Practice Test 59 - Algebra 2

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

Start practice test

1. A student models the area (in square inches) of a circular sticker as a function of its radius $r$ (in inches): $A(r)=\pi r^2$. In the real situation, the radius can be measured to the nearest hundredth of an inch, and the sticker radius cannot exceed 5 inches. What is an appropriate realistic domain for $A(r)$?
2. Use the identity $(a+b)^2=a^2+2ab+b^2$ to compute $101^2$ mentally by writing $101=100+1$.
3. Four functions are defined for $x\ge 0$: linear $f(x)=6x$, quadratic $g(x)=x^2$, polynomial $h(x)=x^4$, and exponential $p(x)=1.5^x$. Order them by eventual growth rate (for sufficiently large $x$).
4. To solve $x^4 - 5x^2 - 36 = 0$ efficiently, which structural approach should be applied?
5. What is the degree of the polynomial $P(x)\cdot Q(x)$? (This relies on closure under multiplication and degree behavior.) Let $P(x)= -2x^3 + x - 5$ and $Q(x)=3x^4 - 7x^2 + 1$.
6. Compute $(4 + 3i)(4 - 3i)$. (Use the key property $(a+bi)(a-bi)=a^2+b^2$.)
7. Consider the function $f(x) = 6x - 4$. Does $f(x)$ have a constant rate of change? If so, what is the rate and what type of function is it?
8. Use conjugates to divide and write in standard form $a+bi$: $\frac{1+2i}{3-i}.$
9. A polynomial function $t(x)$ is defined such that its factored form is $t(x) = -2(x + 4)(x - 1)^4(x - 3)$. When sketching this function's graph, which combination of features requires the most careful attention to avoid common graphing errors?
10. A culture of bacteria starts with 500 cells and grows by 12% each hour. 1) Write an equation for the population $P$ in terms of time $t$ (hours). 2) Choose appropriate axes labels and a reasonable scale to graph the model for $0\le t\le 10$. Which choice is correct?​
11. A savings account earns 12% annual interest. To compare with a credit card that charges 1.2% monthly interest, which form of $(1.12)^t$ reveals the equivalent monthly rate?
12. Simplify using $i^2=-1$: $i^3 + i^4.$
13. A lab solution has hydrogen ion concentration $[H^+]=3.2\times 10^{-6}$ mol/L. Using $\text{pH}=-\log_{10}([H^+])$, create an equation and find the pH. Round to the nearest tenth and interpret your answer.
14. Write the system of linear equations as a single matrix equation $AX=B$ (use variable order $x, y$): $\begin{cases} 3x+2y=8 \\ x-5y=-3 \end{cases}$
15. Solve for $x$: $\log_{10}(x)=2$.​
16. Let $z=-2+7i$ and $w=1-1i$. What is the distance between the points representing $z$ and $w$ on the complex plane?
17. Perform the operation and simplify to lowest terms (like fraction arithmetic with polynomials). $\displaystyle \frac{3}{x-1}-\frac{1}{x+1}$
18. Show that $x^2-4x+5=0$ has exactly 2 complex solutions counting multiplicity (as every quadratic $ax^2+bx+c=0$ does in $\mathbb{C}$). Use the discriminant $b^2-4ac$ to determine the type, then find the solutions. Which option is correct?
19. Convert to rectangular form $a+bi$: $4\left(\cos 120^\circ + i\sin 120^\circ\right).$ Use $a=r\cos\theta$ and $b=r\sin\theta$.
20. Based on the graph shown, which statement correctly describes the long-term behavior of these three functions representing different bacterial colony growth models?

Practice tests