Final Attestation Exam for the B.Ed. in Mathematics 6B01501
Q1.
Evaluate $\int \frac{\sin\sqrt{\theta}}{\sqrt{\theta}\cos^3\sqrt{\theta}}\,d\theta$.
$\tan^2(\sqrt{\theta})+C$
$\sec^2(\sqrt{\theta})+C$
$\tan(\sqrt{\theta})+C$
$-\tan^2(\sqrt{\theta})+C$
$\tan^2(\theta)+C$
Q2.
Find where $g(x)=x^2\sqrt{5-x}$ is increasing/decreasing (domain $x\le5$).
Decreasing on $(-\infty,0)$ and $(4,5)$; increasing on $(0,4)$.
Decreasing on $(-\infty,4)$; increasing on $(4,5)$.
Decreasing on $(-\infty,5)$ only.
Increasing on $(-\infty,0)$ and $(4,5)$; decreasing on $(0,4)$.
Increasing on $(0,5)$ only.
Q3.
Two vertices in a graph are called adjacent if, and only if, which condition holds?
They are both endpoints of two different edges.
They are connected by an edge.
They have the same degree.
They are both isolated.
They lie in the same graph.
Q4.
A circle of radius $r$ rests on a horizontal line and is tangent to an inclined segment of length $\alpha$ that makes an angle $\theta$ with the horizontal. Express $r$ in terms of $\alpha$ and $\theta$.
$r=\dfrac{\alpha\cos\theta}{1-\sin\theta}$
$r=\dfrac{\alpha\sin\theta}{1+\sin\theta}$
$r=\dfrac{\alpha}{1-\sin\theta}$
$r=\dfrac{1-\sin\theta}{\sin\theta}$
$r=\dfrac{\alpha\sin\theta}{1-\sin\theta}$
Q5.
A pond contains $1{,}000{,}000$ gal of water. Water containing $0.01$ g of chemical per gallon flows in at $300$ gal/min, and the well-mixed solution flows out at the same rate. If $y(t)$ is the amount of chemical in the pond (in grams), which differential equation models $y$?
$\displaystyle \frac{dy}{dt}=3-\frac{y}{1000000}$
$\displaystyle \frac{dy}{dt}=3-\frac{3}{10000}y$
$\displaystyle \frac{dy}{dt}=0.01-\frac{3}{10000}y$
$\displaystyle \frac{dy}{dt}=300-\frac{3}{10000}y$
$\displaystyle \frac{dy}{dt}=\frac{3}{10000}y-3$
Q6.
Water drains from the conical tank at $5$ ft$^3$/min. The cone has height $10$ ft and top radius $4$ ft (see figure). How fast is the water level changing when $h=6$ ft?
$\dfrac{dh}{dt}=-\dfrac{125}{144\pi}$ ft/min
$\dfrac{dh}{dt}=-\dfrac{125}{72\pi}$ ft/min
$\dfrac{dh}{dt}=-\dfrac{5}{144\pi}$ ft/min
$\dfrac{dh}{dt}=-\dfrac{5}{4\pi}$ ft/min
$\dfrac{dh}{dt}=-\dfrac{25}{24\pi}$ ft/min
Q7.
Determine the order of the differential equation and whether it is linear or nonlinear: $\displaystyle \frac{d^3y}{dt^3}+t\frac{dy}{dt}+(\cos^2 t)y=t^3$.
First-order linear
Second-order nonlinear
Second-order linear
Third-order nonlinear
Third-order linear
Q8.
Healthy eighteen-year-old women have systolic blood pressure mean $120$ mm Hg and standard deviation $12$ mm Hg. A sample of $50$ freshman women on the day of the final exam has average blood pressure $125.2$. To test whether exam stress elevates blood pressure, which option gives the correct hypotheses, test statistic, and conclusion?
$H_0:\mu=120$ versus $H_1:\mu<120$. The test statistic is $z\approx -3.064$, so reject $H_0$ and conclude blood pressure is lower during finals.
$H_0:\mu=120$ versus $H_1:\mu>120$. The test statistic is $z=\dfrac{125.2-120}{12/50}\approx 21.667$, so reject $H_0$.
