AP Precalculus Unit 4: Functions Involving Parameters, Vectors and Matrices

Not assessed on the AP exam, but part of the course framework. Parametric functions and planar motion; parametrizing lines, circles and other implicitly defined curves; conic sections; vectors and vector-valued functions; matrices, determinants and inverses; and matrices as linear transformations and as models of change between states.

CED topics in this unit: 4.1, 4.2, 4.3, 4.4, 4.5, 4.6, 4.7, 4.8, 4.9, 4.10, 4.11, 4.12, 4.13, 4.14. (Not examined.)

Key formulas and facts

• A parametric function gives a point $(x(t),y(t))$ for each $t$; the direction of travel is the direction of increasing $t$.
• Average rate of change of $y$ with respect to $x$ between $t=a$ and $t=b$: $\frac{y(b)-y(a)}{x(b)-x(a)}$.
• Circle of radius $r$ centered at $(h,k)$: $x=h+r\cos t$, $y=k+r\sin t$, $0\le t\le2\pi$. Segment from $P$ to $Q$: $P+t(Q-P)$, $0\le t\le1$.
• Conics: parabola $y=a(x-h)^2+k$; ellipse $\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1$ (parametrized by $x=h+a\cos t$, $y=k+b\sin t$); hyperbola $\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1$.
• Vectors: $\|\langle a,b\rangle\|=\sqrt{a^2+b^2}$; $\vec u\cdot\vec v=u_1v_1+u_2v_2=\|\vec u\|\|\vec v\|\cos\theta$; a unit vector in the direction of $\vec v$ is $\frac{\vec v}{\|\vec v\|}$.
• $2\times2$ matrix $A=\begin{bmatrix}a&b\\c&d\end{bmatrix}$: $\det A=ad-bc$; if $\det A\ne0$, $A^{-1}=\frac1{ad-bc}\begin{bmatrix}d&-b\\-c&a\end{bmatrix}$.
• A linear transformation $L(\vec v)=A\vec v$ scales areas by $|\det A|$; composing transformations multiplies their matrices; the inverse transformation uses $A^{-1}$.
• Transition matrix model: if $\vec x_n$ holds the amounts in each state, then $\vec x_{n+1}=A\vec x_n$ and $\vec x_n=A^n\vec x_0$.

Worked examples

• Parametric motion: $x(t)=2t+1$, $y(t)=t^2$; from $t=1$ to $t=3$ the point moves from $(3,1)$ to $(7,9)$, so the average rate of change of $y$ with respect to $x$ is $\frac{8}{4}=2$.
• Ellipse: $\frac{x^2}{9}+\frac{y^2}{4}=1$ is traced by $x=3\cos t$, $y=2\sin t$.
• Dot product: $\langle3,4\rangle\cdot\langle-4,3\rangle=0$, so the vectors are perpendicular; $\|\langle3,4\rangle\|=5$.
• Inverse: $\begin{bmatrix}2&1\\5&3\end{bmatrix}^{-1}=\begin{bmatrix}3&-1\\-5&2\end{bmatrix}$, since the determinant is $1$.
• Area scaling: $\begin{bmatrix}3&1\\0&2\end{bmatrix}$ maps the unit square to a parallelogram of area $6$.
• Rotation by $\frac\pi2$ counterclockwise has matrix $\begin{bmatrix}0&-1\\1&0\end{bmatrix}$; it sends $\langle1,0\rangle$ to $\langle0,1\rangle$.
• Transition: with $A=\begin{bmatrix}0.9&0.2\\0.1&0.8\end{bmatrix}$ and $\vec x_0=\langle100,0\rangle$, $\vec x_1=\langle90,10\rangle$ and $\vec x_2=\langle83,17\rangle$.

Common traps

• Treating $t$ as a coordinate: a parametric graph in the $xy$-plane does not show $t$ directly.
• Multiplying matrices entry by entry, or assuming $AB=BA$.
• Writing the inverse of a $2\times2$ matrix without dividing by the determinant, or forgetting that a matrix with determinant $0$ has no inverse.
• Mixing up the order of a composition: applying $A$ and then $B$ is the matrix $BA$.
• Forgetting that this unit is not on the AP exam: study Units 1 to 3 first.

Practice questions

113 multiple-choice and 4 free-response questions, each tagged with its CED topics and with "No calculator" or "Graphing calculator". Try each one before opening the answer. Each line of a free-response scoring guide is one point, and every free-response question here is out of 6. Because this unit is not on the exam, the free-response questions are not tied to an exam question type. A free account opens the worked answers and scoring guides of 20 questions a day; Pro opens every one.

Question 1 · Multiple choice

Find the point on the curve $x(t)=t^2+2$, $y(t)=3t-1$ where $t=2$.

1. $(6, 7)$
2. $(6, 5)$
3. $(5, 6)$
4. $(2, 5)$

Question 2 · Multiple choice

The curve $x(t)=3t$, $y(t)=t^2$ for $-2\le t\le2$ passes through the point $(-3,1)$. For what value of $t$ is the curve at that point?

1. $3$
2. $-3$
3. $-1$
4. $1$

Question 3 · Multiple choice

A point moves along the curve $x(t)=\cos t$, $y(t)=\sin t$ for $0\le t\le\pi$. Which describes its path?

1. It moves clockwise along the upper half of the unit circle, from $(-1,0)$ to $(1,0)$.
2. It moves counterclockwise along the upper half of the unit circle, from $(1,0)$ to $(-1,0)$.
3. It moves counterclockwise once around the whole unit circle, starting and ending at $(1,0)$.
4. It moves counterclockwise along the lower half of the unit circle, from $(1,0)$ to $(-1,0)$.

