Final Attestation Exam for the B.Ed. in Mathematics 6B01501
Q1.
Evaluate the indefinite integral: $$\int \sec^3 x\tan x\,dx$$
$$\sec x + C$$
$$\frac{1}{3}\tan^3 x + C$$
$$\sec^3 x + C$$
$$\frac{1}{4}\sec^4 x + C$$
$$\frac{1}{3}\sec^3 x + C$$
Q2.
Evaluate the integral: $\int \frac{x^2}{1+x^3}\,dx$
$\frac{x^3}{1+x^3}+C$
$\frac13\ln|1+x^3|+C$
$\frac13\ln|x+x^3|+C$
$\arctan x+C$
$\ln|1+x^3|+C$
Q3.
Let $p$ be ``DATAENDFLAG is off,' $q$ be ``ERROR equals $0$,' and $r$ be ``SUM is less than $1000$.' Translate the statement into symbolic notation: DATAENDFLAG is off, ERROR equals $0$, and SUM is less than $1000$.
$p \lor q \lor r$
$p \land (q \lor r)$
$\neg p \land q \land r$
$p \land q \land r$
$p \lor (q \land r)$
Q4.
Evaluate the definite integral: $$\int_{1/2}^{1} \frac{y+4}{y^2+y}\,dy$$
$$\ln\left(\frac{27}{4}\right)$$
$$\ln\left(\frac{9}{4}\right)$$
$$\ln\left(\frac{4}{27}\right)$$
$$\frac{27}{4}$$
$$\ln\left(\frac{27}{8}\right)$$
Q5.
Let $A=\begin{bmatrix}1&-2\\-2&5\end{bmatrix}$ and suppose $AB=\begin{bmatrix}-1&2&-1\\6&-9&3\end{bmatrix}$. Determine the first and second columns of $B$.
First column $=\begin{bmatrix}7\\4\end{bmatrix}$, second column $=\begin{bmatrix}-8\\5\end{bmatrix}$
First column $=\begin{bmatrix}-7\\-4\end{bmatrix}$, second column $=\begin{bmatrix}-8\\-5\end{bmatrix}$
First column $=\begin{bmatrix}4\\7\end{bmatrix}$, second column $=\begin{bmatrix}-5\\-8\end{bmatrix}$
First column $=\begin{bmatrix}7\\4\end{bmatrix}$, second column $=\begin{bmatrix}8\\5\end{bmatrix}$
First column $=\begin{bmatrix}7\\4\end{bmatrix}$, second column $=\begin{bmatrix}-8\\-5\end{bmatrix}$
Q6.
Determine the order of the differential equation and whether it is linear or nonlinear: $\displaystyle \frac{d^2y}{dt^2}+\sin(t+y)=\sin t$.
Third-order nonlinear
Second-order linear
Second-order nonlinear
First-order nonlinear
First-order linear
Q7.
During orientation week, a movie was shown twice. Among 6000 freshmen, 850 attended the first showing, 690 attended the second showing, and 4700 attended neither showing. How many students attended both showings?
150
310
540
1300
240
Q8.
Surface area: revolve $y=\sqrt{2x-x^2}$, $\frac12\le x\le \frac32$, about the $x$-axis.
$2\pi$
$4\pi$
$\pi$
$2$
$2\pi^2$
Q9.
For the conic $x^2-y^2-2x+4y=4$, identify the conic and give its center, vertices, foci, and asymptotes.
Hyperbola centered at $(1,2)$; vertices $(1,1),(1,3)$; foci $(1,2\pm\sqrt{2})$; asymptotes $y-2=\pm(x-1)$
Hyperbola centered at $(1,2)$; vertices $(0,2),(2,2)$; foci $(1\pm\sqrt{2},2)$; asymptotes $y-2=\pm(x-1)$
Parabola with vertex $(1,2)$ and axis horizontal
Hyperbola centered at $(-1,-2)$; vertices $(-2,-2),(0,-2)$; foci $(-1\pm\sqrt{2},-2)$; asymptotes $y+2=\pm(x+1)$
Ellipse centered at $(1,2)$; vertices $(0,2),(2,2)$; foci $(1\pm\sqrt{2},2)$
Q10.
An economy has two sectors, Goods and Services. Each year, Goods sells $80\%$ of its output to Services and retains the rest, while Services sells $70\%$ of its output to Goods and retains the rest. Find equilibrium prices for the annual outputs of the two sectors.
