Practice Test 5 - Precalculus

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

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1. For the system of equations (variable order $[x\;y\;z]^T$) $\begin{cases} 2x+y-z=4\\ 3x-2y+5z=0\\ -x+4y+z=-3 \end{cases}$ what is entry $a_{23}$ of the coefficient matrix $A$?
2. Circle $O$ has radius $3$, and an external point $P$ satisfies $OP=5$. Tangent segments from $P$ touch the circle at points of tangency. Using the given circle and point, how many tangent lines can be drawn from point $P$ to circle $O$?​​​
3. In triangle $ABC$, $a=13$, $b=14$, and the included angle is $\angle C=60^\circ$. What is the value of side $c$?
4. If vector $\vec{v} = (a, b)$ where $a > 0$ and $b < 0$, and scalar multiplication $c\vec{v}$ results in a vector pointing into the second quadrant, which of the following must be true about scalar $c$?
5. Vector $\mathbf{v}$ has magnitude $10$. How does $\| -\tfrac{1}{2}\mathbf{v} \|$ compare to $\|\mathbf{v}\|$?​
6. Use the cosine angle subtraction formula to derive a special case: if you set $A=B=\theta$ in $\cos(A-B)$, what expression do you get for $\cos(0)$ in terms of $\sin(\theta)$ and $\cos(\theta)$?​
7. A circle $O$ has center $O$ and radius $6$. An external point $P$ satisfies $OP=10$. A tangent from $P$ touches the circle at $T$. For the circle described, what is the length of the tangent segment $PT$?
8. If $A=\begin{bmatrix}2 & -1\\0 & 3\end{bmatrix}$ and $B=\begin{bmatrix}-6 & 3\\0 & -9\end{bmatrix}$ satisfy $kA=B$ for some real scalar $k$, what is $k$?​
9. Using the 30-60-90 triangle on the unit circle (hypotenuse $1$), what is the exact value of $\cos\left(\pi/3\right)$?
10. In right triangle $\triangle ABC$, $\angle C = 90^\circ$ and $AB$ is the hypotenuse. If $BC = 12$ and $AB = 13$, what is the value of $\cos(\angle B)$?
11. A point $P$ on the unit circle corresponds to an angle $\theta$ in standard position and has coordinates $P=(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})$. Based on the special right triangle connection to the unit circle, what is the exact value of $\theta$?
12. An angle $\theta=\frac{11\pi}{6}$ is drawn in standard position on the unit circle. For the angle described, what is the value of $\cos(\theta)$?
13. For vectors $\mathbf{u}=\langle 2,0\rangle$ and $\mathbf{v}=\langle 0,2\rangle$, how does $\lVert \mathbf{u}+\mathbf{v} \rVert$ compare to $\lVert \mathbf{u} \rVert + \lVert \mathbf{v} \rVert$?
14. Given that $\sin(\theta) = \frac{3}{5}$ and $\theta$ is in Quadrant II, which expression correctly represents $\cos(\theta) + \tan(\theta)$?
15. Use the sine addition formula with $A=B=\theta$ to derive a double-angle identity. What is $\sin(2\theta)$?
16. In a right triangle, the acute angles $\alpha$ and $\beta$ are complementary ($\alpha+\beta=90^\circ$). Which statement must be true?​
17. Using the Pythagorean identity $\sin^2(\theta)+\cos^2(\theta)=1$, if $\cos(\theta)=\frac{5}{13}$ and $\theta$ is in Quadrant III, what is $\sin(\theta)$?​​
18. A surveyor measures triangle sides $a = 80\text{ m}$, $b = 65\text{ m}$, and $c = 50\text{ m}$ (opposite angles $A$, $B$, and $C$ respectively). Using the measurements, which law should be applied first to find an angle of the triangle?
19. Vector $\mathbf{v}$ has magnitude $12$ and direction $45^\circ$ from the positive $x$-axis. What are the magnitude and direction of $\frac{1}{2}\mathbf{v}$?
20. Let $M=\begin{bmatrix}2 & 4\\ 1 & 2\end{bmatrix}$. Does matrix $M$ have an inverse? Why or why not? (Use the determinant to decide.)

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