Question 4 · Multiple choice

A curve is given by $x(t)=t+1$ and $y(t)=t^2$. Which equation in $x$ and $y$ describes the same curve?

1. $y=(x+1)^2$
2. $y=x^2-1$
3. $y=(x-1)^2$
4. $y=x^2+1$

Question 5 · Multiple choice

For $0\le t\le5$, a curve has $x(t)=2t-4$. What is the set of $x$-values the curve covers?

1. $[0,5]$
2. $[-4,5]$
3. $[-4,6]$
4. $[-2,6]$

Question 6 · Multiple choice

A curve is given by $x(t)=t^2-4t$ and $y(t)=t+1$ for $0\le t\le4$. What is the least $x$-value on the curve?

1. $2$
2. $-4$
3. $0$
4. $-3$

Question 7 · Multiple choice

The curve $x(t)=t^2-9$, $y(t)=t^2-2t$ crosses the $y$-axis at one value of $t>0$. At which point does it cross?

1. $(0, 15)$
2. $(3, 0)$
3. $(0, 3)$
4. $(0, -3)$

Question 8 · Multiple choice

A curve is given by $x(t)=e^{0.5t}$ and $y(t)=\ln(t+1)$ for $t\ge0$. What is the $y$-coordinate of the point where $x=5$?

1. $1.169$
2. $1.792$
3. $0.959$
4. $1.440$

Question 9 · Multiple choice

A ball's position is $x(t)=4t$ and $y(t)=-5t^2+20t$ meters at time $t$ seconds, $0\le t\le4$. What is the greatest height $y$ the ball reaches?

1. $40$
2. $8$
3. $20$
4. $15$

Question 10 · Multiple choice

A ball is thrown so that $x(t)=12t$ and $y(t)=-16t^2+32t+5$, in feet, $t$ seconds after it leaves the hand. How far has it traveled horizontally when it lands ($y=0$)?

1. $24.000$
2. $25.748$
3. $1.748$
4. $2.146$

Question 11 · Multiple choice

A particle moves with $x(t)=3+2t$ and $y(t)=1-t$. How far is it from its starting point ($t=0$) at $t=4$?

1. $2\sqrt{5}$
2. $12$
3. $8$
4. $4\sqrt{5}$

Question 12 · Multiple choice

A particle moves in the plane with $x(t)=t^2-6t$ and $y(t)=2t$ for $0<t<5$. On which interval is it moving to the left?

1. $(0,5)$
2. $(0,2)$
3. $(3,5)$
4. $(0,3)$

Question 13 · Multiple choice

Drone A is at $(2t,\,3t)$ and drone B is at $(8-2t,\,t+4)$ at time $t$ minutes. At what time are the drones at the same point?

1. $1$
2. $2$
3. $4$
4. $3$

Question 14 · Multiple choice

A rider on a carousel is at $x(t)=5\cos\left(\frac{\pi t}{4}\right)$, $y(t)=5\sin\left(\frac{\pi t}{4}\right)$ meters after $t$ seconds. How many seconds does one full turn take?

1. $2\pi$
2. $16$
3. $4$
4. $8$

Question 15 · Multiple choice

The table gives a particle's position at five times. Between which two consecutive times does the particle move both to the left and downward?

1. From $t=1$ to $t=2$
2. From $t=2$ to $t=3$
3. From $t=0$ to $t=1$
4. From $t=3$ to $t=4$

Question 16 · Multiple choice

A ball's position is $x(t)=9t$ and $y(t)=1.5+14t-4.9t^2$ meters after $t$ seconds. What is the horizontal position $x$ when the ball is at its highest point?

1. $25.714$
2. $12.857$
3. $6.429$
4. $1.429$

Question 17 · Multiple choice

For $x(t)=t^2$ and $y(t)=3t-1$, what is the average rate of change of $x$ with respect to $t$ on $1\le t\le3$?

1. $4$
2. $8$
3. $3$
4. $\frac{2}{3}$

Question 18 · Multiple choice

For $x(t)=2t+1$ and $y(t)=t^3$, what is the average rate of change of $y$ with respect to $x$ from $t=0$ to $t=2$?

1. $\frac{1}{2}$
2. $8$
3. $4$
4. $2$

Question 19 · Multiple choice

The table gives values of a parametric function. What is the average rate of change of $y$ with respect to $x$ from $t=2$ to $t=5$?

1. $6$
2. $\frac{1}{2}$
3. $2$
4. $4$

Question 20 · Multiple choice

On an interval of $t$, the $x$-coordinate of a parametric curve increases by $6$ and the $y$-coordinate decreases by $3$. What is the average rate of change of $y$ with respect to $x$ on that interval?

1. $-\frac{1}{2}$
2. $-2$
3. $\frac{1}{2}$
4. $2$

Question 21 · Multiple choice

For $x(t)=t+2$ and $y(t)=t^2-4t$, what is the average rate of change of $y$ with respect to $x$ from $t=0$ to $t=4$?

1. $4$
2. $0$
3. $-4$
4. $1$

Question 22 · Multiple choice

A particle moves along $x(t)=3t$, $y(t)=6-t$. What is the average rate of change of $y$ with respect to $x$ over any interval of time?

1. $3$
2. $-3$
3. $\frac{1}{3}$
4. $-\frac{1}{3}$

Question 23 · Multiple choice

For $x(t)=\sqrt{t+1}$ and $y(t)=\ln(t+2)$, what is the average rate of change of $y$ with respect to $x$ from $t=0$ to $t=3$?