Any positive multiple of $\begin{bmatrix}7\\8\end{bmatrix}$
Any positive multiple of $\begin{bmatrix}1\\1\end{bmatrix}$
Any positive multiple of $\begin{bmatrix}8\\7\end{bmatrix}$
Any positive multiple of $\begin{bmatrix}2\\7\end{bmatrix}$
Any positive multiple of $\begin{bmatrix}4\\5\end{bmatrix}$
Q11.
Evaluate the indefinite integral: $$\int 7\cos^7 t\,dt$$
$$\sin^7 t + C$$
$$7\sin t+7\sin^3 t+\frac{21}{5}\sin^5 t+\sin^7 t+C$$
$$7\sin t-\frac{7}{3}\sin^3 t+\frac{7}{5}\sin^5 t-\frac{1}{7}\sin^7 t+C$$
$$7\sin t-7\sin^3 t+\frac{21}{5}\sin^5 t-\sin^7 t+C$$
$$7\cos t-7\cos^3 t+\frac{21}{5}\cos^5 t-\cos^7 t+C$$
Q12.
Evaluate the indefinite integral: $$\int \frac{dt}{t^3+t^2-2t}$$
$$\frac{1}{6}\ln|t+2|+\frac{1}{3}\ln|t-1|+\frac{1}{2}\ln|t|+C$$
$$\frac{1}{t^2+t-2}+C$$
$$\frac{1}{6}\ln|t+2|+\frac{1}{3}\ln|t-1|-\frac{1}{2}\ln|t|+C$$
$$\ln|t^3+t^2-2t|+C$$
$$\frac{1}{6}\ln|t+2|-\frac{1}{3}\ln|t-1|-\frac{1}{2}\ln|t|+C$$
Q13.
Let $T:\mathbb{R}^2\to\mathbb{R}^4$ be linear with $T(e_1)=(3,1,3,1)$ and $T(e_2)=(-5,2,0,0)$. Find the standard matrix of $T$.
$\begin{bmatrix}3&-5\\1&2\\3&0\\1&0\end{bmatrix}$
$\begin{bmatrix}3&1&3&1\\-5&2&0&0\end{bmatrix}$
$\begin{bmatrix}3&-5\\1&2\\0&3\\1&0\end{bmatrix}$
$\begin{bmatrix}3&1\\-5&2\\3&0\\1&0\end{bmatrix}$
$\begin{bmatrix}-5&3\\2&1\\0&3\\0&1\end{bmatrix}$
Q14.
Determine the order of the differential equation and whether it is linear or nonlinear: $\displaystyle (1+y^2)\frac{d^2y}{dt^2}+t\frac{dy}{dt}+y=e^t$.
Second-order nonlinear
First-order linear
Third-order nonlinear
Second-order linear
First-order nonlinear
Q15.
Find the inverse of $\begin{bmatrix}8&6\\5&4\end{bmatrix}$.
$\begin{bmatrix}4&-6\\-5&8\end{bmatrix}$
$\frac{1}{2}\begin{bmatrix}4&-5\\-6&8\end{bmatrix}$
$\frac{1}{2}\begin{bmatrix}4&-6\\-5&8\end{bmatrix}$
$\frac{1}{2}\begin{bmatrix}8&-6\\-5&4\end{bmatrix}$
$\frac{1}{2}\begin{bmatrix}4&6\\5&8\end{bmatrix}$
Q16.
Let $p$ be ``DATAENDFLAG is off,' $q$ be ``ERROR equals $0$,' and $r$ be ``SUM is less than $1000$.' Translate the statement into symbolic notation: DATAENDFLAG is off; however, ERROR is not $0$ or SUM is greater than or equal to $1000$.
$p \land (\neg q \lor r)$
$p \lor (\neg q \lor \neg r)$
$(p \land \neg q) \lor \neg r$
$p \land (\neg q \lor \neg r)$
$\neg p \land (\neg q \lor \neg r)$
Q17.
Lucy runs two independent online scams. The probability she is arrested for the first is 0.1, for the second is 1/30, and the probability she is arrested for both is 0.0025. What is the probability she avoids incarceration (i.e., is not arrested for either)?
0.90
0.8667
157/1200
1043/1200
0.8750
Q18.
Let $A$ and $B$ be events. If $P(A\cup B)=0.3$ and $P(A\cap B^c)=0.1$, what is $P(B)$?
0.2
0.3
0.25
0.1
0.4
Q19.
