Exam

Q1. For the series $\displaystyle \sum_{n=1}^{\infty}\frac{(x-1)^{2n-2}}{(2n-1)!}$, find the radius of convergence and determine where the series converges absolutely and conditionally.

$R=2$; absolute on $(-1,3)$; diverges outside this interval

$R=\infty$; the series converges absolutely for all real $x$; there is no conditional convergence

$R=1$; absolute on $(0,2)$; conditional at $x=0$ and $x=2$

$R=0$; converges only at $x=1$

$R=\infty$; the series converges conditionally for all real $x$

Q2. Find where $f(x)=x-6\sqrt{x-1}$ is increasing/decreasing (domain $x\ge1$).

Decreasing on $(1,9)$; increasing on $(9,\infty)$.

Decreasing on $(1,10)$; increasing on $(10,\infty)$.

Increasing on $(1,10)$; decreasing on $(10,\infty)$.

Increasing on $(1,\infty)$ only.

Decreasing on $(1,\infty)$ only.

Q3. Which statement is the negation of: "If $n$ is prime, then $n$ is odd or $n=2$"?

If $n$ is odd, then $n$ is prime.

$n$ is not prime and $n\ne2$.

$n$ is prime, $n$ is not odd, and $n\ne2$.

$n$ is prime and $n=2$.

$n$ is prime and $n$ is odd.

Q4. Let $X$ and $Y$ have joint pdf $f_{X,Y}(x,y)=2e^{-(x+y)}$ on $0\le x\le y$ and $y\ge 0$. Which option is correct for
(a) $P(Y<1\mid X<1)$,
(b) $P(Y<1\mid X=1)$,
(c) $f_{Y\mid X}(y\mid x)$,
and
(d) $E(Y\mid X=x)$?

(a) $\dfrac{e-1}{e+1}$(b) $\dfrac{1}{2}$(c) $f_{Y\mid X}(y\mid x)=e^{-(y+x)}$ for $y\ge x$(d) $E(Y\mid X=x)=x+2$

(a) $\dfrac{1}{3}$(b) $0$(c) $f_{Y\mid X}(y\mid x)=e^{-(y-x)}$ for $y\ge x$(d) $E(Y\mid X=x)=1-x$

(a) $\dfrac{e-1}{e+1}$(b) $0$(c) $f_{Y\mid X}(y\mid x)=e^{-(y-x)}$ for $y\ge x$(d) $E(Y\mid X=x)=x+1$

(a) $\dfrac{e-1}{e}$(b) $0$(c) $f_{Y\mid X}(y\mid x)=e^{-y}$ for $y\ge 0$(d) $E(Y\mid X=x)=x$

(a) $\dfrac{1}{2}$(b) $e^{-1}$(c) $f_{Y\mid X}(y\mid x)=2e^{-(x+y)}$ for $y\ge x$(d) $E(Y\mid X=x)=1$

Q5. For $f(x)=x^{4/3}$ on $[-1,8]$, find the absolute minimum and maximum values and where they occur.

Min $1$ at $x=-1$; Max $16$ at $x=8$

Min $0$ at $x=0$; Max $16$ at $x=8$

Min $0$ at $x=0$; Max $1$ at $x=-1$

Min $0$ at $x=-1$; Max $16$ at $x=8$

Min $0$ at $x=0$; Max $16$ at $x=-1$

Q6. Assume $R$ and $S$ are relations on a set $A$. Consider the statement:
"If $R$ and $S$ are symmetric, then $R\cap S$ is symmetric."
Which option is correct?

False, because intersections destroy symmetry.

True only when $R$ and $S$ are reflexive.

True, because if $(a,b)$ is in both $R$ and $S$, then $(b,a)$ is in both $R$ and $S$, so it is in $R\cap S$.

False, because $(b,a)$ may fail to be in one of the relations.

True only when $R\cup S$ is symmetric.

Q7. Find the equation of the plane containing the intersecting lines $x=-1+t$, $y=2+t$, $z=1-t$ and $x=1-4s$, $y=1+2s$, $z=2-2s$.

$y+z=3$

$x+y=1$

$y-z=3$

$x+z=0$

$x+y+z=3$

Q8. Given the sample
$
y_1=2.3,\ y_2=1.9,\ y_3=4.6,
$
assume the observations come from
$
f_Y(y;\theta)=\frac{y^3e^{-y/\theta}}{6\theta^4}, \qquad y\ge 0.
$
Find the maximum likelihood estimate of $\theta$.