1. $0.805$
2. $1.091$
3. $0.916$
4. $0.305$

Question 24 · Multiple choice

For $x(t)=2^t$ and $y(t)=3\sin t$ ($t$ in radians), what is the average rate of change of $y$ with respect to $x$ from $t=1$ to $t=2$?

1. $9.829$
2. $2.626$
3. $0.102$
4. $0.203$

Question 25 · Multiple choice

Which pair $(x(t),y(t))$, $0\le t\le2\pi$, traces the circle with center $(2,-1)$ and radius $3$?

1. $(-2+3\cos(t), 1+3\sin(t))$
2. $(2+3\cos(t), -1+3\sin(t))$
3. $(2+9\cos(t), -1+9\sin(t))$
4. $(2+3\cos(t), -1+3\cos(t))$

Question 26 · Multiple choice

Which pair $(x(t),y(t))$, $0\le t\le1$, traces the segment from $(1,2)$ to $(7,-1)$, starting at $(1,2)$?

1. $(7-6t, -1+3t)$
2. $(1+6t, 2-3t)$
3. $(1+7t, 2-t)$
4. $(1+6t, 2+3t)$

Question 27 · Multiple choice

A circle is traced by $x(t)=-3+5\cos t$, $y(t)=2+5\sin t$. What is its center?

1. $(5, 5)$
2. $(-3, 5)$
3. $(3, -2)$
4. $(-3, 2)$

Question 28 · Multiple choice

How does the point $(x(t),y(t))=(3\cos t,\,-3\sin t)$ move for $0\le t\le2\pi$?

1. Counterclockwise once around the circle of radius $3$, starting at $(0,3)$
2. Clockwise once around the circle of radius $3$, starting at $(3,0)$
3. Counterclockwise once around the circle of radius $3$, starting at $(3,0)$
4. Clockwise once around the circle of radius $3$, starting at $(0,-3)$

Question 29 · Multiple choice

A circle is traced by $x(t)=1+2\cos t$, $y(t)=2\sin t$. Which point is on it at $t=\frac\pi3$?

1. $(1+\sqrt{3}, 1)$
2. $(2, \sqrt{3})$
3. $(1, \sqrt{3})$
4. $(2, 1)$

Question 30 · Multiple choice

A segment is traced by $x(t)=-2+8t$, $y(t)=5-4t$, $0\le t\le1$. For what value of $t$ is the point at $(4,2)$?

1. $6$
2. $\frac{3}{4}$
3. $\frac{1}{2}$
4. $\frac{1}{4}$

Question 31 · Multiple choice

How many times does $x(t)=\cos(2t)$, $y(t)=\sin(2t)$, $0\le t\le2\pi$, go around the unit circle?

1. $1$
2. $4$
3. $2$
4. $\frac{1}{2}$

Question 32 · Multiple choice

The line $x(t)=2t$, $y(t)=3-t$ passes through $(0,3)$ in the direction $\langle2,-1\rangle$. Which point is on the line?

1. $(2, 1)$
2. $(4, 2)$
3. $(6, 0)$
4. $(-2, 5)$

Question 33 · Multiple choice

Exactly one of these points satisfies $x^2-xy+y^2=13$. Which one?

1. $(1, 4)$
2. $(1, -4)$
3. $(0, 13)$
4. $(4, 4)$

Question 34 · Multiple choice

How many points of the curve $y^2=x+4$ have $x=5$?

1. $1$
2. $2$
3. $3$
4. $0$

Question 35 · Multiple choice

The circle $x^2+y^2=169$ has two points with $x=5$. What is the set of their $y$-coordinates?

1. $\{-5,5\}$
2. $\{-144,144\}$
3. $\{-12,12\}$
4. $\{12\}$

Question 36 · Multiple choice

The curve $x^3+y^2=8$ is symmetric about which of the following?

1. Both axes
2. The $y$-axis only
3. The origin only
4. The $x$-axis only

Question 37 · Multiple choice

The part of the curve $2x+3y^2=14$ with $y>0$ is the graph of which function?

1. $y=\sqrt{\frac{14-2x}{3}}$
2. $y=\sqrt{\frac{14+2x}{3}}$
3. $y=\frac{14-2x}{3}$
4. $y=\sqrt{14-2x}$

Question 38 · Multiple choice

How high does the curve $x^2+9y^2=36$ reach? Give its greatest $y$-value.

1. $2$
2. $4$
3. $\frac{1}{3}$
4. $6$

Question 39 · Multiple choice

At which $y$-values does the curve $x^2+y^2-6y=0$ meet the $y$-axis?

1. $\{0,3\}$
2. $\{0,6\}$
3. $\{-6,6\}$
4. $\{6\}$

Question 40 · Multiple choice

Both $(2,3)$ and $(3,2)$ lie on the curve $x^3+y^3=35$. Find the $y$-value on this curve that goes with $x=1.5$.

1. $5.624$
2. $3.199$
3. $3.162$
4. $31.625$

Question 41 · Multiple choice

What kind of curve is the graph of $4x^2+9y^2=36$?

1. An ellipse that is not a circle
2. A circle
3. A hyperbola
4. A parabola

Question 42 · Multiple choice

What kind of curve is the graph of $y^2-4x^2=16$?

1. A circle
2. An ellipse that is not a circle
3. A hyperbola
4. A parabola

Question 43 · Multiple choice

What kind of curve is the graph of $x=2y^2-3y+1$?

1. A parabola
2. A hyperbola
3. An ellipse that is not a circle
4. A circle

Question 44 · Multiple choice

What is the vertex of the parabola $y=2(x-3)^2-5$?

