Practice Test 2 - Precalculus

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

Given the magnitudes and directions, a force $\mathbf{F}_1$ is $8\text{ N}$ at $0^\circ$ (due east) and a force $\mathbf{F}_2$ is $15\text{ N}$ at $90^\circ$ (due north), with angles measured from the positive $x$ -axis. What is the magnitude of the resultant force $\mathbf{F}_1+\mathbf{F}_2$ (direction not needed)?
A sector of a circle with radius $r=4$ is formed by a central angle of $\theta=\frac{\pi}{2}$ radians. Using the given information, what is the area of the sector? (Use $A=\frac{1}{2}r^2\theta$ .)
An angle of $5\pi/6$ is drawn in standard position on the unit circle. Using the 30-60-90 reference triangle (reference angle $\pi/6$ ) and applying quadrant signs, what is the exact value of $\cos(5\pi/6)$ ?
In triangle $ABC$ , $a=9$ , $b=12$ , and the included angle $C=120^\circ$ are given. What is the length of side $c$ ?
On a circle with radius $r=5$ , an angle $\theta$ in standard position starting from the positive $x$ -axis subtends an arc of length $s=\frac{5\pi}{2}$ . Based on the information given, what is the radian measure of $\theta$ ?
In right triangle $ABC$ , angle $C$ is the right angle. If $\sin A = \frac{5}{13}$ and $\cos A = \frac{12}{13}$ , what is the value of $\sin B + \cos B$ ?
Circle $O$ has center $O$ and radius $5$ . From an external point $P$ , a tangent line touches the circle at $T$ . For the circle described, what is the angle (in degrees) between the radius $OT$ and the tangent line $PT$ at the point of tangency?
A circle $O$ has center $O$ and radius $5$ . An external point $P$ is such that $OP=13$ . From $P$ , a tangent segment $PT$ is drawn to the circle, touching the circle at $T$ . For the circle described, what is the length of the tangent segment $PT$ ?
Let $M$ be a $2\times 2$ matrix and $I_2$ the $2\times 2$ identity matrix: $M=\begin{bmatrix}0&-3\\2&1\end{bmatrix},\quad I_2=\begin{bmatrix}1&0\\0&1\end{bmatrix}.$ Based on the matrices defined, what is $MI_2$ ?
Matrix $R = \begin{bmatrix} x & 2x \\ -x & 3x \end{bmatrix}$ represents a scaled version of matrix $S = \begin{bmatrix} 1 & 2 \\ -1 & 3 \end{bmatrix}$ . If the determinant of $R$ equals 20, what is the value of $x^2$ ?
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 $\tan(\theta)$ ?
Using the circle described: an angle $\theta$ in standard position measures $\frac{3\pi}{4}$ radians on a unit circle (radius $r=1$ ), starting from the positive $x$ -axis. What is the length $s$ of the arc intercepted by the angle?
In this scenario, three forces act on a sign: $\vec F_1=10\ \text{N}$ east, $\vec F_2=10\ \text{N}$ west, and $\vec F_3=12\ \text{N}$ north. What are the magnitude and direction of the net force $\vec F_{\text{net}}$ ?
Using the unit circle, if $\cos(t) = -\frac{3}{5}$ and $\sin(t) > 0$ , what is the value of $\sin(t + 2\pi) \cdot \cos(t - 4\pi)$ ?
Given $\mathbf{v}=\langle 3,4\rangle$ and scalar $c=-1$ , which statement correctly describes $c\mathbf{v}$ ?
Compare $\sin(20^\circ)$ and $\cos(70^\circ)$ . (Note that $20^\circ$ and $70^\circ$ are complementary.) Which statement is true?
The voltage in an AC circuit is modeled by $V(t)=10\sin(120\pi t)$ volts, where $t$ is in seconds. Using the model, which equation correctly represents finding the times in $0\le t\le 0.05$ when the voltage is $5$ volts?
If $\|\mathbf{v}\|=8$ , what is $\| -3\mathbf{v} \|$ ?
For the system of equations $\begin{cases} -x+2y+4z=3\\ 5x-y+z=-7\\ 2x+3y-2z=1 \end{cases}$ what is the entry $a_{23}$ of the coefficient matrix $A$ (row 2, column 3) when the variable order is $x,y,z$ ?
A student incorrectly concludes that since $\sin^2(\theta) + \cos^2(\theta) = 1$ , it follows that $\sin(\theta) + \cos(\theta) = 1$ for all $\theta$ . Which counterexample best demonstrates the flaw in this reasoning?

Practice tests