$\hat{\theta}=\dfrac{8.8}{4}\approx 2.2000$

$\hat{\theta}=\dfrac{8.8}{3}\approx 2.9333$

$\hat{\theta}=\dfrac{8.8}{12^2}\approx 0.0611$

$\hat{\theta}=\dfrac{15}{11}\approx 1.3636$

$\hat{\theta}=\dfrac{11}{15}\approx 0.7333$

Q9. Find the volume of the solid bounded below by the hemisphere $\rho=1$, $z\ge 0$, and above by the cardioid of revolution $\rho=1+\cos\phi$.

$\frac{5\pi}{3}$

$\frac{11\pi}{12}$

$\frac{11\pi}{6}$

$\frac{3\pi}{2}$

$\frac{8\pi}{3}$

Q10. Let $B$ be a $4\times 4$ matrix with $\det B=-1$. Compute $\det(B^5)$.

$1$

$-5$

$-1$

$-3$

$5$

Q11. Use Green's Theorem to find the area enclosed by the limaçon $x=2\cos t-\cos 2t$, $y=2\sin t$, $0\le t\le 2\pi$.

$6\pi$

$4\pi$

$3\pi$

$8\pi$

$2\pi$

Q12. Find the area cut from the first quadrant by the cardioid $r=1+\sin\theta$.

$1+\frac{\pi}{4}$

$1+\frac{\pi}{8}$

$\frac{3\pi}{8}$

$1+\frac{3\pi}{8}$

$2+\frac{3\pi}{8}$

Q13. Diagonalize $A=\begin{pmatrix}5&-3&0&9\\0&3&1&-2\\0&0&2&0\\0&0&0&2\end{pmatrix}$ if possible. Which statement is correct?

$A$ is not diagonalizable because $\lambda=2$ is repeated.

$A$ is diagonalizable with $P=\begin{pmatrix}1&3&-1&-1\\0&2&-1&2\\0&0&1&0\\0&0&0&1\end{pmatrix}$ and $D=\begin{pmatrix}3&0&0&0\\0&5&0&0\\0&0&2&0\\0&0&0&2\end{pmatrix}$, so $A=PDP^{-1}$.

$A$ is diagonalizable with $P=I$ and $D=\operatorname{diag}(5,3,2,2)$.

$A$ is diagonalizable with $P=\begin{pmatrix}1&3&-1&-1\\0&2&-1&2\\0&0&1&0\\0&0&0&1\end{pmatrix}$ and $D=\begin{pmatrix}5&0&0&0\\0&3&0&0\\0&0&2&0\\0&0&0&2\end{pmatrix}$, so $A=PDP^{-1}$.

$A$ is diagonalizable with $P=\begin{pmatrix}1&3&1&-1\\0&2&-1&2\\0&0&1&0\\0&0&0&1\end{pmatrix}$ and $D=\begin{pmatrix}5&0&0&0\\0&3&0&0\\0&0&2&0\\0&0&0&2\end{pmatrix}$, so $A=PDP^{-1}$.

Q14. Solve the initial value problem $\mathbf{x}^{\prime}=\begin{pmatrix}2&-1\\3&-2\end{pmatrix}\mathbf{x}$, $\mathbf{x}(0)=\begin{pmatrix}2\\-1\end{pmatrix}$.

$\mathbf{x}(t)=\begin{pmatrix}\frac{7e^{t}-3e^{-t}}{2}\\\frac{7e^{t}-9e^{-t}}{2}\end{pmatrix}$

$\mathbf{x}(t)=\begin{pmatrix}\frac{7e^{-t}-3e^{t}}{2}\\\frac{7e^{-t}-9e^{t}}{2}\end{pmatrix}$

$\mathbf{x}(t)=\begin{pmatrix}\frac{7e^{t}-3e^{-t}}{2}\\\frac{7e^{t}-3e^{-t}}{2}\end{pmatrix}$

$\mathbf{x}(t)=\begin{pmatrix}\frac{7e^{t}+3e^{-t}}{2}\\\frac{7e^{t}+9e^{-t}}{2}\end{pmatrix}$

$\mathbf{x}(t)=\begin{pmatrix}\frac{7e^{2t}-3e^{-t}}{2}\\\frac{7e^{2t}-9e^{-t}}{2}\end{pmatrix}$

Q15. A chemical engineer wants to study the effect of temperature, pressure, and catalyst concentration on yield. If she uses 2 temperatures, 3 pressures, and 2 catalyst levels, how many runs are needed to observe each temperature-pressure-catalyst combination exactly twice?

