Practice Test 10 - Algebra 2

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

Subtract and simplify to lowest terms. (The difference of rational expressions is still a rational expression.) Subtract: $\dfrac{x+1}{x^2-4}-\dfrac{3}{x-2}$ (assume denominators are nonzero).
A medication dose starts at 240 mg. Each hour, 20% of the medication is eliminated (so 80% remains). Model the amount remaining with an exponential equation and solve for how many hours it takes until 60 mg remains. Let $t$ = time in hours.
A complex number $z$ has modulus $5$ and is located in the third quadrant such that the reference angle is $\frac{\pi}{3}$ . If $z = a + bi$ in rectangular form, what is the value of $a^2 + b^2$ ?
An account grows by 12% per year, so its value after $t$ years is multiplied by $(1.12)^t$ . Rewrite $(1.12)^t$ in an equivalent form that reveals the monthly growth factor (assume 12 months per year).
Which matrix equation $AX=B$ correctly represents the system (use variable order $x, y, z$ )? $\begin{cases} 6x-2y=1\\ -y+4z=-8\\ 3x+z=5 \end{cases}$ (Remember: missing variables must have coefficient $0$ in $A$ .)
Solve the system and state the solution set. $\begin{cases} x + 2y = 10 \\ 2x + 4y = 20 \end{cases}$
Translate the explicit formula $a_n=3\cdot 2^{n-1}$ into an equivalent recursive definition (domain: positive integers).
The sequence $5,\ 15,\ 45,\ 135,\dots$ is used to model the number of views a video gets each day, where day 1 has 5 views and each day has the same multiplicative change from the previous day. Write both a recursive and an explicit formula for the sequence (integer domain $n\ge 1$ ).
On the complex plane, the distance between $z_1=a+bi$ and $z_2=c+di$ is $|z_1-z_2|=\sqrt{(a-c)^2+(b-d)^2}$ (same as the 2D distance formula), and the midpoint is $\dfrac{z_1+z_2}{2}$ . For $z_1=2+5i$ and $z_2=6-1i$ , what are the distance and the midpoint?
Function $c$ is described as "a linear function with slope $-\frac{1}{2}$ passing through point $(4, 3)$ ". The graph shows function $d(x)$ . Compare the $x$ -intercepts of these two functions.
Generate a Pythagorean triple using the identity $(x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2$ with $x=4$ and $y=1$ . Which triple $(a,b,c)$ is correct (order of $a,b$ may be swapped)?
An event planner is ordering chairs and tables for a fundraiser. Let $x$ be the number of chairs and $y$ be the number of tables. Constraints: - Each chair costs $8 and each table costs $25; budget at most $400: $8x+25y\le 400$ . - Seating requirement: each table requires 4 chairs, so there must be at least 4 chairs per table: $x\ge 4y$ . - At least 6 tables are needed: $y\ge 6$ . - Chairs and tables must be whole numbers and nonnegative. Which constraint does $(x,y)=(20,7)$ violate?
Check whether $f(x)=\dfrac{2x+1}{3}$ and $g(x)=\dfrac{3x-1}{2}$ are inverses by computing both $f(g(x))$ and $g(f(x))$ and simplifying to see if each equals $x$ .
Consider the rational expression $\frac{x^2+3x-2}{x-1}.$ If you view the entire numerator as a single entity (a chunk), which statement best describes how the expression is structured?
A number and its reciprocal have a sum of $\frac{10}{3}$ . What equation represents this context? Solve it and interpret the solutions. Let $x$ = the number (assume $x\ne 0$ ).
Distinguish between terms and factors: In the expression $2x(x-3)^2 + 5$ , which list correctly gives the terms, and also gives the factors of the first term?
When deriving the quadratic formula from $ax^2 + bx + c = 0$ using completing the square, the intermediate step $a(x + \frac{b}{2a})^2 = \frac{b^2 - 4ac}{4a}$ is reached. What is the next step that leads directly to the quadratic formula?
Use the quadratic formula to solve: $2x^2-3x-5=0$
Describe the end behavior of $f(x)=\frac{3x-1}{x^2+4}.$ Which statement is correct?
Convert $3\left(\cos 60^\circ+i\sin 60^\circ\right)$ to rectangular form $a+bi$ using $a=r\cos\theta$ and $b=r\sin\theta$ .

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