Practice Test 11 - Precalculus

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

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1. A triangle with vertices at $A(2, 1)$, $B(4, 1)$, and $C(2, 3)$ is transformed by the matrix $\begin{pmatrix} 1 & -1 \\ 2 & 0 \end{pmatrix}$. What is the ratio of the area of the transformed triangle to the area of the original triangle?
2. For the matrices $A=\begin{pmatrix}2&0\\1&3\end{pmatrix},\quad B=\begin{pmatrix}1&4\\0&-1\end{pmatrix},$ what is the result of the calculation $AB$?
3. In right triangle $ABC$, the right angle is at $C$. The legs are $AC=12$ and $BC=5$, and the hypotenuse is $AB=13$. What is the value of $\cos(\angle A)$?
4. Let $R = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ and $S = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$. Compute $RS$ and $SR$, then determine which property explains why $(RS)^2 \neq R^2S^2$.
5. A farm ships apples and oranges to two markets. Rows represent products (row 1 = Apples, row 2 = Oranges) and columns represent markets (column 1 = Market A, column 2 = Market B). The shipment matrix $S$ (in crates) is $S=\begin{bmatrix}15 & 20\\ 10 & 25\end{bmatrix}$ If each crate sells for $2 at both markets, which calculation gives the revenue matrix (in dollars) for each product and market?
6. On two circles, one with radius $5$ and one with radius $10$, the same central angle of $\theta=\frac{\pi}{3}$ radians intercepts arcs along each circle. Based on the proportional relationship $s=r\theta$, if the radius is doubled while the angle remains constant, how does the arc length change?
7. In right triangle $\triangle ABC$, $\angle C = 90^\circ$ and $AB$ is the hypotenuse. If $AB = 13$ and $AC = 5$, what is the exact length of leg $BC$?
8. For matrices $A,B,C$ of compatible dimensions, what is the expanded form of $A(B+C)$ (property being examined: distributivity of multiplication over addition)?
9. In triangle $DEF$, $DE = 8$, $EF = 6$, and $DF = 10$. Point $G$ is the foot of the perpendicular from $E$ to side $DF$. Using both the standard area formula and the trigonometric area formula, what is $EG$?
10. For vectors $\mathbf{u}=\langle 3,0\rangle$ and $\mathbf{v}=\langle 0,4\rangle$, what is the magnitude $\lVert \mathbf{u}+\mathbf{v} \rVert$? Give your answer in simplest form.
11. Vector $\vec{m}$ has initial point $(3, -2)$ and terminal point $(8, -5)$. Vector $\vec{n}$ has initial point $(-1, 4)$ and terminal point $(2, 6)$. What is $\vec{m} - \vec{n}$ expressed in component form?
12. The temperature in a greenhouse is modeled by $T(t)=10\cos\left(\frac{\pi t}{12}\right)+20$, where $t$ is hours after midnight and $0\le t<24$. Using the model, when is the temperature $15^\circ\text{C}$ during the day? Give all solutions in the interval $[0,24)$.
13. Vectors $\vec v$ and $\vec w$ are drawn tail-to-tail from the origin on a coordinate plane, with $\vec v=\langle 4,1\rangle$ and $\vec w=\langle 1,3\rangle$. Using the tip-to-tip method, what are the components of $\vec v-\vec w$?
14. Given the velocities, an airplane has 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 magnitude of the plane's ground velocity $|\vec v_{\text{ground}}|$?
15. Given $M=\begin{bmatrix}4 & 0 & -2\\1 & -3 & 5\end{bmatrix}$ (a $2\times3$ matrix), what is $\tfrac{1}{2}M$?​
16. Which of the following correctly demonstrates why $\sin^4(x) + \cos^4(x) + 2\sin^2(x)\cos^2(x) = 1$ using the Pythagorean identity?
17. In this scenario, a drone's velocity relative to the air is $\vec v_{\text{drone}}=\langle 12,\,5\rangle\text{ m/s}$ (where $+x$ is east and $+y$ is north). The wind velocity is $\vec v_{\text{wind}}=\langle -3,\,0\rangle\text{ m/s}$. What is the drone's velocity relative to the ground, $\vec v_{\text{ground}}$?
18. A point moves counterclockwise around the unit circle in the coordinate plane starting from $(1,0)$, sweeping through an angle of $\theta=\frac{3\pi}{4}$ radians in standard position. Using the unit circle relationship $\cos(\theta)=x$ and $\sin(\theta)=y$, what is the value of $\cos(\theta)$?​
19. Using the angle subtraction formula for sine, find the exact value of $\sin(15^\circ)$ by writing it as $\sin(45^\circ-30^\circ)$.
20. For the system of equations (variable order $[x\;y\;z]^T$) $\begin{cases} x-3y+2z=0\\ 4x+z=5\\ -2y+3z=-6 \end{cases}$ what is the coefficient matrix $A$ in $Ax=b$?

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