Practice Test 9 - Precalculus

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

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1. For vectors $\mathbf{u}=\langle 3,4\rangle$ and $\mathbf{v}=\langle -1,2\rangle$, using component-wise addition, what is $\mathbf{u}+\mathbf{v}$ (in component form)?​​
2. An object moves from position $(6, 1)$ to position $(2, 7)$ in the coordinate plane. Which vector correctly represents the displacement from the initial point to the terminal point?
3. For $A=\begin{bmatrix}-3 & 2 & 0\\1 & -4 & 5\end{bmatrix}$, which matrix represents $0\cdot A$?
4. Regarding matrix multiplication, the property being examined is associativity. Does $(AB)C=A(BC)$ for all matrices $A,B,C$ (with compatible dimensions)?
5. In right triangle $\triangle ABC$, $\angle C = 90^\circ$. If $AC = 12$ and $BC = 5$, what is the exact value of $\tan(\angle A)$?
6. For vector $\vec v = \langle 3,4\rangle$ drawn as an arrow from the origin to $(3,4)$ on a coordinate plane, what are the components of $2\vec v$?
7. A physics student incorrectly writes the velocity of a moving object as $v = 25$ m/s northeast. What is the primary error in this vector representation, and how should it be corrected?
8. Given the system of equations $\begin{cases} 2x - y + z = 1 \\ - x + 3y - 2z = -4 \\ 4x + 0y + 5z = 6 \end{cases}$ write the system as a matrix equation $Ax=b$ (using variable order $(x,y,z)$).
9. In triangle $ABC$, side $a=9$, side $b=12$, and side $c=15$. What is the measure of $\angle C$ (in degrees)?
10. If $\vec w=\langle -3,2\rangle$, what is the opposite vector $-\vec w$ used in rewriting $\vec v-\vec w$ as $\vec v+(-\vec w)$?
11. In right triangle $MNO$, the right angle is at $N$. The hypotenuse is $MO = 13$, and $MN = 5$. What is the length of $NO$?
12. Three position vectors are drawn from the origin in a coordinate plane: $\vec{r_1}$ to point $(3, 4)$, $\vec{r_2}$ to point $(-3, -4)$, and $\vec{r_3}$ to point $(4, -3)$. A student claims that $||\vec{r_1}|| = ||\vec{r_2}|| = ||\vec{r_3}||$ and concludes all three vectors are equal. What is the error in this reasoning?
13. Given the transformation $T(\mathbf{v})=A\mathbf{v}$ with $A=\begin{bmatrix}3 & 0\\1 & -2\end{bmatrix}$ and a general column vector $\mathbf{v}=\begin{bmatrix}x\\y\end{bmatrix}$, what is the general form of $A\mathbf{v}$?
14. For the inverse function, evaluate $\sin(\arcsin(1/2))$.
15. Using periodic behavior, the smallest positive number $p$ such that $\tan(\theta+p)=\tan(\theta)$ for all $\theta$ is called the period of tangent. What is the period of $\tan(\theta)$?
16. Given the magnitudes and directions, vector $\mathbf{u}$ has magnitude $13$ at $0^\circ$ and vector $\mathbf{v}$ has magnitude $13$ at $90^\circ$ (angles from the positive $x$-axis). What are the magnitude and direction of $\mathbf{u}+\mathbf{v}$?
17. In triangle $PQR$, $PQ = 15$, $QR = 20$, and $\angle PQR = 120°$. A point $S$ is chosen on side $PR$ such that $QS$ bisects $\angle PQR$. What is the ratio of the area of triangle $PQS$ to the area of triangle $QRS$?
18. Point $P$ on the unit circle corresponds to an angle of $7\pi/6$ in standard position. Using the 30-60-90 reference triangle and quadrant signs, what are the coordinates of $P$?
19. Let $A$ and $B$ be $2\times 2$ matrices $A=\begin{bmatrix}1 & 0\\ -2 & 3\end{bmatrix},\quad B=\begin{bmatrix}4 & -1\\ 2 & 0\end{bmatrix}.$ Calculate $2A+B$.
20. A circle $O$ has center $O$ and radius $10$. From an external point $P$, a tangent segment $PT$ is drawn to the circle, touching at $T$. Suppose the tangent length is $PT=24$. For the circle described, what is the distance $OP$ from the center to the external point?

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