1. $(3, -5)$
2. $(3, 5)$
3. $(2, -5)$
4. $(-3, -5)$

Question 45 · Multiple choice

What is the length of the major axis of the ellipse $\frac{(x-1)^2}{25}+\frac{(y+2)^2}{9}=1$?

1. $6$
2. $5$
3. $25$
4. $10$

Question 46 · Multiple choice

The graph of $x^2+6x+y^2-8y=11$ is a circle. How long is its radius?

1. $36$
2. $\sqrt{11}$
3. $11$
4. $6$

Question 47 · Multiple choice

Which point is a vertex of the hyperbola $\frac{x^2}{16}-\frac{y^2}{9}=1$?

1. $(4, 0)$
2. $(16, 0)$
3. $(3, 0)$
4. $(0, 3)$

Question 48 · Multiple choice

Which way does the parabola $x=-(y-1)^2+4$ open?

1. Downward
2. Upward
3. To the left
4. To the right

Question 49 · Multiple choice

Which pair $(x(t),y(t))$, $0\le t\le2\pi$, traces the ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1$?

1. $(4\cos(t), 3\cos(t))$
2. $(16\cos(t), 9\sin(t))$
3. $(3\cos(t), 4\sin(t))$
4. $(4\cos(t), 3\sin(t))$

Question 50 · Multiple choice

An ellipse centered at $(-3,2)$ has equation $\frac{(x+3)^2}{36}+\frac{(y-2)^2}{4}=1$. Which pair $(x(t),y(t))$, $0\le t\le2\pi$, traces it once?

1. $(3+6\cos(t), -2+2\sin(t))$
2. $(-3+36\cos(t), 2+4\sin(t))$
3. $(-3+6\cos(t), 2+2\sin(t))$
4. $(-3+2\cos(t), 2+6\sin(t))$

Question 51 · Multiple choice

Which pair $(x(t),y(t))$, with $t$ any real number, traces the whole parabola $y=x^2-3$?

1. $(2t, 2t^2-3)$
2. $(t+1, t^2+2t-2)$
3. $(t-1, t^2-3)$
4. $(t+1, t^2-2)$

Question 52 · Multiple choice

Which pair $(x(t),y(t))$ satisfies $x^2-y^2=1$ for every $t$ where it is defined?

1. $(\sec(t), \sin(t))$
2. $(\tan(t), \sec(t))$
3. $(\cos(t), \sin(t))$
4. $(\sec(t), \tan(t))$

Question 53 · Multiple choice

Eliminate the parameter: the curve $x(t)=4\cos t$, $y(t)=7\sin t$ satisfies which equation?

1. $\frac{x^2}{16}+\frac{y^2}{49}=1$
2. $\frac{x^2}{4}+\frac{y^2}{7}=1$
3. $\frac{x^2}{16}-\frac{y^2}{49}=1$
4. $\frac{x^2}{49}+\frac{y^2}{16}=1$

Question 54 · Multiple choice

Which pair $(x(t),y(t))$, $0\le t\le2\pi$, traces the circle $x^2+y^2=49$ once clockwise, starting at $(0,7)$?

1. $(7\cos(t), 7\sin(t))$
2. $(7\sin(t), -7\cos(t))$
3. $(7\sin(t), 7\cos(t))$
4. $(-7\sin(t), 7\cos(t))$

Question 55 · Multiple choice

An ellipse is traced by $x(t)=1+2\cos t$, $y(t)=3+6\sin t$. How long is its vertical axis?

1. $12$
2. $3$
3. $6$
4. $4$

Question 56 · Multiple choice

Which equation describes the curve $x(t)=2+t$, $y(t)=t^2-1$?

1. $y=(x-2)^2+1$
2. $y=x^2-1$
3. $y=(x+2)^2-1$
4. $y=(x-2)^2-1$

Question 57 · Multiple choice

What is the magnitude of $\langle-5,12\rangle$?

1. $17$
2. $\sqrt{119}$
3. $13$
4. $7$

Question 58 · Multiple choice

If $\vec u=\langle2,-3\rangle$ and $\vec v=\langle-1,4\rangle$, what is $3\vec u-2\vec v$?

1. $\langle 8, -17\rangle$
2. $\langle 4, -1\rangle$
3. $\langle 5, -17\rangle$
4. $\langle 8, -1\rangle$

Question 59 · Multiple choice

What is $\langle3,-2\rangle\cdot\langle4,5\rangle$?

1. $12$
2. $-2$
3. $22$
4. $2$

Question 60 · Multiple choice

What is the unit vector in the direction of $\langle6,-8\rangle$?

1. $\langle \frac{3}{4}, -1\rangle$
2. $\langle \frac{3}{5}, -\frac{4}{5}\rangle$
3. $\langle \frac{3}{7}, -\frac{4}{7}\rangle$
4. $\langle -\frac{4}{5}, \frac{3}{5}\rangle$

Question 61 · Multiple choice

What is the angle between $\langle1,1\rangle$ and $\langle1,0\rangle$?

1. $\frac{\pi}{6}$
2. $\frac{\pi}{3}$
3. $\frac{\pi}{4}$
4. $\frac{\pi}{2}$

Question 62 · Multiple choice

For what value of $k$ are $\langle k,3\rangle$ and $\langle2,-4\rangle$ perpendicular?

1. $6$
2. $-\frac{3}{2}$
3. $\frac{3}{2}$
4. $-6$

Question 63 · Multiple choice

To the nearest thousandth of a degree, what is the angle between $\langle3,1\rangle$ and $\langle-1,4\rangle$?

1. $85.601$
2. $4.399$
3. $57.529$
4. $75.964$

Question 64 · Multiple choice

Point $P$ is $(2,5)$ and point $Q$ is $(-1,9)$. What is the vector $\overrightarrow{PQ}$?