$H_0:\mu=120$ versus $H_1:\mu>120$. The test statistic is $z=\dfrac{125.2-120}{12/\sqrt{50}}\approx 3.064$, with one-sided $P$-value $\approx 0.0011$. Reject $H_0$; there is strong evidence that final-exam stress elevates mean blood pressure.
$H_0:\mu=125.2$ versus $H_1:\mu>125.2$. The test statistic is $z=\dfrac{125.2-120}{12/\sqrt{50}}\approx 3.064$, so reject $H_0$.
$H_0:\mu=120$ versus $H_1:\mu\ne120$. The test statistic is $z\approx 3.064$, with one-sided $P$-value $\approx 0.0011$. Fail to reject $H_0$.
Q9.
Compute $\begin{bmatrix}8&3&-4\\5&1&2\end{bmatrix}\begin{bmatrix}1\\1\\1\end{bmatrix}$.
$\begin{bmatrix}-7\\8\end{bmatrix}$
$\begin{bmatrix}1\\8\end{bmatrix}$
$\begin{bmatrix}7\\6\end{bmatrix}$
$\begin{bmatrix}7\\8\end{bmatrix}$
$\begin{bmatrix}15\\8\end{bmatrix}$
Q10.
Let $T$ be the congruence modulo 3 relation on $\mathbb{Z}$:
$ m T n \iff 3 \mid (m-n) $.
Which set gives five integers $n$ such that $n T 0$?
${-6,-3,0,3,6}$
${-4,-1,2,5,8}$
${-6,-2,0,3,7}$
${-5,-2,1,4,7}$
${-3,0,1,3,6}$
Q11.
Solve the matrix equation $Ax=b$ for $A=\begin{bmatrix}3&-7&-2\\-3&5&1\\6&-4&0\end{bmatrix}$ and $b=\begin{bmatrix}-7\\5\\2\end{bmatrix}$.
$x=\begin{bmatrix}-3\\4\\-6\end{bmatrix}$
$x=\begin{bmatrix}3\\4\\-6\end{bmatrix}$
$x=\begin{bmatrix}3\\4\\6\end{bmatrix}$
$x=\begin{bmatrix}3\\-4\\6\end{bmatrix}$
$x=\begin{bmatrix}4\\3\\-6\end{bmatrix}$
Q12.
Find the tangent plane and a normal line to the surface $x^2-xy-y^2-z=0$ at the point $(1,1,-1)$.
Tangent plane: $x+3y-z+1=0$; normal line: $x=1+t$, $y=1-3t$, $z=-1-t$
Tangent plane: $x-3y-z+1=0$; normal line: $x=1+t$, $y=1-3t$, $z=-1-t$
Tangent plane: $x-3y+z+1=0$; normal line: $x=1+t$, $y=1-3t$, $z=-1-t$
Tangent plane: $x-3y-z=0$; normal line: $x=1+t$, $y=1-3t$, $z=-1-t$
Tangent plane: $x-3y-z+1=0$; normal line: $x=1+t$, $y=1+3t$, $z=-1-t$
Q13.
Which parametric equations describe the line through $(1,1,1)$ parallel to the $z$-axis?
$x=1+t,\ y=1,\ z=1$
$x=t,\ y=1,\ z=1+t$
$x=1,\ y=1,\ z=1+t$
$x=1+t,\ y=1+t,\ z=1+t$
$x=1,\ y=1+t,\ z=1$
Q14.
Solve the Cauchy-Euler equation $t^2y^{\prime\prime}+ty^{\prime}+y=0$ for $t>0$.
$y=C_1t+C_2t^{-1}$
$y=C_1\cos(\ln t)+C_2\sin(\ln t)$
$y=C_1\cos t+C_2\sin t$
$y=t\left(C_1\cos(\ln t)+C_2\sin(\ln t)\right)$
$y=t^{-1}\left(C_1\cos(\ln t)+C_2\sin(\ln t)\right)$
Q15.
Assume the domain is all people, and let $T(x,y)$ mean "$x$ trusts $y$."
Which pair gives a correct formalization and its negation for "Everybody trusts somebody"?