Q16. Find the area inside the cardioid $r=a(1+\cos\theta)$, where $a>0$.

$3\pi a^2$

$2\pi a^2$

$\frac{3\pi a}{2}$

$\frac{\pi a^2}{2}$

$\frac{3\pi a^2}{2}$

Q17. 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 $0$.

$p \land \neg q$

$\neg(p \land q)$

$p \land q$

$\neg p \land q$

$p \lor \neg q$

Q18. Evaluate $\displaystyle \lim_{x\to 9}\frac{\sin(\sqrt{x}-3)}{x-9}$.

$\frac{1}{3}$

$\frac{1}{6}$

$\frac{1}{12}$

DNE

Q19. Determine the order of the differential equation and whether it is linear or nonlinear: $\displaystyle t^2\frac{d^2y}{dt^2}+t\frac{dy}{dt}+2y=\sin t$.

First-order linear

Second-order linear

Third-order linear

First-order nonlinear

Second-order nonlinear

Q20. 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 $=\begin{bmatrix}4\\7\end{bmatrix}$, 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 $=\begin{bmatrix}-8\\-5\end{bmatrix}$

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 $=\begin{bmatrix}-8\\-5\end{bmatrix}$

Q21. An engineer models the proportion $y$ of a task completed with the pdf $f_Y(y;\theta)=\theta y^{\theta-1}$, $0<y<1$, $\theta>0$. In a previous project, the observed proportions were $0.77$, $0.82$, $0.92$, $0.94$, and $0.98$. Find the maximum likelihood estimate of $\theta$.

$\hat{\theta}=\dfrac{5}{\sum y_i}\approx 1.1287$

$\hat{\theta}=\dfrac{\sum \ln y_i}{5}\approx -0.1251$

$\hat{\theta}=\dfrac{1}{\bar y}\approx 1.1274$

$\hat{\theta}=\bar y=0.8860$

$\hat{\theta}=-\dfrac{5}{\sum \ln y_i}\approx 7.9965$

Q22. Find the tangent plane to the level surface $x^2+y^2+z=4$ at $P_0=(1,1,2)$ and parametric equations for the normal line there.

Tangent plane: $2x+2y+z=6$; normal line: $x=1+t$, $y=1+2t$, $z=2+2t$

Tangent plane: $2x+2y-z=6$; normal line: $x=1+2t$, $y=1+2t$, $z=2-t$

Tangent plane: $2x+2y+z=6$; normal line: $x=1+2t$, $y=1+2t$, $z=2+t$

Tangent plane: $2x+2y+z=4$; normal line: $x=1+2t$, $y=1+2t$, $z=2+t$

Tangent plane: $x+y+z=4$; normal line: $x=1+t$, $y=1+t$, $z=2+t$

Q23. Find the general solution of $x_1^{\prime}=4x_1-3x_2$, $x_2^{\prime}=8x_1-6x_2$.

$x_1=C_1+3C_2e^{-2t}$, $x_2=2C_1+4C_2e^{-2t}$

$x_1=3C_1+C_2e^{2t}$, $x_2=4C_1+2C_2e^{2t}$

$x_1=3C_1+tC_2$, $x_2=4C_1+2tC_2$

$x_1=3C_1+C_2e^{-2t}$, $x_2=4C_1+2C_2e^{-2t}$

$x_1=3C_1+2C_2e^{-2t}$, $x_2=4C_1+C_2e^{-2t}$

Q24. In a town, $\frac{2}{3}$ of the adult men are married to $\frac{3}{5}$ of the adult women. All marriages are monogamous, and there are at least 100 adult men. What is the least possible number of adult women?

135

120

108

112

100

Q25. Differentiate with respect to $t$: $y=\frac{1+\ln t}{1-\ln t}$.

$\frac{2}{t(1+\ln t)^2}$

$\frac{1}{t(1-\ln t)^2}$

$\frac{2}{t(1-\ln t)^2}$

$\frac{2}{(1-\ln t)^2}$

$\frac{2}{t(1-\ln t)}$

Q26. Let the domain be the set of geometric figures in the plane. Let $\operatorname{Square}(x)$ mean "$x$ is a square" and $\operatorname{Rect}(x)$ mean "$x$ is a rectangle."
Rewrite the statement without variables or quantifiers, and indicate whether it is true or false:
$\exists x$ such that $\operatorname{Rect}(x)\land\neg\operatorname{Square}(x)$

"Every rectangle is not a square"; true

"There exists a square that is not a rectangle"; true

"There exists a rectangle that is not a square"; true

"Every square is a rectangle"; true

"There exists a rectangle that is not a square"; false

Q27. Find the general solution of $y^{\prime\prime}+2y^{\prime}+y=2e^{-t}$.

