Practice Test 6 - Precalculus

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

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1. An angle of $\theta=\frac{9\pi}{4}$ radians is drawn in standard position on the unit circle in the coordinate plane. (This is one full rotation of $2\pi$ plus an additional $\frac{\pi}{4}$.) For the angle described, which coordinates represent the terminal point of this angle on the unit circle?
2. Points $A$, $B$, and $C$ form a triangle where $A(-2, 5)$, $B(3, -1)$, and $C(1, 7)$. If $\vec{AB} + \vec{BC} = \vec{AD}$, what are the coordinates of point $D$?
3. A store sells 2 products (row 1 = Product 1, row 2 = Product 2) in 3 weeks (column 1 = Week 1, column 2 = Week 2, column 3 = Week 3). Quantities sold matrix $Q$ (in units) is $Q=\begin{bmatrix}10 & 15 & 20\\ 5 & 10 & 15\end{bmatrix}$ The price per unit for each week is given by the column vector $p$ (in dollars per unit): $p=\begin{bmatrix}2\\ 3\\ 4\end{bmatrix}$ Based on the matrices, what does the matrix product $Qp$ represent in this context?
4. If $M$ and $N$ are $2 \times 2$ matrices with $M + N = \begin{pmatrix} 4 & 6 \\ 2 & 8 \end{pmatrix}$ and $MN \neq NM$, which statement about the distributive property is most accurate?
5. A vector has initial point $A(5, -1)$ and terminal point $B(0, 3)$. For the vector described, what is the component form of $\overrightarrow{AB}$?
6. In right triangle $ABC$, the right angle is at $C$. The legs are $AC=9$ cm and $BC=12$ cm, and the hypotenuse is $AB=c$. What is the exact length of side $c$?
7. Consider the system: $3x + 2y = 5$ $-x + 4y = 3$ If this system is represented as $A\vec{x} = \vec{b}$ where $\vec{x} = \begin{pmatrix} x \\ y \end{pmatrix}$, which of the following correctly represents the matrix equation?
8. In the right triangle, the acute angles are complementary. If one acute angle is $25^\circ$, which statement correctly uses the complementary angle relationship to rewrite $\sin(25^\circ)$?
9. For vectors $\vec v=\langle 5,3\rangle$ and $\vec w=\langle 2,1\rangle$, what are the components of $\vec v-\vec w$? (Recall: vector subtraction means adding the opposite, $\vec v-\vec w=\vec v+(-\vec w)$.)
10. A vector $\vec{w}$ is represented by a directed line segment that is 6 units long and makes a $30°$ angle with the positive $x$-axis. If the same vector is translated so its initial point moves from $(2, 1)$ to $(-1, 4)$, what properties of the vector representation remain unchanged?
11. On the unit circle (radius $r=1$), an angle $\theta$ in standard position intercepts an arc that is exactly one quarter of the circle. Based on the information given, what is the radian measure of $\theta$?
12. Triangle $JKL$ has sides $JK=11$ and $JL=13$, and the included angle between them is $\angle KJL=45^\circ$. For the triangle described, what is the area of the triangle?
13. On the unit circle, an angle of $4\pi/3$ is drawn in standard position. Using the reference angle $\pi/3$ from a 30-60-90 triangle and applying quadrant signs, what is the exact value of $\sin(4\pi/3)$?
14. On a unit circle (radius $r=1$), an angle $\theta$ in standard position starts at the positive $x$-axis and measures $\frac{5\pi}{6}$ radians. According to the circle described, what is the length $s$ of the arc intercepted by the angle?
15. A wind velocity vector is written as $\vec{v}=\langle 5,12\rangle$ (in mph). What is the correct notation for the magnitude of vector $\vec{v}$?
16. Point $A$ is located at $( -1, 4)$ and point $B$ is located at $(6, 1)$. What is the component form of vector $\overrightarrow{AB}$?​
17. A regular hexagon is inscribed in the unit circle with one vertex at $(1, 0)$. If a vertex of this hexagon corresponds to the angle $\frac{\pi}{3}$, what is the value of $\sin(\pi + \frac{\pi}{3}) - \cos(2\pi - \frac{\pi}{3})$?
18. Circle $O$ has center $O$ and radius $4$. Point $P$ is outside the circle, and a tangent segment $PT$ touches the circle at $T$. To construct a tangent from $P$ to circle $O$, what must be true at the point of tangency $T$?​​
19. Let $A=\begin{bmatrix}3 & 5\\ 2 & 4\end{bmatrix}$. Based on the determinant, does matrix $A$ have an inverse? Why or why not?
20. The function $h(x) = |\sin(x)|$ has which of the following properties?

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