1. $\langle -3, 14\rangle$
2. $\langle 1, 14\rangle$
3. $\langle -3, 4\rangle$
4. $\langle 3, -4\rangle$

Question 65 · Multiple choice

A vector has magnitude $10$ and makes an angle of $120^\circ$ with the positive $x$-axis. What are its components?

1. $\langle -5, 5\sqrt{3}\rangle$
2. $\langle 5, 5\sqrt{3}\rangle$
3. $\langle -5\sqrt{3}, 5\rangle$
4. $\langle -5, -5\sqrt{3}\rangle$

Question 66 · Multiple choice

An object's position is $\vec p(t)=\langle t^2,\,3t-1\rangle$. Where is it at $t=2$?

1. $\langle 2, 5\rangle$
2. $\langle 4, 6\rangle$
3. $\langle 4, 5\rangle$
4. $\langle 5, 4\rangle$

Question 67 · Multiple choice

For $\vec p(t)=\langle2t+1,\,t^2\rangle$, what is the displacement from $t=1$ to $t=3$?

1. $\langle 7, 9\rangle$
2. $\langle 4, 8\rangle$
3. $\langle 4, 4\rangle$
4. $\langle 2, 4\rangle$

Question 68 · Multiple choice

For $\vec p(t)=\langle t^3,\,4t\rangle$, what is the average rate of change of $\vec p$ (the average velocity) from $t=0$ to $t=2$?

1. $\langle 4, 4\rangle$
2. $\langle 2, 2\rangle$
3. $\langle 12, 4\rangle$
4. $\langle 8, 8\rangle$

Question 69 · Multiple choice

For $\vec p(t)=\langle3t^2,\,4t\rangle$, how far is the object from the origin at $t=1$?

1. $\sqrt{7}$
2. $25$
3. $7$
4. $5$

Question 70 · Multiple choice

For $\vec p(t)=\langle\cos(\pi t),\,\sin(\pi t)\rangle$, what is $\vec p\left(\frac12\right)$?

1. $\langle -1, 0\rangle$
2. $\langle 0, 1\rangle$
3. $\langle 1, 0\rangle$
4. $\langle \frac{1}{2}, \frac{1}{2}\rangle$

Question 71 · Multiple choice

Two objects are at $\vec r_1(t)=\langle t,\,2t\rangle$ and $\vec r_2(t)=\langle6-t,\,t+3\rangle$. How far apart are they at $t=1$?

1. $20$
2. $2\sqrt{13}$
3. $2\sqrt{5}$
4. $6$

Question 72 · Multiple choice

For $\vec p(t)=\langle2^t,\,t^2-1\rangle$ with $t>0$, when is the object on the $x$-axis?

1. $1$
2. $2$
3. $0$
4. $-1$

Question 73 · Multiple choice

For $\vec p(t)=\langle e^{0.2t},\,\ln(t+1)\rangle$, what is the magnitude of the displacement from $t=0$ to $t=5$?

1. $3.256$
2. $2.483$
3. $3.510$
4. $0.497$

Question 74 · Multiple choice

If $A=\begin{bmatrix}2&-1\\0&3\end{bmatrix}$ and $B=\begin{bmatrix}1&4\\-2&5\end{bmatrix}$, what is $A+B$?

1. $\begin{bmatrix}1&-5\\2&-2\end{bmatrix}$
2. $\begin{bmatrix}2&-4\\0&15\end{bmatrix}$
3. $\begin{bmatrix}3&5\\-2&8\end{bmatrix}$
4. $\begin{bmatrix}3&3\\-2&8\end{bmatrix}$

Question 75 · Multiple choice

If $A=\begin{bmatrix}2&-1\\0&3\end{bmatrix}$ and $B=\begin{bmatrix}1&4\\-2&5\end{bmatrix}$, what is $3A-B$?

1. $\begin{bmatrix}7&1\\-2&14\end{bmatrix}$
2. $\begin{bmatrix}5&-7\\-2&4\end{bmatrix}$
3. $\begin{bmatrix}5&1\\2&4\end{bmatrix}$
4. $\begin{bmatrix}5&-7\\2&4\end{bmatrix}$

Question 76 · Multiple choice

If $A=\begin{bmatrix}2&-1\\0&3\end{bmatrix}$ and $B=\begin{bmatrix}1&4\\-2&5\end{bmatrix}$, what is $AB$?

1. $\begin{bmatrix}4&3\\6&15\end{bmatrix}$
2. $\begin{bmatrix}2&11\\-4&17\end{bmatrix}$
3. $\begin{bmatrix}2&-4\\0&15\end{bmatrix}$
4. $\begin{bmatrix}4&3\\-6&15\end{bmatrix}$

Question 77 · Multiple choice

Matrix $A$ has 2 rows and 3 columns, and matrix $B$ has 3 rows and 2 columns. What are the dimensions of $AB$?

1. $2\times2$
2. $3\times3$
3. $2\times3$
4. $AB$ is not defined

Question 78 · Multiple choice

If $A=\begin{bmatrix}1&3\\-2&4\end{bmatrix}$ and $B=\begin{bmatrix}5&0\\2&-1\end{bmatrix}$, what is the entry in row 2, column 1 of $AB$?

1. $18$
2. $8$
3. $-2$
4. $-10$

Question 79 · Multiple choice

If $A=\begin{bmatrix}2&-1\\0&3\end{bmatrix}$ and $B=\begin{bmatrix}1&4\\-2&5\end{bmatrix}$, what is $BA$?