Suppose $P(A\cap B)=0.1$, $P((A\cup B)^c)=0.3$, and $P(A)=0.2$. Compute $$P\big((A\cap B)\mid (A\cup B)^c\big).$$
0.1
0.2
1/3
0
0.3
Q20.
For the parametric curve $x=2\cos t$, $y=2\sin t$ at $t=\frac{\pi}{4}$, find the tangent line and the value of $\frac{d^2y}{dx^2}$ at this point.
Tangent: $y=-x+2\sqrt{2}$; $\frac{d^2y}{dx^2}=-\sqrt{2}$
Tangent: $y=x$; $\frac{d^2y}{dx^2}=-\sqrt{2}$
Tangent: $y=-x+2\sqrt{2}$; $\frac{d^2y}{dx^2}=\sqrt{2}$
Tangent: $y=-x+2\sqrt{2}$; $\frac{d^2y}{dx^2}=-\frac{\sqrt{2}}{2}$
Tangent: $y=-x+\sqrt{2}$; $\frac{d^2y}{dx^2}=-\sqrt{2}$
Q21.
Let $p$ be ``DATAENDFLAG is off,' $q$ be ``ERROR equals $0$,' and $r$ be ``SUM is less than $1000$.' Translate the statement into symbolic notation: DATAENDFLAG is off but ERROR is not equal to $0$.
$\neg p \land q$
$p \land q$
$p \land \neg q$
$p \lor \neg q$
$\neg(p \land q)$
Q22.
A spherical raindrop evaporates at a rate proportional to its surface area. If $V(t)$ is the volume of the raindrop, which differential equation can model $V$ as a function of time?
$\displaystyle \frac{dV}{dt}=-kV^{2/3}$, where $k>0$
$\displaystyle \frac{dV}{dt}=-kV$
$\displaystyle \frac{dV}{dt}=-\frac{k}{V^{2/3}}$
$\displaystyle \frac{dV}{dt}=-kV^{3/2}$
$\displaystyle \frac{d^2V}{dt^2}=-kV^{2/3}$
Q23.
An urn contains one red chip and one white chip. One chip is drawn. If it is red, that chip plus two additional red chips are put into the urn. If it is white, the chip is returned and nothing else is added. Then a second chip is drawn. What is the probability that both draws are red?
5/16
3/4
1/4
1/2
3/8
Q24.
Let $p$ be ``DATAENDFLAG is off,' $q$ be ``ERROR equals $0$,' and $r$ be ``SUM is less than $1000$.' Translate the statement into symbolic notation: Either DATAENDFLAG is on or it is the case that both ERROR equals $0$ and SUM is less than $1000$.
$\neg p \lor (q \land r)$
$(\neg p \lor q) \land r$
$p \lor (q \land r)$
$\neg p \land (q \land r)$
$\neg(p \lor q \land r)$
Q25.
Determine the order of the differential equation and whether it is linear or nonlinear: $\displaystyle \frac{dy}{dt}+ty^2=0$.
Second-order nonlinear
Second-order linear
First-order linear
Third-order nonlinear
First-order nonlinear
Q26.
Evaluate the definite integral: $$\int_{0}^{\pi} 8\sin^4 x\,dx$$
$$4\pi$$
$$\pi$$
$$\frac{3\pi}{2}$$
$$3\pi$$
$$2\pi$$
Q27.
A team has a 10% chance to win game 1 and a 30% chance to win game 2. The probability of losing both games is 65%. What is the probability of winning exactly one game?
0.30
0.05
0.35
0.40
0.25
Q28.
Determine the order of the differential equation and whether it is linear or nonlinear: $\displaystyle \frac{d^4y}{dt^4}+\frac{d^3y}{dt^3}+\frac{d^2y}{dt^2}+\frac{dy}{dt}+y=1$.
First-order linear
Second-order linear
Fourth-order nonlinear
Third-order linear
Fourth-order linear
Q29.
Evaluate the indefinite integral: $$\int \cos^2 x\,dx$$
$$\frac{x}{2}+\frac{\cos 2x}{4}+C$$
$$\frac{x}{2}+\frac{\sin 2x}{4}+C$$
$$\frac{\sin^2 x}{2}+C$$
$$\frac{x}{2}-\frac{\sin 2x}{4}+C$$
$$x+\frac{\sin 2x}{2}+C$$
Q30.
