Practice Test 10 - Precalculus

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

Start practice test

1. In right triangle $JKL$, the right angle is at $K$. If $JK = 9$ and $KL = 12$, what is the exact length of the hypotenuse $JL$?
2. Using the cofunction identity for complementary angles, which equation correctly expresses the relationship between sine and cosine?​
3. In right triangle $ABC$, the right angle is at $C$. The hypotenuse is $AB=13$ m and one leg is $BC=5$ m. What is the exact length of the other leg $AC$?
4. An angle of $\theta=\frac{4\pi}{3}$ radians is drawn in standard position on the unit circle in the coordinate plane. For the angle described, what is the reference angle for $\theta$?​
5. For vectors $\vec v=\langle 5,3\rangle$ and $\vec w=\langle 2,1\rangle$, use component-wise subtraction (equivalently, $\vec v-\vec w=\vec v+(-\vec w)$) to find the components of $\vec v-\vec w$ in $\langle a,b\rangle$ form.
6. Given the velocities, an airplane flies with airspeed $\vec v_{\text{plane}}=300\text{ km/h}$ due east, and the wind is $\vec v_{\text{wind}}=50\text{ km/h}$ due south. What is the velocity of the airplane relative to the ground (magnitude and direction, as an angle south of east)?
7. Let $A$ and $B$ be $2\times 2$ matrices: $A=\begin{bmatrix}1 & 0\\2 & 1\end{bmatrix},\quad B=\begin{bmatrix}0 & 1\\1 & 0\end{bmatrix}.$ Given the matrices above, which statement about $AB$ and $BA$ is true?​​
8. Given the velocities, a drone's ground velocity is the vector sum $\vec v_{\text{ground}}=\vec v_{\text{drone/air}}+\vec v_{\text{wind}}$. The drone flies at $\vec v_{\text{drone/air}}=10\ \text{m/s}$ due north, and the wind is $\vec v_{\text{wind}}=10\ \text{m/s}$ due south. What is the velocity of the drone relative to the ground?
9. An abstract periodic function has a maximum value of $5$ and a minimum value of $-3$. Based on the periodic function described, what is the amplitude?
10. If the company wants to create a matrix showing the number of cities that can be reached directly from each city, which operation should be performed?
11. A research team collects data in matrix $D = \begin{bmatrix} 0.6 & -0.3 \\ 0.9 & 0.4 \end{bmatrix}$. To convert the data to percentages and then apply a confidence factor of 0.8, they perform scalar multiplication. What is the sum of the absolute values of all entries in the final matrix?
12. Use the angle addition formulas to simplify $\sin\left(x+\dfrac{\pi}{4}\right)+\sin\left(x-\dfrac{\pi}{4}\right)$. What is the simplified expression?​
13. For the transformation matrix $F=\begin{bmatrix}1&0\\0&-1\end{bmatrix}$ (reflection across the $x$-axis), what is the result of applying $F$ to the vector $\begin{bmatrix}2\\5\end{bmatrix}?$
14. When constructing the inverse sine function $\arcsin(x)$, we restrict $\sin(x)$ to the domain $[-\pi/2,\pi/2]$, where it is always increasing. Why is this restriction necessary for the inverse function to exist?
15. The number of daylight hours in a location varies periodically throughout the year. The maximum is $15$ hours on day $172$ and the minimum is $9$ hours on day $355$. For the phenomenon described, what is the midline of the daylight function (in hours)?
16. A vector has initial point $A(5, -1)$ and terminal point $B(0, 3)$. For the vector described, what is the component form of $\overrightarrow{AB}$?​
17. A circle has radius $r=5$. A central angle $\theta$ (in radians) intercepts an arc of length $s$. If the radius is doubled while the angle remains constant, how does the arc length change?​
18. Consider the equation $\cos(x) = \frac{1}{2}$ where $x$ is a special angle. If $\alpha$ and $\beta$ are the two solutions in $[0, 2\pi)$ with $\alpha < \beta$, what is the value of $\tan(\pi - \alpha) + \tan(\pi - \beta)$?
19. In right triangle $GHI$, the right angle is at $H$. If $GH = 8$ and $GI = 17$ (the hypotenuse), what is the length of $HI$?
20. A scaling transformation is defined by $S=\begin{bmatrix}2 & 0\\0 & -1\end{bmatrix}.$ Where does the point $\mathbf{v}=\begin{bmatrix}-3\\4\end{bmatrix}$ map under this transformation (i.e., what is $S\mathbf{v}$)?

Practice tests