$\exists x\,\forall y\,T(x,y)$; negation: $\forall x\,\exists y\,\neg T(x,y)$
$\forall x\,\exists y\,T(x,y)$; negation: $\exists x\,\forall y\,\neg T(x,y)$
$\forall x\,\exists y\,T(x,y)$; negation: $\forall x\,\exists y\,\neg T(x,y)$
$\exists x\,\exists y\,T(x,y)$; negation: $\forall x\,\forall y\,\neg T(x,y)$
$\forall x\,\forall y\,T(x,y)$; negation: $\exists x\,\exists y\,\neg T(x,y)$
Q16.
A display case contains 35 gems, 10 real diamonds and 25 fake diamonds. Four gems are removed at random, one at a time without replacement. What is the probability that the last gem stolen is the second real diamond among the four stolen?
$\dfrac{675}{5236}$
$\dfrac{5}{37}$
$\dfrac{225}{1309}$
$\dfrac{45}{374}$
$\dfrac{9}{70}$
Q17.
A coin is tossed four times. Let
$A$ be the event that exactly two heads appear,
$B$ be the event that heads and tails alternate,
and
$C$ be the event that the first two tosses are heads.
Which subset relation is true?
$B \subseteq A$
$A \subseteq B$
$C \subseteq B$
None of $A$, $B$, $C$ is a subset of another
$C \subseteq A$
Q18.
Five independent samples are drawn from a normal distribution with known $\sigma$. For each sample, the interval $(\bar y-0.96\cdot \sigma/\sqrt{n},\ \bar y+1.06\cdot \sigma/\sqrt{n})$ is constructed. What is the probability that at least four of the five intervals contain the unknown mean $\mu$?
$0.4718$ approximately
$0.5932$ approximately
$0.5014$ approximately
$0.6869$ approximately
$0.3435$ approximately
Q19.
A triangle has sides $a=2$ and $b=3$ with included angle $C=60^\circ$. Find the length of side $c$.
7
5
$\sqrt{13}$
$\sqrt{19}$
$\sqrt{7}$
Q20.
A sequence satisfies
$c_k=c_{k-1}+6c_{k-2}$ for all integers $k\ge 2$,
with $c_0=0$ and $c_1=3$.
Which explicit formula is correct?
$c_n=\frac{1}{5}\left(3^n-(-2)^n\right)$
$c_n=3\left(3^n-2^n\right)$
$c_n=\frac{3}{5}\left(3^n+(-2)^n\right)$
$c_n=3^n-(-2)^n$
$c_n=\frac{3}{5}\left(3^n-(-2)^n\right)$
Q21.
Let $T:\mathbb{R}^3\to\mathbb{R}^2$ be linear with $T(e_1)=(1,3)$, $T(e_2)=(4,-7)$, and $T(e_3)=(-5,4)$. Find the standard matrix of $T$.
$\begin{bmatrix}1&4&-5\\3&-7&4\end{bmatrix}$
$\begin{bmatrix}1&-5&4\\3&4&-7\end{bmatrix}$
$\begin{bmatrix}1&4\\3&-7\\-5&4\end{bmatrix}$
$\begin{bmatrix}1&4&-5\\-3&7&-4\end{bmatrix}$
$\begin{bmatrix}1&3\\4&-7\\-5&4\end{bmatrix}$
Q22.
Without solving the problem, determine an interval in which the solution of the initial value problem $y'+(\tan t)y=\sin t$, $y(\pi)=0$, is certain to exist and be unique.
$(\pi,2\pi)$
$(0,\pi)$
$(-\infty,\infty)$
$\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)$
$\left(\dfrac{\pi}{2},\dfrac{3\pi}{2}\right)$
Q23.
Find all point(s) on $y=\dfrac{x}{x-2}$ where the tangent line is perpendicular to $y=2x+3$.
$(4,2)$ only
$(2,1)$ only
$(0,0)$ and $(4,2)$
$(1,-1)$ and $(3,3)$
$(0,0)$ only
Q24.
Assume that all matrices mentioned below have appropriate sizes. Which statements are true?