Let $A=\begin{bmatrix}2&5\\-3&1\end{bmatrix}$ and $B=\begin{bmatrix}4&-5\\3&k\end{bmatrix}$. For what value of $k$ does $AB=BA$?
$k=5$
$k=\frac{15}{2}$
$k=-5$
$k=3$
No such real value of $k$ exists
Q31.
Evaluate the indefinite integral: $$\int \sin^2\theta\cos 3\theta\,d\theta$$
$$\frac{1}{6}\sin 3\theta-\frac{1}{20}\sin 5\theta + C$$
$$-\frac{1}{4}\cos\theta+\frac{1}{6}\cos 3\theta-\frac{1}{20}\cos 5\theta + C$$
$$-\frac{1}{2}\sin\theta+\frac{1}{3}\sin 3\theta-\frac{1}{10}\sin 5\theta + C$$
$$\frac{1}{4}\sin\theta+\frac{1}{6}\sin 3\theta+\frac{1}{20}\sin 5\theta + C$$
$$-\frac{1}{4}\sin\theta+\frac{1}{6}\sin 3\theta-\frac{1}{20}\sin 5\theta + C$$
Q32.
Solve the matrix equation $Ax=b$ for $A=\begin{bmatrix}3&-7&-2\\-3&5&1\\6&-4&0\end{bmatrix}$ and $b=\begin{bmatrix}-7\\5\\2\end{bmatrix}$.
$x=\begin{bmatrix}3\\4\\6\end{bmatrix}$
$x=\begin{bmatrix}3\\4\\-6\end{bmatrix}$
$x=\begin{bmatrix}4\\3\\-6\end{bmatrix}$
$x=\begin{bmatrix}3\\-4\\6\end{bmatrix}$
$x=\begin{bmatrix}-3\\4\\-6\end{bmatrix}$
Q33.
Two fair dice are rolled. What is $P(\text{first die} \ge 4 \mid \text{sum}=8)$?
3/5
2/5
1/2
4/5
1/5
Q34.
Evaluate the indefinite integral: $$\int \sec^3 x\tan^3 x\,dx$$
$$\frac{1}{4}\sec^4 x-\frac{1}{2}\sec^2 x + C$$
$$\frac{1}{5}\tan^5 x-\frac{1}{3}\tan^3 x + C$$
$$\frac{1}{3}\sec^3 x\tan^2 x + C$$
$$\frac{1}{5}\sec^5 x-\frac{1}{3}\sec^3 x + C$$
$$\frac{1}{5}\sec^5 x+\frac{1}{3}\sec^3 x + C$$
Q35.
Find $\lim_{x\to\infty}(\ln x)^{1/x}$.
$\ln x$
$1$
$0$
$\infty$
$e$
Q36.
A fair coin is tossed three times. Find $P(\text{at least two heads} \mid \text{at most two heads})$.
3/7
2/7
4/7
3/8
1/2
Q37.
Let $h$ be ``John is healthy,' $w$ be ``John is wealthy,' and $s$ be ``John is wise.' Translate the statement into symbolic notation: John is neither wealthy nor wise, but he is healthy.
$h \lor \neg w \lor \neg s$
$\neg h \land \neg w \land \neg s$
$h \land \neg w \land \neg s$
$h \land \neg(w \land s)$
$\neg h \lor (\neg w \land \neg s)$
Q38.
Let $p$ be ``DATAENDFLAG is off,' $q$ be ``ERROR equals $0$,' and $r$ be ``SUM is less than $1000$.' Translate the statement into symbolic notation: DATAENDFLAG is on and ERROR equals $0$ but SUM is greater than or equal to $1000$.
$\neg p \land q \land \neg r$
$\neg(p \land q \land r)$
$p \land q \land \neg r$
$\neg p \lor q \land \neg r$
$\neg p \land q \land r$
Q39.
Evaluate $\int_{0}^{1}\frac{4}{\sqrt{4-s^{2}}}\,ds$.
$\frac{\pi}{2}$
$\frac{2\pi}{3}$
$\pi$
$\frac{\pi}{3}$
$\frac{4\pi}{3}$
Q40.
Over its natural domain, find extrema of $y=\sqrt{x^2-1}$.
No extrema.
Absolute maximum $0$ at $x=\pm1$; no minimum.
Absolute minimum $0$ at $x=\pm1$; no maximum.
Absolute minimum $-1$ at $x=0$; no maximum.
Absolute maximum at $x=\pm1$; absolute minimum at infinity.
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