$y=(C_1+C_2t+t)e^{-t}$

$y=C_1e^{-t}+C_2e^{-2t}+t^2e^{-t}$

$y=(C_1+C_2t+t^2)e^{-t}$

$y=(C_1+C_2t+2t^2)e^{-t}$

$y=(C_1+C_2t+t^2)e^{t}$

Q28. At time $t$ (seconds), two particles on a line have positions $s_1=3t^3-12t^2+18t+5$ and $s_2=-t^3+9t^2-12t$ (meters). When do they have the same velocity?

$t=1$ and $t=2.5$

Never

$t=2.5$ only

$t=1$ only

$t=0$ and $t=1$

Q29. Find the general solution of $2y^{\prime\prime\prime}-4y^{\prime\prime}-2y^{\prime}+4y=0$.

$y=C_1e^{2t}+C_2e^{-t}+C_3e^{-2t}$

$y=C_1\cos t+C_2\sin t+C_3e^{2t}$

$y=C_1e^{-2t}+C_2e^t+C_3e^{-t}$

$y=C_1e^{2t}+C_2e^t+C_3e^{-t}$

$y=(C_1+C_2t)e^{2t}+C_3e^{-t}$

Q30. Find the total area bounded by $y=\frac{x^3}{3}-x$ and $y=\frac{x}{3}$ from $x=-2$ to $x=3$.

$\frac{9}{4}$

$\frac{19}{2}$

$\frac{19}{3}$

$\frac{13}{4}$

$\frac{19}{4}$

Q31. Compute the adjugate of $A=\begin{pmatrix}3&5&4\\1&0&1\\2&1&1\end{pmatrix}$.

$\operatorname{adj}(A)=\begin{pmatrix}-1&-1&5\\1&5&1\\1&7&-5\end{pmatrix}$

$\operatorname{adj}(A)=\begin{pmatrix}-1&-1&5\\1&-5&1\\1&7&-5\end{pmatrix}$

$\operatorname{adj}(A)=\begin{pmatrix}-1&-1&5\\1&-5&1\\-1&-7&5\end{pmatrix}$

$\operatorname{adj}(A)=\begin{pmatrix}1&-1&5\\1&-5&1\\1&7&-5\end{pmatrix}$

$\operatorname{adj}(A)=\begin{pmatrix}-1&1&1\\-1&-5&7\\5&1&-5\end{pmatrix}$

Q32. Which observation completes the induction step for proving that $n(n^2+5)$ is divisible by 6 for every integer $n\ge 0$?

The expression is prime for all $n$.

The difference is $k^2+k+2$, which is always divisible by 6.

The difference between the $(k+1)$ case and the $k$ case is $3(k^2+k+2)$, and $k^2+k+2$ is even.

It is enough to check $n=0$.

The difference is $3k+5$, which is always even.

Q33. One focus of a hyperbola is at $(0,-7)$ and the corresponding directrix is the line $y=-1$. Find an equation of the hyperbola if its eccentricity is $e=2$.

$\displaystyle \frac{x^2}{48}-\frac{(y-1)^2}{16}=1$

$\displaystyle \frac{(y-1)^2}{48}-\frac{x^2}{16}=1$

$\displaystyle \frac{(y+1)^2}{16}-\frac{x^2}{48}=1$

$\displaystyle \frac{(y-1)^2}{16}-\frac{x^2}{48}=1$

$\displaystyle \frac{(y+7)^2}{16}-\frac{x^2}{48}=1$

Q34.
For persons infected with a certain form of malaria, the remission time $Y$ has pdf $f_Y(y)=\tfrac19 y^2$ for $0\le y\le 3$. What is the probability that remission lasts longer than one year?

$\dfrac{8}{9}$

$\dfrac{2}{3}$

$\dfrac{17}{18}$

$\dfrac{26}{27}$

$\dfrac{1}{27}$

Q35. For testing $H_0:\mu=42.9$ versus $H_1:\mu\ne 42.9$ with $\bar y=45.1$, $n=16$, $\sigma=3.2$, and $\alpha=0.01$, which option gives the correct decision rule, test statistic, and conclusion?

