Practice Test 56 - Algebra 2

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

Start practice test

1. A factory's daily profit (in dollars) from producing $n$ portable chargers is modeled by $P(n)=-2n^2+180n-200$. The factory can produce at most 80 chargers per day, and it cannot produce a negative number of chargers. What is an appropriate realistic domain for $P(n)$?
2. Order these polynomials by eventual growth rate as $x\to\infty$ (fastest to slowest): $f(x)=3x^2$, $g(x)=x^5$, $h(x)=100x^3$, $k(x)=0.1x^4$.
3. The population of bacteria in a petri dish is modeled by $P(h) = 500 \cdot 3^{0.5h}$, where $h$ is the number of hours after the experiment begins. After how many hours will the bacterial population first triple from its initial value?
4. Convert $3\left(\cos 120^\circ+i\sin 120^\circ\right)$ to rectangular form $a+bi$ using $a=r\cos\theta$ and $b=r\sin\theta$.
5. Which method is most efficient for solving the quadratic equation $x^2 = 81$?
6. What is $f^{-1}(x)$ if $f(x)=\dfrac{x+1}{x-1}$, where $x\ne 1$? (Note: $f^{-1}$ means inverse function, not reciprocal.)
7. Use the Remainder Theorem to find the remainder when $p(x)=x^3-4x^2+5x-2$ is divided by $(x-3)$. (Recall: remainder $=p(a)$ when dividing by $(x-a)$.)
8. Using the fact that $(a+bi)(a-bi)=a^2+b^2$, compute $(2 + 5i)(2 - 5i)$.
9. Seeing structure can lead to different but equivalent rewrites. Which choice shows a correct factoring of $x^4-1$ using the difference of squares pattern (and continuing to factor when possible)?
10. Use the geometric series formula to verify the sum. Which expression equals $3+6+12+24?$
11. Use the Rational Zeros Theorem to list all possible rational zeros of $P(x)=6x^4-x^3-7x^2+x+1.$ (List candidates $p/q$ in lowest terms.)
12. Use the geometric series formula $S_n=\dfrac{a(1-r^n)}{1-r}$ (for $r\ne 1$) to find the sum $2+2(1.1)+2(1.1)^2+\cdots+2(1.1)^9.$
13. In optics, the thin lens equation is $\dfrac{1}{f}=\dfrac{1}{a}+\dfrac{1}{b}$, where $f$ is focal length, $a$ is object distance, and $b$ is image distance. Solve for $f$ in terms of $a$ and $b$.
14. Use the Binomial Theorem to compute $103^3$ by rewriting it as $(100+3)^3$.​
15. A medication dose starts at 240 mg. Each hour, 20% of the medication is eliminated (so 80% remains). Write and solve an exponential equation to find how many hours it takes until 60 mg remains. Let $t$ = time in hours.
16. The sequence $1,3,9,27,81,\dots$ is a function with domain $n\in\{1,2,3,\dots\}$. Which recursive definition matches this sequence?
17. Represent the system as $AX=B$ using variable order $x, y, z$: $\begin{cases} 2x+y-z=5\\ x-3y+2z=1\\ 4x+y+z=9 \end{cases}$ Which matrix equation matches the system?
18. A student claims the system (variable order $x, y$) $\begin{cases} 4x-y=2\\ -3x+6y=9 \end{cases}$ can be written as $\begin{bmatrix}4&-1\\-3&6\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}2\\9\end{bmatrix}.$ Which statement best verifies the connection between the system and the matrix equation?​
19. A savings account balance is modeled by $B(t) = 1200(1.04)^t$, where $B$ is the balance in dollars and $t$ is time in years. What does the base $1.04$ represent in this context?
20. Given the explicit formula $a_n = 9 + 4(n-1)$ (domain: positive integers), write an equivalent recursive definition for the sequence.

Practice tests