1. $\begin{bmatrix}2&11\\-4&17\end{bmatrix}$
2. $\begin{bmatrix}4&3\\-6&15\end{bmatrix}$
3. $\begin{bmatrix}2&-4\\0&15\end{bmatrix}$
4. $\begin{bmatrix}2&11\\4&17\end{bmatrix}$

Question 80 · Multiple choice

What is $\begin{bmatrix}3&-2\\1&4\end{bmatrix}\begin{bmatrix}2\\5\end{bmatrix}$?

1. $\begin{bmatrix}-4\\22\end{bmatrix}$
2. $\begin{bmatrix}16\\22\end{bmatrix}$
3. $\begin{bmatrix}-4\\-18\end{bmatrix}$
4. $\begin{bmatrix}6\\5\end{bmatrix}$

Question 81 · Multiple choice

If $A=\begin{bmatrix}1.2&0.5\\0.3&0.8\end{bmatrix}$, what is the entry in row 1, column 2 of $A^3$?

1. $2.208$
2. $0.125$
3. $0.957$
4. $1.595$

Question 82 · Multiple choice

What is $\det\begin{bmatrix}4&-2\\3&5\end{bmatrix}$?

1. $23$
2. $-26$
3. $14$
4. $26$

Question 83 · Multiple choice

Matrix $C=\begin{bmatrix}4&3\\5&4\end{bmatrix}$ has determinant $1$. Which matrix is $C^{-1}$?

1. $\begin{bmatrix}4&-3\\-5&4\end{bmatrix}$
2. $\begin{bmatrix}4&-5\\-3&4\end{bmatrix}$
3. $\begin{bmatrix}\frac{1}{4}&\frac{1}{3}\\\frac{1}{5}&\frac{1}{4}\end{bmatrix}$
4. $\begin{bmatrix}-4&3\\5&-4\end{bmatrix}$

Question 84 · Multiple choice

For what value of $k$ does $\begin{bmatrix}k&6\\2&3\end{bmatrix}$ have no inverse?

1. $1$
2. $-4$
3. $9$
4. $4$

Question 85 · Multiple choice

A triangle has area $4$. The transformation with matrix $\begin{bmatrix}3&0\\1&-2\end{bmatrix}$ is applied to it. What is the area of the image?

1. $24$
2. $12$
3. $-24$
4. $6$

Question 86 · Multiple choice

What is the inverse of $\begin{bmatrix}2&4\\1&3\end{bmatrix}$?

1. $\begin{bmatrix}\frac{1}{2}&\frac{1}{4}\\1&\frac{1}{3}\end{bmatrix}$
2. $\begin{bmatrix}\frac{3}{2}&-2\\-\frac{1}{2}&1\end{bmatrix}$
3. $\begin{bmatrix}3&-4\\-1&2\end{bmatrix}$
4. $\begin{bmatrix}\frac{3}{2}&2\\\frac{1}{2}&1\end{bmatrix}$

Question 87 · Multiple choice

Solve $\begin{bmatrix}2&1\\1&1\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}5\\3\end{bmatrix}$. What is $(x,y)$?

1. $(1, 2)$
2. $(8, 1)$
3. $(2, 1)$
4. $(3, 0)$

Question 88 · Multiple choice

Matrices $A$ and $B$ are $2\times2$ with $\det A=3$ and $\det B=-2$. What is $\det(AB)$?

1. $-5$
2. $6$
3. $-6$
4. $1$

Question 89 · Multiple choice

What is $\det\begin{bmatrix}1.5&-0.4\\2.2&3.1\end{bmatrix}$?

1. $-5.530$
2. $5.530$
3. $3.770$
4. $4.540$

Question 90 · Multiple choice

Which matrix reflects every point of the plane across the $x$-axis?

1. $\begin{bmatrix}0&1\\1&0\end{bmatrix}$
2. $\begin{bmatrix}-1&0\\0&1\end{bmatrix}$
3. $\begin{bmatrix}1&0\\0&-1\end{bmatrix}$
4. $\begin{bmatrix}-1&0\\0&-1\end{bmatrix}$

Question 91 · Multiple choice

A rotation of $90^\circ$ counterclockwise about the origin is applied to the point $(3,1)$. Where does it go?

1. $(1, -3)$
2. $(-1, 3)$
3. $(-3, 1)$
4. $(3, -1)$

Question 92 · Multiple choice

What does the transformation with matrix $\begin{bmatrix}2&0\\0&3\end{bmatrix}$ do to the plane?

1. Rotates the plane and then stretches it by a factor of $6$
2. Stretches horizontally by a factor of $3$ and vertically by a factor of $2$
3. Stretches horizontally by a factor of $2$ and vertically by a factor of $3$
4. Shifts every point $2$ units right and $3$ units up

Question 93 · Multiple choice

The transformation $\begin{bmatrix}1&2\\0&1\end{bmatrix}$ is applied to the point $(1,2)$. What is the image?

1. $(3, 2)$
2. $(1, 4)$
3. $(5, 2)$
4. $(5, 4)$

Question 94 · Multiple choice

Which matrix rotates the plane $\frac\pi3$ radians counterclockwise about the origin?

1. $\begin{bmatrix}\frac{\sqrt{3}}{2}&-\frac{1}{2}\\\frac{1}{2}&\frac{\sqrt{3}}{2}\end{bmatrix}$
2. $\begin{bmatrix}\frac{1}{2}&-\frac{\sqrt{3}}{2}\\\frac{\sqrt{3}}{2}&\frac{1}{2}\end{bmatrix}$
3. $\begin{bmatrix}\frac{1}{2}&\frac{\sqrt{3}}{2}\\-\frac{\sqrt{3}}{2}&\frac{1}{2}\end{bmatrix}$
4. $\begin{bmatrix}\frac{1}{2}&-\frac{\sqrt{3}}{2}\\-\frac{\sqrt{3}}{2}&\frac{1}{2}\end{bmatrix}$

Question 95 · Multiple choice

A linear transformation sends $\langle1,0\rangle$ to $\langle2,1\rangle$ and $\langle0,1\rangle$ to $\langle-1,3\rangle$. What is its matrix?

