Practice Test 4 - Precalculus

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

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1. For the matrices $A=\begin{pmatrix}1&2\\0&1\end{pmatrix} \text{ and } B=\begin{pmatrix}3&0\\4&3\end{pmatrix},$ is $AB=BA$ in general? Why or why not?
2. In this scenario, a swimmer can swim at $\vec v_{\text{swim}}=4\ \text{m/s}$ due north relative to the water, while the river current is $\vec v_{\text{current}}=3\ \text{m/s}$ due east. What is the swimmer's actual velocity relative to the ground (magnitude and direction)?
3. Given $\sin(\theta)=\frac{8}{17}$ and $\theta$ is in Quadrant I, using the Pythagorean identity $\sin^2(\theta)+\cos^2(\theta)=1$, what is $\cos(\theta)$?
4. Given the symmetry of cosine on the unit circle (cosine is an even function), what is $\cos\left(-\frac{\pi}{3}\right)$?
5. A surveyor stands $24$ m from the base of a vertical tower. The angle of elevation from the surveyor to the top of the tower is $45^\circ$. Using the given information, what is the height of the tower?
6. A tournament bracket can be represented using an incidence matrix $M$ where rows represent teams and columns represent matches. Entry $M_{ij} = 1$ if team $i$ plays in match $j$, and $M_{ij} = 0$ otherwise. If we want to find which teams play against each other in any match, we should examine:
7. Vector $\mathbf{v}$ has magnitude $12$ and direction $45^\circ$ from the positive $x$-axis. What are the magnitude and direction of $\frac{1}{2}\mathbf{v}$?​
8. A shear transformation is defined by $H=\begin{pmatrix}1 & 2\\0 & 1\end{pmatrix}.$ Given $\mathbf{v}=\begin{pmatrix}1\\3\end{pmatrix},$ what is the result of the matrix-vector multiplication $H\mathbf{v}$?
9. On the unit circle in the coordinate plane, an angle of $\theta=\frac{3\pi}{2}$ radians is drawn in standard position. The terminal point is $P(x,y)$ where $\cos(\theta)=x$ and $\sin(\theta)=y$, and $\tan(\theta)=\frac{y}{x}$ when defined. For the angle described, what is the value of $\tan(\theta)$?​
10. For the situation described, a rescue boat's velocity relative to the water is $\vec v_{\text{boat}}=15\text{ km/h}$ due west, and the ocean current is $\vec v_{\text{current}}=8\text{ km/h}$ due north. In what direction does the boat actually move relative to the ground (as an angle north of west)?
11. An angle $\theta$ in standard position terminates in Quadrant III on the unit circle and has reference angle $\frac{\pi}{4}$. Based on the unit circle, what is the value of $\tan(\theta)$?
12. Using the 30-60-90 triangle scaled to the unit circle (hypotenuse 1), what is the exact value of $\tan(\pi/6)$?
13. A small café tracks how many sandwiches and salads it has in stock at two locations. Rows represent items (row 1 = Sandwiches, row 2 = Salads) and columns represent locations (column 1 = Downtown, column 2 = Uptown). Inventory matrix $A$ (in units) is $A=\begin{bmatrix}20 & 30\\ 15 & 25\end{bmatrix}$ Units sold during the day are in matrix $B$ (in units): $B=\begin{bmatrix}5 & 10\\ 10 & 15\end{bmatrix}$ Based on the matrices, what does the matrix $A-B$ represent in this context?
14. In triangle JKL, $\angle J = 30°$, $k = 8$, and $j = 5$. A student uses the Law of Sines to find $\angle K$ and gets $\sin K = \frac{8 \sin(30°)}{5} = 0.8$. What should the student consider next?
15. Vector $\vec{a} = \begin{pmatrix} 3 \\ 1 \end{pmatrix}$ is transformed by matrix $L$, and the result is then transformed by matrix $L^{-1}$. If the final result is $\begin{pmatrix} 5 \\ -2 \end{pmatrix}$, what can be concluded about the transformations?
16. A triangle has two fixed sides of lengths $5$ and $7$. The angle $C$ between these two sides can vary. For this triangle described, for what angle $C$ is the area maximized?​​​
17. Circle $O$ has center $O$ and radius $8$. Point $P$ is outside the circle with $OP=17$. Tangent segments $PT$ and $PU$ are drawn from $P$ to the circle, touching at $T$ and $U$. Which of the following is true about the two tangent segments from $P$?
18. Given $A=\begin{bmatrix}1 & 0\\-2 & 3\end{bmatrix}$ and $B=\begin{bmatrix}2 & -1\\4 & 0\end{bmatrix}$, calculate $2A-3B$.
19. Let $A=\begin{bmatrix}2 & -1\\0 & 3\end{bmatrix}$ (a $2\times2$ matrix). What is $-2A$?
20. A hiker walks $5$ units east and then $3$ units north. Treat these as vectors added head-to-tail. What are the components of the resultant displacement vector in the form $\langle x,y\rangle$?

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