I. If $A$ is a $3\times 3$ matrix with three pivot positions, then there exist elementary matrices $E_1,\ldots,E_p$ such that $E_p\cdots E_1A=I$.
II. If $AB=I$, then $A$ is invertible.
III. If $A$ and $B$ are square and invertible, then $AB$ is invertible and $(AB)^{-1}=A^{-1}B^{-1}$.
IV. If $AB=BA$ and $A$ is invertible, then $A^{-1}B=BA^{-1}$.
V. If $A$ is invertible and $r\ne 0$, then $(rA)^{-1}=rA^{-1}$.
I only
I and IV only
II and V only
IV only
I, II, and IV only
Q25.
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) $\dfrac{e-1}{e}$(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) $E(Y\mid X=x)=1$
(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) $\dfrac{e-1}{e+1}$(b) $\dfrac{1}{2}$(c) $f_{Y\mid X}(y\mid x)=e^{-(y+x)}$ for $y\ge x$(d) $E(Y\mid X=x)=x+2$
Q26.
Suppose
$
Y_1=8.3,\ Y_2=4.9,\ Y_3=2.6,\ Y_4=6.5
$
is a random sample of size 4 from the two-parameter uniform pdf
$
f_Y(y;\theta_1,\theta_2)=\frac{1}{2\theta_2}, \qquad \theta_1-\theta_2\le y\le \theta_1+\theta_2.
$
Use the method of moments to estimate $\theta_1$ and $\theta_2$.
$\hat{\theta}_{1e}=4.900, \qquad \hat{\theta}_{2e}\approx 3.6319$
$\hat{\theta}_{1e}=5.575, \qquad \hat{\theta}_{2e}\approx 3.6319$
$\hat{\theta}_{1e}=5.575, \qquad \hat{\theta}_{2e}=2.85$
$\hat{\theta}_{1e}=6.500, \qquad \hat{\theta}_{2e}=2.950$
$\hat{\theta}_{1e}=5.575, \qquad \hat{\theta}_{2e}\approx 4.1940$
Q27.
Compute $\lim_{x\to 0^{+}}\frac{\sqrt{x}}{\sqrt{\sin x}}$.
$\frac12$
$0$
$\sqrt{2}$
$1$
Infinity
Q28.
An urn contains 40 red chips and 60 white chips. Six chips are drawn and discarded, and then a seventh chip is drawn. What is the probability that the seventh chip is red?
$\dfrac{2}{5}$
$\dfrac{1}{2}$
$\dfrac{9}{25}$
$\dfrac{7}{20}$
$\dfrac{3}{10}$
Q29.
Find the volume of the solid enclosed by the cone $z=\sqrt{x^2+y^2}$ between the planes $z=1$ and $z=2$.
$\frac{7\pi}{3}$
$\frac{5\pi}{3}$
$\pi$
$\frac{8\pi}{3}$
$3\pi$
Q30.
Assume $t>0$. Solve the system $t\mathbf{x}^{\prime}=\begin{pmatrix}2&-1\\3&-2\end{pmatrix}\mathbf{x}$.
$\mathbf{x}(t)=C_1t\begin{pmatrix}1\\3\end{pmatrix}+C_2t^{-1}\begin{pmatrix}1\\1\end{pmatrix}$
$\mathbf{x}(t)=C_1t^{-2}\begin{pmatrix}1\\3\end{pmatrix}+C_2t\begin{pmatrix}1\\1\end{pmatrix}$
$\mathbf{x}(t)=C_1t^{-1}\begin{pmatrix}1\\3\end{pmatrix}+C_2t\begin{pmatrix}1\\1\end{pmatrix}$
$\mathbf{x}(t)=C_1t^{-1}\begin{pmatrix}3\\1\end{pmatrix}+C_2t\begin{pmatrix}1\\1\end{pmatrix}$
$\mathbf{x}(t)=C_1t^{-1}\begin{pmatrix}1\\3\end{pmatrix}+C_2t^2\begin{pmatrix}1\\1\end{pmatrix}$
Q31.
Consider the infinite sequence of shaded right triangles in the accompanying diagram. Find the total shaded area.