1. $\begin{bmatrix}2&1\\-1&3\end{bmatrix}$
2. $\begin{bmatrix}2&-1\\1&3\end{bmatrix}$
3. $\begin{bmatrix}1&2\\3&-1\end{bmatrix}$
4. $\begin{bmatrix}2&-1\\3&1\end{bmatrix}$

Question 96 · Multiple choice

A region of area $7$ is transformed by $\begin{bmatrix}1&4\\2&3\end{bmatrix}$. What is the area of the image?

1. $77$
2. $35$
3. $-35$
4. $7$

Question 97 · Multiple choice

Which of these transformations reverses orientation (turns clockwise order of points into counterclockwise order)?

1. $\begin{bmatrix}2&1\\1&3\end{bmatrix}$
2. $\begin{bmatrix}3&0\\0&2\end{bmatrix}$
3. $\begin{bmatrix}0&-1\\1&0\end{bmatrix}$
4. $\begin{bmatrix}2&1\\3&1\end{bmatrix}$

Question 98 · Multiple choice

Transformation $A=\begin{bmatrix}1&2\\0&1\end{bmatrix}$ is applied first, then $B=\begin{bmatrix}0&-1\\1&0\end{bmatrix}$. Which single matrix does both?

1. $\begin{bmatrix}1&1\\1&1\end{bmatrix}$
2. $\begin{bmatrix}2&-1\\1&0\end{bmatrix}$
3. $\begin{bmatrix}0&-2\\1&0\end{bmatrix}$
4. $\begin{bmatrix}0&-1\\1&2\end{bmatrix}$

Question 99 · Multiple choice

What matrix undoes a $90^\circ$ counterclockwise rotation about the origin?

1. $\begin{bmatrix}0&-1\\1&0\end{bmatrix}$
2. $\begin{bmatrix}1&0\\0&-1\end{bmatrix}$
3. $\begin{bmatrix}-1&0\\0&-1\end{bmatrix}$
4. $\begin{bmatrix}0&1\\-1&0\end{bmatrix}$

Question 100 · Multiple choice

$L(\vec v)=A\vec v$ with $A=\begin{bmatrix}2&1\\3&2\end{bmatrix}$. Which $\vec v$ has $L(\vec v)=\langle4,7\rangle$?

1. $\langle 1, 2\rangle$
2. $\langle 15, 26\rangle$
3. $\langle 2, 1\rangle$
4. $\langle -1, 6\rangle$

Question 101 · Multiple choice

The reflection across the line $y=x$ is applied twice. Which matrix gives the result?

1. $\begin{bmatrix}1&0\\0&1\end{bmatrix}$
2. $\begin{bmatrix}-1&0\\0&-1\end{bmatrix}$
3. $\begin{bmatrix}0&-1\\-1&0\end{bmatrix}$
4. $\begin{bmatrix}0&1\\1&0\end{bmatrix}$

Question 102 · Multiple choice

The plane is scaled by $3$ about the origin and then rotated $90^\circ$ counterclockwise. Where does $(1,0)$ go?

1. $(0, 3)$
2. $(-3, 0)$
3. $(0, 1)$
4. $(3, 0)$

Question 103 · Multiple choice

Which statement about the transformation $L(\vec v)=\begin{bmatrix}2&4\\1&2\end{bmatrix}\vec v$ is true?

1. It has no inverse: it maps the whole plane onto a line.
2. It has an inverse, because no entry is $0$.
3. It doubles every area, so its inverse halves areas.
4. It is a rotation, so its inverse is a rotation the other way.

Question 104 · Multiple choice

Rotating the plane $\frac\pi6$ radians counterclockwise three times in a row is the same as which single matrix?

1. $\begin{bmatrix}-1&0\\0&-1\end{bmatrix}$
2. $\begin{bmatrix}0&-1\\1&0\end{bmatrix}$
3. $\begin{bmatrix}0&1\\-1&0\end{bmatrix}$
4. $\begin{bmatrix}\frac{\sqrt{3}}{2}&-\frac{1}{2}\\\frac{1}{2}&\frac{\sqrt{3}}{2}\end{bmatrix}$

Question 105 · Multiple choice

$L(\vec v)=A\vec v$ with $A=\begin{bmatrix}1.1&0.3\\-0.2&0.9\end{bmatrix}$. If $L(\vec v)=\langle2,3\rangle$, what is the first component of $\vec v$?

1. $3.524$
2. $3.100$
3. $1.818$
4. $0.857$

Question 106 · Multiple choice

Each week, bikes move between two docking stations. The transition matrix $A=\begin{bmatrix}0.8&0.3\\0.2&0.7\end{bmatrix}$ gives the fraction of bikes going from the station of each column to the station of each row. This week there are $500$ bikes at each station. How many are at each station next week?

1. $\langle 500, 500\rangle$
2. $\langle 450, 550\rangle$
3. $\langle 575, 425\rangle$
4. $\langle 550, 450\rangle$

Question 107 · Multiple choice

Each week, bikes move between two docking stations. The transition matrix $A=\begin{bmatrix}0.8&0.3\\0.2&0.7\end{bmatrix}$ gives the fraction of bikes going from the station of each column to the station of each row. This week there are $500$ bikes at each station. How many are at each station in two weeks?