Reject $H_0$ if $|z|>2.326$. Here $z=2.75$. Reject $H_0$; there is evidence that $\mu\ne 42.9$.

Reject $H_0$ if $|z|>2.576$. Here $z=\dfrac{45.1-42.9}{3.2/\sqrt{16}}=2.75$. Reject $H_0$; there is evidence that $\mu\ne 42.9$.

Reject $H_0$ if $z<-2.576$. Here $z=-2.75$. Reject $H_0$; there is evidence that $\mu<42.9$.

Reject $H_0$ if $|z|>2.576$. Here $z=\dfrac{45.1-42.9}{3.2}=0.6875$. Fail to reject $H_0$; there is not enough evidence that $\mu\ne 42.9$.

Reject $H_0$ if $|z|>1.960$. Here $z=2.75$. Reject $H_0$; there is evidence that $\mu\ne 42.9$.

Q36. Determine whether $\mathbf{w}=\begin{pmatrix}1\\3\\-4\end{pmatrix}$ is in $\operatorname{Nul}A$, where
$A=\begin{pmatrix}3&-5&-3\\6&-2&0\\-8&4&1\end{pmatrix}$.

Yes, because $A\mathbf{w}=\begin{pmatrix}0\\4\\0\end{pmatrix}$

No, because $A\mathbf{w}=\begin{pmatrix}1\\1\\1\end{pmatrix}$

No, because $A\mathbf{w}=\begin{pmatrix}0\\0\\-4\end{pmatrix}$

Yes, because $A\mathbf{w}=\begin{pmatrix}0\\0\\0\end{pmatrix}$

Yes, because $A\mathbf{w}=\begin{pmatrix}2\\0\\0\end{pmatrix}$

Q37. Company records show that drivers get an average of $32{,}500$ miles on a certain tire. A new polymer is added, and a sample of $15$ drivers reports an average of $33{,}800$ miles. Assume $\sigma=4000$ and test $H_0:\mu=32500$ versus $H_1:\mu>32500$ at $\alpha=0.05$. Which option gives the correct test statistic, critical value, and conclusion?

$z=\dfrac{33800-32500}{4000/15}\approx 4.875$, critical value $z_{0.95}=1.645$, so reject $H_0$; there is enough evidence that the mean mileage increased.

$z=\dfrac{33800-32500}{4000/\sqrt{15}}\approx 1.259$, critical value $z_{0.975}=1.960$, so reject $H_0$; there is enough evidence that the mean mileage increased.

$z=\dfrac{33800-32500}{4000/\sqrt{15}}\approx 1.259$, critical value $z_{0.90}=1.282$, so reject $H_0$ at $\alpha=0.05$.

$z=\dfrac{33800-32500}{4000/\sqrt{15}}\approx 1.259$, critical value $z_{0.95}=1.645$, so reject $H_0$; there is enough evidence that the mean mileage increased.

$z=\dfrac{33800-32500}{4000/\sqrt{15}}\approx 1.259$, critical value $z_{0.95}=1.645$, so fail to reject $H_0$; there is not enough evidence that the mean mileage increased.

Q38. A telephone solicitor calls households in three suburbs. Past contribution rates are 60% for Belle Meade, 55% for Oak Hill, and 35% for Antioch. Her list contains 1000 Belle Meade numbers, 1000 Oak Hill numbers, and 2000 Antioch numbers. If one number is selected at random, what is the probability of getting a donation?

0.4750

0.4625

0.4500

0.4250

0.5000

Q39. Find the directional derivative of $g(x,y)=\dfrac{x-y}{xy+2}$ at the point $(1,-1)$ in the direction of $\mathbf{u}=12\mathbf{i}+5\mathbf{j}$.

$-\dfrac{21}{13}$

$\dfrac{21}{13}$

$21$

$\dfrac{31}{13}$

$\dfrac{11}{13}$

Q40. 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?

$\displaystyle \frac{dV}{dt}=-\frac{k}{V^{2/3}}$

$\displaystyle \frac{dV}{dt}=-kV$

$\displaystyle \frac{d^2V}{dt^2}=-kV^{2/3}$

$\displaystyle \frac{dV}{dt}=-kV^{3/2}$

$\displaystyle \frac{dV}{dt}=-kV^{2/3}$, where $k>0$

Final Attestation Exam for the B.Ed. in Mathematics 6B01501