Exam

Q2.
A fair coin is tossed three times. Let $X$ be the number of heads minus the number of tails. Which pmf is correct?

$P(X=-3)=\tfrac18$, $P(X=-1)=\tfrac38$, $P(X=1)=\tfrac38$, $P(X=3)=\tfrac18$

$P(X=-1)=\tfrac12$, $P(X=1)=\tfrac12$

$P(X=-2)=\tfrac18$, $P(X=0)=\tfrac38$, $P(X=2)=\tfrac38$, $P(X=4)=\tfrac18$

$P(X=-3)=\tfrac18$, $P(X=-1)=\tfrac14$, $P(X=1)=\tfrac12$, $P(X=3)=\tfrac18$

$P(X=-3)=\tfrac14$, $P(X=-1)=\tfrac14$, $P(X=1)=\tfrac14$, $P(X=3)=\tfrac14$

Q3. Find a basis for the set of vectors in $\mathbb{R}^3$ lying in the plane $x+2y+z=0$.

$\left\{(1,2,1)\right\}$

$\left\{(-2,1,1),(-1,1,0)\right\}$

$\left\{(-2,1,0),(-1,0,1)\right\}$

$\left\{(-2,0,1),(0,1,-2)\right\}$

$\left\{(2,-1,0),(1,0,1)\right\}$

Q4. Find the tangent plane and a normal line to the surface $x\ln y+y\ln z=x$ at the point $(2,1,e)$.

Tangent plane: $-ex+3ey+z-2e=0$; normal line: $x=2+et$, $y=1+3et$, $z=e+t$

Tangent plane: $-ex+3ey+z-e=0$; normal line: $x=2-et$, $y=1+3et$, $z=e+t$

Tangent plane: $-ex+3ey-z-2e=0$; normal line: $x=2-et$, $y=1+3et$, $z=e+t$

Tangent plane: $-ex+3ey+z-2e=0$; normal line: $x=2-et$, $y=1+3et$, $z=e+t$

Tangent plane: $ex+3ey+z-2e=0$; normal line: $x=2-et$, $y=1+3et$, $z=e+t$

Q5. Compute $\lim_{x\to \pi/2} \frac{\sec x}{\tan x}$.

$0$

$1$

Infinity

Does not exist

$-1$

Q6. Surface area: revolve $y=\sqrt{x}$, $\frac34\le x\le \frac{15}{4}$, about the $x$-axis.

$\frac{56\pi}{3}$

$\frac{28\pi}{3}$

$28\pi$

$\frac{28}{3}$

$\frac{14\pi}{3}$

Q7. Solve the system
$-5x_1+2x_2=9$,
$3x_1-x_2=-4$.

$(x_1,x_2)=(1,-7)$

$(x_1,x_2)=(-1,7)$

$(x_1,x_2)=(1,7)$

$(x_1,x_2)=(7,1)$

$(x_1,x_2)=(-1,-7)$

Q8. Assume that sequence converges and find its limit. $$\sqrt{1},\;\sqrt{1+\sqrt{1}},\;\sqrt{1+\sqrt{1+\sqrt{1}}},\;\ldots$$

$$2$$

$$\frac{1+\sqrt{5}}{2}$$

$$\frac{1-\sqrt{5}}{2}$$

$$\sqrt{2}$$

$$1+\sqrt{2}$$

Q9. Determine a suitable form for $Y(t)$ if the method of undetermined coefficients is to be used for $y^{(4)}-2y^{\prime\prime}+y=e^t+\sin t$.

$Y(t)=At^2e^t+B\cos t+C\sin t$

$Y(t)=At^2e^t+t(B\cos t+C\sin t)$

$Y(t)=Ate^t+B\cos t+C\sin t$

$Y(t)=At^2e^t+B\sin t$

$Y(t)=Ae^t+B\cos t+C\sin t$

Q10. Rewrite the statement without variables or quantifiers, and indicate whether it is true or false:
$\exists x$ such that $\operatorname{Real}(x)\land\neg\operatorname{Int}(x)$

"Some integers are not real numbers"; false

"There exists an integer that is not real"; true

"There exists a real number that is not an integer"; true

"Every real number is not an integer"; true

"There exists a real number that is not an integer"; false

Q11. 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

Q12. Which statement is the negation of: "If $x$ is nonnegative, then $x$ is positive or $x=0$"?

$x$ is nonnegative or $x$ is positive.

$x$ is negative and $x\ne0$.

$x$ is nonnegative, $x$ is not positive, and $x\ne0$.

$x$ is nonnegative and $x=0$.

If $x=0$, then $x$ is nonnegative.

Q13. Let $S$ be the set of students at your school, let $M$ be the set of movies that have ever been released, and let $V(s,m)$ mean "student $s$ has seen movie $m$." Rewrite the statement
$\exists s\in S$ such that $V(s,\text{Casablanca})$.

At least one student at your school has seen Casablanca.

There is a movie that every student at your school has seen, namely Casablanca.

Every student at your school has seen Casablanca.

No student at your school has seen Casablanca.

Some movie has been seen by a student at your school.