1. $\langle 550, 450\rangle$
2. $\langle 587.5, 412.5\rangle$
3. $\langle 575, 425\rangle$
4. $\langle 600, 400\rangle$

Question 108 · Multiple choice

Each week, bikes move between two docking stations. The transition matrix $A=\begin{bmatrix}0.8&0.3\\0.2&0.7\end{bmatrix}$ gives the fraction of bikes going from the station of each column to the station of each row. There are $1000$ bikes in all. Which distribution stays the same from week to week?

1. $\langle 600, 400\rangle$
2. $\langle 550, 450\rangle$
3. $\langle 700, 300\rangle$
4. $\langle 500, 500\rangle$

Question 109 · Multiple choice

Each week, bikes move between two docking stations. The transition matrix $A=\begin{bmatrix}0.8&0.3\\0.2&0.7\end{bmatrix}$ gives the fraction of bikes going from the station of each column to the station of each row. What does the entry $0.3$ (row 1, column 2) mean?

1. Each week, $30\%$ of the bikes at station 2 stay there.
2. Each week, $30\%$ of the bikes at station 2 move to station 1.
3. Station 2 holds $30\%$ of all the bikes.
4. Each week, $30\%$ of the bikes at station 1 move to station 2.

Question 110 · Multiple choice

A transition matrix $A=\begin{bmatrix}0.9&0.4\\0.1&0.6\end{bmatrix}$ moves customers between two plans each month (columns: from, rows: to). This month the plans have $620$ and $380$ customers. How many did each plan have last month?

1. $\langle 620, 380\rangle$
2. $\langle 560, 440\rangle$
3. $\langle 710, 290\rangle$
4. $\langle 440, 560\rangle$

Question 111 · Multiple choice

A transition matrix $A=\begin{bmatrix}0.85&0.25\\0.15&0.75\end{bmatrix}$ (columns: from, rows: to) acts on $\vec x_0=\langle1000,0\rangle$. What is the first component of $\vec x_{10}=A^{10}\vec x_0$?

1. $196.874$
2. $627.267$
3. $625.000$
4. $372.733$

Question 112 · Multiple choice

A student's choice of lunch each day follows $A=\begin{bmatrix}0.7&0.4\\0.3&0.6\end{bmatrix}$, where column $j$ gives the chances of tomorrow's choice after choice $j$ today (choice 1: cafeteria, choice 2: packed lunch). The student packs a lunch today. What is the chance that the student eats in the cafeteria two days from now?

1. $0.52$
2. $0.48$
3. $0.28$
4. $0.4$

Question 113 · Multiple choice

A student's choice of lunch each day follows $A=\begin{bmatrix}0.7&0.4\\0.3&0.6\end{bmatrix}$, where column $j$ gives the chances of tomorrow's choice after choice $j$ today (choice 1: cafeteria, choice 2: packed lunch). In the long run, what fraction of days does the student eat in the cafeteria?

1. $\frac{4}{7}$
2. $\frac{7}{10}$
3. $\frac{1}{2}$
4. $\frac{3}{7}$

Question 114 · Free response (6 points)

A particle moves in the plane so that its position at time $t$ seconds is $(x(t),y(t))$, where $x(t)=t^2-4t$ and $y(t)=2t+1$, for $0\le t\le5$. Answer without a calculator.

1. (A(i)) Find the position of the particle at $t=1$. [1 point]
2. (A(ii)) Find the time $t>0$ at which the particle is on the $y$-axis, and its position then. [1 point]
3. (B) Find the average rate of change of $y$ with respect to $x$ from $t=1$ to $t=5$. [2 points]
4. (C) On what interval of $t$ is the particle moving to the left? Give a reason. [2 points]

Question 115 · Free response (6 points)

A curve in the plane is given by $9x^2+4y^2-36x+8y+4=0$. Answer without a calculator.

1. (A) Write the equation in the form $\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1$ and give the center of the curve. [2 points]
2. (B) Find the lengths of the horizontal and vertical semi-axes. [1 point]
3. (C) Write a pair of parametric equations, $0\le t\le2\pi$, that traces the curve once. [1 point]
4. (D) Using your equations from part C, find the point on the curve when $t=\frac\pi6$. [2 points]

Question 116 · Free response (6 points)

An object moves in the plane with position vector $\vec p(t)=\langle3t,\,8t-t^2\rangle$ meters at time $t$ seconds, $0\le t\le6$. A graphing calculator is allowed. Give decimal answers to three decimal places.

1. (A) Find $\vec p(2)$ and the distance of the object from the origin at $t=2$. [2 points]
2. (B) Find the displacement vector from $t=0$ to $t=4$ and its magnitude. [2 points]
3. (C) Find the angle, in degrees, between $\vec p(2)$ and the positive $x$-axis. [2 points]

Question 117 · Free response (6 points)

Two parks share a fleet of $600$ rental scooters. Each day the scooters move between the parks according to $A=\begin{bmatrix}0.6&0.2\\0.4&0.8\end{bmatrix}$ (column $j$: where the scooters at park $j$ end the day; row 1: park 1, row 2: park 2). Today each park has $300$ scooters. Answer without a calculator.

1. (A(i)) Find $\det A$. [1 point]
2. (A(ii)) Find the number of scooters at each park tomorrow. [1 point]
3. (B) Find the number of scooters at each park the day after tomorrow. [1 point]
4. (C) Find the distribution of the $600$ scooters that does not change from day to day. [2 points]
5. (D) Find $A^{-1}$. [1 point]