Practice Test 12 - Precalculus

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

Let $A$ be a $2\times3$ matrix and let $O$ be the $3\times2$ zero matrix (all entries are $0$ ). Based on the zero matrix absorption property under multiplication, what is $A\cdot O$ ?
A circle has radius $r=6$ cm. An angle of $\theta=\frac{\pi}{3}$ radians at the center intercepts an arc. Using the relationship between circumference and radian measure, determine the arc length $s$ by reasoning that $\frac{s}{2\pi r}=\frac{\theta}{2\pi}$ . For the circle described, what is $s$ ?
A circle $O$ has center $O(0,0)$ and radius $5$ . Point $P$ is located at $(13,0)$ , and a tangent segment $PT$ is drawn from $P$ to the circle, touching the circle at point $T$ . For the circle described, what is the length of the tangent segment $PT$ ?
Let $A$ be the $2\times 3$ matrix $A=\begin{bmatrix}2 & -1 & 0\\ 3 & 4 & -2\end{bmatrix}.$ What is $3A$ ?
For vectors $\vec v=\langle -2,6\rangle$ and $\vec w=\langle 3,-1\rangle$ , rewrite $\vec v-\vec w$ as $\vec v+(-\vec w)$ and find the components of $\vec v-\vec w$ .
On a unit circle (radius $r=1$ ), an angle $\theta$ in standard position starts on the positive $x$ -axis and measures $\theta=\frac{\pi}{4}$ radians. Using the circle described, what is the length $s$ of the arc intercepted by the angle?
If $f(x) = \sin(x) + \cos(x)$ and $g(x) = \sin(x) - \cos(x)$ , which statement about the symmetry properties of these functions is correct?
A ramp rises $6$ ft vertically from the ground to a platform. The ramp makes a $30^\circ$ angle with the ground, forming a right triangle. How long is the ramp (in feet)?
A vector $\vec{v}$ is transformed by matrix $P = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$ , then by matrix $Q = \begin{pmatrix} \frac{1}{2} & \frac{\sqrt{3}}{2} \\ \frac{\sqrt{3}}{2} & -\frac{1}{2} \end{pmatrix}$ . Which single transformation is equivalent to this composition?
In right triangle $JKL$ , the right angle is at $K$ . If $JK=6$ and $KL=8$ , what is the value of $\sin(\angle J)$ ?
Consider the composition of two transformations: first a reflection across the line $y = x$ represented by matrix $R = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ , followed by a scaling transformation $S = \begin{pmatrix} 3 & 0 \\ 0 & 2 \end{pmatrix}$ . A circle with area $\pi$ square units undergoes this composition. What is the area of the final image?
A velocity vector $\mathbf{v}$ has magnitude $50\ \text{km/h}$ due east. What are the magnitude and compass direction of $-\mathbf{v}$ ?
A coordinate-plane diagram shows $\vec w$ is collinear with $\vec v$ , points in the same direction as $\vec v$ , and appears to be twice as long. If $\vec w = k\vec v$ , what is the value of $k$ ?
An angle of $\theta=-\frac{\pi}{3}$ radians is drawn in standard position on the unit circle in the coordinate plane (measured clockwise from the positive $x$ -axis). For the angle described, what is the value of $\sin(\theta)$ ?
Given the $2\times2$ zero matrix $O=\begin{bmatrix}0&0\\0&0\end{bmatrix}$ , what is $\det(O)$ ?
Compare $\sin\left(\pi/6\right)$ and $\sin\left(\pi/3\right)$ using the 30-60-90 triangle and their unit circle coordinates. Which statement is true?
Vectors $\mathbf{u}$ and $\mathbf{v}$ are drawn tail-to-tail from the same initial point. Using the parallelogram rule, which vector represents $\mathbf{u}+\mathbf{v}$ ?
The directed line segment from $M(7, -1)$ to $N(2, 4)$ represents a vector. For the vector described, what is the component form?
For the matrices $A=\begin{pmatrix}1&2\\3&4\end{pmatrix}$ and $B=\begin{pmatrix}0&1\\2&-1\end{pmatrix}$ , is $AB=BA$ in general? Why or why not?
A sector of a circle with radius $r=4$ cm is formed by a central angle of $\theta=\frac{\pi}{2}$ radians. Using the given information, what is the area of the sector?

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