Practice Test 78 - Algebra 2

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

The system of equations $\begin{cases} 2x + 3y - z = 7 \\ x - 2y + 4z = -1 \\ 3x + y + 2z = 8 \end{cases}$ can be written as $A\vec{x} = \vec{b}$ . If matrix $A$ has an inverse, and $A^{-1}$ has the entry $\frac{2}{15}$ in position $(1,3)$ , what is the contribution of the third equation to the value of $x$ ?
Is the rate of change constant for the function $g(x)=x^2+1$ ? Why or why not (no calculus needed)?
A signal is represented by the complex number $6+8i$ (rectangular form). Convert it to polar form $r(\cos\theta+i\sin\theta)$ , where $r=\sqrt{a^2+b^2}$ and $\theta=\arctan\left(\frac{b}{a}\right)$ (in degrees, with quadrant adjustment).
What function undoes $f(x)=\dfrac{x-4}{2}$ ? (That is, find $f^{-1}(x)$ .)
A student is planning a weekly study schedule. Let $x$ = hours of math study and $y$ = hours of science study. Constraints: - Total study time is at most 12 hours: $x+y\le 12$ . - Math must be at least 2 hours: $x\ge 2$ . - Science must be at least 3 hours: $y\ge 3$ . - Non-negativity: $x\ge 0$ , $y\ge 0$ . Determine which of the following schedules are viable: $(2,10)$ , $(1,8)$ , $(6,7)$ .
Use the conjugate method to simplify and write in $a+bi$ form: $\frac{2-i}{3+2i}.$
On the complex plane, the points $z_1=-1+3i$ and $z_2=5-1i$ correspond to coordinates $(-1,3)$ and $(5,-1)$ . Using $|z_1-z_2|=\sqrt{(a-c)^2+(b-d)^2},$ what is the distance between $z_1$ and $z_2$ ?
A parametric equation system includes the expression $4\cos^2(\theta) + 12\cos(\theta)\sin^2(\theta) - 8\cos^3(\theta)$ . To analyze the $\theta$ dependence, which structural interpretation is most revealing?
The matrix equation $\begin{bmatrix}2&-1\\3&4\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}5\\7\end{bmatrix}$ represents what system of linear equations?
Using the properties $\log_b(xy)=\log_b(x)+\log_b(y),\quad \log_b\left(\frac{x}{y}\right)=\log_b(x)-\log_b(y),\quad \log_b(x^p)=p\log_b(x),$ write the expression as a single logarithm (condense): $2\log_5(a)-\tfrac{1}{2}\log_5(b)+\log_5(c).$
Verify with a complex example that the identity $a^2-b^2=(a+b)(a-b)$ works in $\mathbb{C}$ . Let $a=1+i$ and $b=2-i$ . Which option correctly computes both sides and shows they match?
A quadratic function $f(x) = x^2 - 6x + k$ has zeros at $x = 2$ and $x = 4$ . What is the factored form of this expression?
Subtract and simplify to lowest terms. Subtract: $\dfrac{3}{x^2-4}-\dfrac{1}{x-2}$ (assume denominators are nonzero).
A lab culture starts with 500 cells and doubles every 4 hours. Write an exponential model for the number of cells after $t$ hours and determine when the culture will reach 20,000 cells. Round to the nearest tenth of an hour.
Use the Binomial Theorem to compute $103^3$ by rewriting it as $(100+3)^3$ .
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 the expression as a single logarithm: $3\log_5(a)-\tfrac12\log_5(b)+\log_5(c).$
Verify that the identity $a^3+b^3=(a+b)(a^2-ab+b^2)$ is true for all real $a$ and $b$ by expanding the right-hand side.
A company's weekly profit (in thousands of dollars) from producing $x$ hundred items is modeled by $P(x)=-2(x-3)^2+18.$ Identify the absolute maximum of $P(x)$ and interpret it in context.
Function $f$ is given by the equation $f(x)=2^x$ . Function $g$ is represented numerically by the table: $\begin{array}{c|ccccc} x & 0 & 1 & 2 & 3 & 4\\ \hline g(x) & 3 & 6 & 12 & 24 & 48 \end{array}$ Which function has the larger y-intercept?
Identify the zeros and vertical asymptotes of $f(x)=\frac{x^2+x-6}{x^2-16}.$

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