Practice Test 7 - Precalculus

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

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1. When matrix $A=\begin{bmatrix}1&1\\2&2\end{bmatrix}$ is applied as a transformation of the plane, how does it affect areas?
2. A drone's displacement is recorded as vector $\vec{d}$ with $|\vec{d}| = 50$ meters. Later, the same displacement is represented using different notation as $\mathbf{d}$. A student writes $||\mathbf{d}|| = 50$ meters and claims this is incorrect notation. Evaluate the student's claim about vector magnitude notation.
3. Based on the description, $\mathbf{u}$ is $3$ units east and $\mathbf{v}$ is $4$ units north (so $\mathbf{u}=\langle 3,0\rangle$ and $\mathbf{v}=\langle 0,4\rangle$). How does $\lVert \mathbf{u}+\mathbf{v} \rVert$ compare to $\lVert \mathbf{u} \rVert+\lVert \mathbf{v} \rVert$?​
4. Using the periodicity property, $\sin(\theta+2\pi)=\sin(\theta)$, what is $\sin\left(\frac{17\pi}{6}\right)$?
5. For the system of equations (variable order $x, y, z$) $\begin{cases} 2x + y - z = 0\\ -x + 3y + 2z = 7\\ 4x - 2y + z = -1 \end{cases}$ what is the entry $a_{23}$ of the coefficient matrix $A$ in $Ax = b$?
6. Given the velocities, a person walks at $\vec v_{\text{person/train}}=1.5\ \text{m/s}$ east relative to a train, and the train moves at $\vec v_{\text{train/ground}}=10\ \text{m/s}$ east relative to the ground. How fast is the person moving relative to the ground?
7. If $kA=B,\quad A=\begin{bmatrix}1 & -2\\ 3 & 0\end{bmatrix},\quad B=\begin{bmatrix}4 & -8\\ 12 & 0\end{bmatrix},$ what is the scalar $k$?
8. Given the restriction that defines $\arcsin(x)$ (inverse of $\sin(x)$ on $[-\pi/2,\pi/2]$), evaluate $\arcsin(\sqrt{3}/2)$ in radians.
9. A circle is tangent to both coordinate axes in the first quadrant. If a tangent line from point $(8, 0)$ to this circle has slope $m = -\frac{3}{4}$, what is the radius of the circle?
10. Based on the unit circle, the point at angle $\theta$ is $(\cos(\theta),\sin(\theta))$ and the point at angle $-\theta$ reflects across the $x$-axis. If $\sin\left(\frac{\pi}{6}\right)=\frac{1}{2}$, what is $\sin\left(-\frac{\pi}{6}\right)$?
11. In triangle $ABC$, $a=9$, $b=12$, and $c=15$ are given. What is the measure of angle $C$ (in degrees)?
12. On the unit circle in the coordinate plane, point $P$ is located at coordinates $\left(-\frac{1}{2},\frac{\sqrt{3}}{2}\right)$. An angle $\theta$ in standard position has its terminal side passing through $P$. Using the given information, what is the value of $\sin(\theta)$?
13. In triangle $ABC$, sides $a=8$ and $b=10$ are known, and the included angle $C=60^\circ$ is known. What is the length of side $c$ (opposite $\angle C$)?
14. For the transformation matrix $R=\begin{pmatrix}0&-1\\1&0\end{pmatrix}$, which represents a rotation of the coordinate plane, what is the image of the point $(3,4)$ under $T(\mathbf{v})=R\mathbf{v}$? (This transformation has $\det(R)=1$, so areas are preserved.)
15. Using the cosine angle subtraction formula, find the exact value of $\cos(15^\circ)$ by writing $15^\circ = 45^\circ - 30^\circ$.
16. Circle $O$ has center $O$ and radius $4$. Point $P$ is an external point with $OP=5$. Two tangent segments from $P$ touch the circle at $T_1$ and $T_2$. For the circle described, which of the following is true about the two tangent segments $PT_1$ and $PT_2$?
17. A triangle has two fixed sides of lengths 5 and 7. For the triangle described, for what included angle $C$ (between the two given sides) is the area maximized?
18. If $\vec w=\langle -4, 3 \rangle$, what is the opposite vector $-\vec w$ (the vector with the same magnitude but opposite direction), which is used when rewriting $\vec v-\vec w$ as $\vec v+(-\vec w)$?
19. A drone moves according to three displacement vectors: $\vec{v_1} = \langle 6, 0 \rangle$ m, $\vec{v_2} = \langle 0, 8 \rangle$ m, and $\vec{v_3} = \langle -6, 0 \rangle$ m. Using the end-to-end method, what is the magnitude of the total displacement after all three moves?
20. For vectors $\mathbf{u}=\langle -6,2\rangle$ and $\mathbf{v}=\langle 1,-5\rangle$, what is $\mathbf{u}+\mathbf{v}$ (in component form)?

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