Practice Test 9 - Algebra 2

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

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1. Determine whether the system has one solution, no solution, or infinitely many solutions: $\begin{cases} 2x + y = 5\\ 4x + 2y = 12 \end{cases}$
2. A gym charges members according to the cost function $C = 25m + 40$, where $C$ is the total cost in dollars and $m$ is the number of months of membership. In the function, what does the parameter $40$ represent in context?
3. For which values of $x$ is the expression $\frac{x^2 - 9}{x^2 + x - 12} \div \frac{x + 3}{x - 3}$ undefined?
4. Find the polar form of $-5i$ as $r(\cos\theta + i\sin\theta)$ with $\theta$ in degrees, where $r=\sqrt{a^2+b^2}$ and $\theta$ is the argument measured from the positive real axis.
5. A geometric series has first term $a=12$, common ratio $r=-\tfrac{1}{3}$, and $n=5$ terms. Find $S_5$.
6. A medicine amount in the bloodstream is recorded every hour: Hour: 0, 1, 2, 3 Amount (mg): 80, 68, 57.8, 49.13 Classify: exponential growth, exponential decay, or neither. If exponential, what percent does it change per hour?
7. Solve $6\cdot 10^{t/3}=15$ for $t$. Express the solution as a logarithm and then approximate using technology.
8. For $f(x)=\frac{2x^2-5}{x^2+4},$ what is the horizontal asymptote?​
9. A gym membership costs $25 per month plus a one-time sign-up fee of $60. Let $m$ be the number of months. Write an equation for total cost $C$ (dollars) in terms of $m$ and choose appropriate axes labels and scale to graph for $0\le m\le 12$. Which choice is correct?​
10. Which set of points on the complex plane represents all complex numbers $z$ such that $|z|=4$? (Here $|z|$ is the distance from the origin.)
11. Convert the system to matrix form $AX=B$ (use variable order $x, y$): $$\begin{cases} 4x-y=2\\ -3x+6y=15 \end{cases}$$
12. Two points on the complex plane are $z_1=-4+1i$ and $z_2=2-5i$. Using the coordinate-geometry distance formula written for complex numbers, $|z_1-z_2|=\sqrt{(a-c)^2+(b-d)^2}$, what is the distance between them?
13. A quadratic expression in the form $x^2 + bx + c$ has been factored as $(x + 4)(x - 7)$. What is the sum of the coefficients $b + c$?
14. Solve $f(x)=16$ where $f(x)=2x^3$, and express the result as $f^{-1}(16)$.​​​
15. If $\log_2(x)=7$, find $x$ by rewriting in exponential form.
16. Possible rational zeros of $P(x)=x^3-4x^2+x+6$ are $\pm 1,\pm 2,\pm 3,\pm 6$. Which of these candidates are actual zeros? (You may test by substitution.)
17. Use $p(a)$ to find the remainder for division by $(x-a).$ If $p(x)=x^4-x^2+2x-6,$ what is the remainder when dividing by $(x-2)?$
18. Graph the rational function $f(x)=\frac{(x-2)(x+1)}{(x-4)(x+3)}$ identifying zeros, vertical asymptotes, and the horizontal asymptote (end behavior).
19. What is the horizontal asymptote of the exponential function $p(x)=2^x+4$? (Connect to the parent function $y=2^x$.)
20. A physics student needs to determine the acceleration of an object given the formula $F = ma + kv^2$, where $F$ is the total force, $m$ is mass, $a$ is acceleration, $k$ is a drag coefficient, and $v$ is velocity. Which expression correctly isolates the acceleration?

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