$\frac{1}{2}$
$\frac{2}{3}$
$\frac{3}{4}$
$1$
$\frac{5}{8}$
Q32.
Evaluate $\displaystyle \lim_{x\to 0^+}\frac{\sin x}{\sin\sqrt{x}}$.
0
1
DNE
$\infty$
$\frac{1}{2}$
Q33.
For the equation $2x+y=7$, determine which statements are true:
(I) For all real numbers $x$, there exists a real number $y$ such that the equation is true.
(II) There exists a real number $x$ such that for all real numbers $y$, the equation is true.
I only
I is false and II is true
Neither I nor II
II only
Both I and II
Q34.
Differentiate with respect to $x$: $y=\ln\left(\frac{(x^2+1)^5}{\sqrt{1-x}}\right)$.
$\frac{10x}{x^2+1}+\frac{1}{2(1-x)}$
$\frac{10}{x^2+1}+\frac{1}{2(1-x)}$
$\frac{5x}{x^2+1}+\frac{1}{2(1-x)}$
$\frac{10x}{x^2+1}+\frac{1}{1-x}$
$\frac{10x}{x^2+1}-\frac{1}{2(1-x)}$
Q35.
Find the Wronskian of two solutions of $(\cos t)y^{\prime\prime}+(\sin t)y^{\prime}-ty=0$ without solving the equation.
$W(t)=C/\cos t$
$W(t)=C\sin t$
$W(t)=C\cos t$
$W(t)=Ce^t\cos t$
$W(t)=C\sec t$
Q36.
Let $\mathbf{u}=\mathbf{i}\times\mathbf{j}$ and $\mathbf{v}=\mathbf{j}\times\mathbf{k}$. Find $\lVert\mathbf{v}\times\mathbf{u}\rVert$ and a unit vector in the direction of $\mathbf{v}\times\mathbf{u}$ (if defined).
$\lVert\mathbf{v}\times\mathbf{u}\rVert=1$; direction $-\mathbf{j}$
$\lVert\mathbf{v}\times\mathbf{u}\rVert=1$; direction $\mathbf{k}$
$\lVert\mathbf{v}\times\mathbf{u}\rVert=0$; direction undefined
$\lVert\mathbf{v}\times\mathbf{u}\rVert=1$; direction $\mathbf{j}$
$\lVert\mathbf{v}\times\mathbf{u}\rVert=\sqrt{2}$; direction $-\mathbf{j}$
Q37.
Suppose $H_0:\mu=240$ is tested against $H_1:\mu<240$ at level $\alpha=0.01$ using a sample of size $25$ from a normal distribution with $\sigma=50$. If the true mean is $\mu=220$, what proportion of the time will the procedure fail to recognize that the mean has dropped to $220$?
$\beta(220)\approx 0.3721$
$\beta(220)\approx 0.9900$
$\beta(220)\approx 0.6279$
$\beta(220)\approx 0.1867$
$\beta(220)\approx 0.5000$
Q38.
Which formula is logically equivalent to $(p \to r) \leftrightarrow (q \to r)$?
$r \lor (\neg p \land \neg q)$
$r \lor (p \land q)$
$r \lor (p \land \neg q) \lor (\neg p \land q)$
$r \lor (p \land q) \lor (\neg p \land \neg q)$
$\neg r \lor (p \land q) \lor (\neg p \land \neg q)$
Q39.
Find the inverse of $\begin{bmatrix}8&6\\5&4\end{bmatrix}$.
$\frac{1}{2}\begin{bmatrix}4&-5\\-6&8\end{bmatrix}$
$\frac{1}{2}\begin{bmatrix}4&-6\\-5&8\end{bmatrix}$
$\frac{1}{2}\begin{bmatrix}4&6\\5&8\end{bmatrix}$
$\frac{1}{2}\begin{bmatrix}8&-6\\-5&4\end{bmatrix}$
$\begin{bmatrix}4&-6\\-5&8\end{bmatrix}$
Q40.
Evaluate $\displaystyle \lim_{x\to 9}\frac{\sin(\sqrt{x}-3)}{x-9}$.
0
$\frac{1}{12}$
DNE
$\frac{1}{3}$
$\frac{1}{6}$
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