? Express as a logarithm and evaluate using a calculator.
18. Use the logarithm properties $$\log_b(xy)=\log_b(x)+\log_b(y),\quad \log_b\left(\frac{x}{y}\right)=\log_b(x)-\log_b(y),\quad \log_b(x^p)=p\log_b(x)$$ to write as a single logarithm: $\log_2(5)+\log_2(3)-\log_2(4).$
19. Divide $P(x)=x^3+x^2-5x+2$ by $(x-1)$ using synthetic division (often faster than long division for $(x-c)$). Read the quotient and remainder from the bottom row. Which is correct? (Check with the Remainder Theorem: remainder $=P(1)$.)
20. A circuit current is represented by the complex number $-3-3i$ amps in rectangular form $a+bi$. Convert it to polar form $r(\cos\theta+i\sin\theta)$ with $\theta$ in degrees. Use $r=\sqrt{a^2+b^2}$ and $\theta=\arctan\left(\frac{b}{a}\right)$ with quadrant adjustment. (Argument is measured from the positive real axis.)

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