Practice Test 1 - Precalculus

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

A periodic function has maximum value 5 and minimum value $-3$ . For the periodic function described, what is the amplitude?
A hiker walks $6$ km due east (direction $0^\circ$ from the positive $x$ -axis), then walks $8$ km due north (direction $90^\circ$ ). Using the information provided, what are the magnitude and direction of the hiker's resultant displacement (direction as an angle from the positive $x$ -axis)?
For the system of equations $\begin{cases} 3x - 2y = 5\\ 4x + y = -6 \end{cases}$ what is the coefficient matrix $A$ in $Ax=b$ (using variable order $x, y$ )?
Given the vector $\mathbf{v}=\langle 3,4\rangle$ and scalar multiplication by $k=2$ , what are the components of $2\mathbf{v}$ ?
Two concentric circles have radii $r$ and $3r$ respectively. If the same central angle intercepts an arc of length $12$ on the smaller circle, what is the ratio of the area of the sector on the larger circle to the area of the sector on the smaller circle?
Using both periodicity and symmetry, evaluate $\sin\left(-\frac{13\pi}{6}\right)$ . (Recall $\sin(\theta+2\pi)=\sin(\theta)$ and $\sin(-\theta)=-\sin(\theta)$ .)
Given the matrix equation $\begin{bmatrix}1 & -3\\ 2 & 5\end{bmatrix}\begin{bmatrix}x\\ y\end{bmatrix}=\begin{bmatrix}4\\ -1\end{bmatrix},$ which system of equations does it represent?
A triangular garden has two sides of lengths $15\text{ m}$ and $20\text{ m}$ that meet at an included angle of $45^\circ$ . Using the given information, what is the area of the triangle?
A rotating object sweeps out an angle $\theta$ in standard position starting from the positive $x$ -axis on a circle of radius $r=3$ . If the intercepted arc length is $s=2\pi$ , what is the radian measure of $\theta$ ?
Vector $\mathbf{v}$ has magnitude $5$ and points in some direction on the coordinate plane. What is the magnitude of $-3\mathbf{v}$ ?
Using the periodicity property of cosine on the unit circle (one full rotation), what is $\cos\left(\theta+2\pi\right)$ ?
In triangle $ABC$ , sides $a=8$ and $b=15$ and the included angle $C=90^\circ$ are given. Which law should be used first to solve this triangle for side $c$ ?
In right triangle $\triangle JKL$ , $\angle L = 90^\circ$ , $JL = 8$ in, and $KL = 15$ in. Side $JK$ is the hypotenuse. Based on the triangle described, what is the exact length of side $JK$ ?
For vector $\mathbf{v}=\langle 5,0\rangle$ and scalar $k=\dfrac{1}{2}$ , what is the effect of multiplying $\mathbf{v}$ by $k$ on its length and direction?
Angles are measured in degrees counterclockwise from the positive $x$ -axis. Vector $\mathbf{u}$ has magnitude $10$ at direction $0^\circ$ , and vector $\mathbf{v}$ has magnitude $10$ at direction $90^\circ$ . Using the information provided, what is the magnitude of $\mathbf{u}+\mathbf{v}$ ? (Direction is not needed.)
An angle $\theta$ is drawn in standard position on the unit circle in the coordinate plane with measure $\theta=\frac{7\pi}{6}$ . The terminal point on the unit circle is $P(x,y)$ , where $\cos(\theta)=x$ and $\sin(\theta)=y$ . For the angle described, what is the value of $\sin(\theta)$ ?
Two vectors are given: $\vec{u}=\langle 2,3\rangle$ and $\vec{w}=\langle 2,3\rangle$ . For the vectors described, which statement about the vector is true?
For vectors $\mathbf{u}=\langle 3,0\rangle$ and $\mathbf{v}=\langle 0,4\rangle$ , what is the magnitude $\|\mathbf{u}+\mathbf{v}\|$ ? Give your answer in simplest form.
A circle has circumference $C=2\pi r$ , and one full revolution corresponds to $2\pi$ radians. Using the relationship between circumference and radian measure, how does the formula $s=r\theta$ derive from the proportionality of arc length to angle?
A boat's velocity relative to the water is $\langle 5,0\rangle$ m/s and the current's velocity is $\langle 0,2\rangle$ m/s. Using the vectors given, what are the components of the resultant velocity $\mathbf{v}_{\text{result}}$ ?

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