Q14. For the subspace
$\left\{\begin{pmatrix}4s\\-3t\\-t\end{pmatrix}: s,t\in\mathbb{R}\right\}$,
find a basis and its dimension.

A basis is $\left\{\begin{pmatrix}4\\-3\\-1\end{pmatrix},\begin{pmatrix}0\\1\\0\end{pmatrix}\right\}$ and the dimension is $2$.

A basis is $\left\{\begin{pmatrix}0\\-3\\-1\end{pmatrix}\right\}$ and the dimension is $1$.

A basis is $\left\{\begin{pmatrix}4\\0\\0\end{pmatrix},\begin{pmatrix}0\\-3\\-1\end{pmatrix}\right\}$ and the dimension is $2$.

A basis is $\left\{\begin{pmatrix}4\\0\\0\end{pmatrix}\right\}$ and the dimension is $1$.

A basis is $\left\{\begin{pmatrix}4\\0\\0\end{pmatrix},\begin{pmatrix}0\\-3\\-1\end{pmatrix},\begin{pmatrix}0\\0\\1\end{pmatrix}\right\}$ and the dimension is $3$.

Q15. Find the general solution of $2y^{\prime\prime}+3y^{\prime}+y=t^2+3\sin t$.

$y=C_1e^{-t}+C_2e^{-t/2}+t^2-6t+14-\dfrac{9}{10}\cos t-\dfrac{3}{10}\sin t$

$y=C_1e^{-t}+C_2e^{-t/2}+t^2+6t+14-\dfrac{9}{10}\cos t-\dfrac{3}{10}\sin t$

$y=C_1e^{t}+C_2e^{t/2}+t^2-6t+14-\dfrac{9}{10}\cos t-\dfrac{3}{10}\sin t$

$y=C_1e^{-t}+C_2e^{-t/2}+\dfrac{1}{2}t^2-6t+14-\dfrac{9}{10}\cos t-\dfrac{3}{10}\sin t$

$y=C_1e^{-t}+C_2e^{-t/2}+t^2-6t+14+\dfrac{9}{10}\cos t-\dfrac{3}{10}\sin t$

Q16. Assume $t>0$. Solve the system $t\mathbf{x}^{\prime}=\begin{pmatrix}2&-1\\3&-2\end{pmatrix}\mathbf{x}$.

$\mathbf{x}(t)=C_1t^{-1}\begin{pmatrix}3\\1\end{pmatrix}+C_2t\begin{pmatrix}1\\1\end{pmatrix}$

$\mathbf{x}(t)=C_1t^{-1}\begin{pmatrix}1\\3\end{pmatrix}+C_2t^2\begin{pmatrix}1\\1\end{pmatrix}$

$\mathbf{x}(t)=C_1t^{-1}\begin{pmatrix}1\\3\end{pmatrix}+C_2t\begin{pmatrix}1\\1\end{pmatrix}$

$\mathbf{x}(t)=C_1t\begin{pmatrix}1\\3\end{pmatrix}+C_2t^{-1}\begin{pmatrix}1\\1\end{pmatrix}$

$\mathbf{x}(t)=C_1t^{-2}\begin{pmatrix}1\\3\end{pmatrix}+C_2t\begin{pmatrix}1\\1\end{pmatrix}$

Q17. A politician previously had $65\%$ male supporters. A new poll of $120$ current supporters found that $72$ were men. Using $\alpha=0.05$, test the one-sided hypothesis that the proportion of male supporters has remained the same, with alternative that it has decreased. Which option is correct?

Test $H_0:p=0.60$ versus $H_1:p<0.65$. With $\hat p=0.65$, the statistic is $z\approx 1.148$. Fail to reject $H_0$.

Test $H_0:p=0.65$ versus $H_1:p<0.65$. With $\hat p=0.60$, the statistic is $z\approx -1.148$. The $P$-value is about $0.0250$, so reject $H_0$.

Test $H_0:p=0.65$ versus $H_1:p<0.65$. With $\hat p=0.60$, the statistic is $z=\dfrac{0.60-0.65}{0.65(0.35)/120}\approx -26.37$, so reject $H_0$.

Test $H_0:p=0.65$ versus $H_1:p<0.65$. With $\hat p=72/120=0.60$, the statistic is $z=\dfrac{0.60-0.65}{\sqrt{0.65(0.35)/120}}\approx -1.148$. The $P$-value is about $0.1254$, so fail to reject $H_0$.

Test $H_0:p=0.65$ versus $H_1:p\ne0.65$. With $\hat p=0.60$, the statistic is $z=\dfrac{0.60-0.65}{\sqrt{0.60(0.40)/120}}\approx -1.118$, so reject $H_0$ at $\alpha=0.05$.

Q18. If $(0.57, 0.63)$ is a $50\%$ confidence interval for $p$, determine $\dfrac{k}{n}$ and the sample size $n$.

$\dfrac{k}{n}=0.60,\ n=120$

$\dfrac{k}{n}=0.60,\ n=121$

$\dfrac{k}{n}=0.63,\ n=120$

$\dfrac{k}{n}=0.50,\ n=100$

$\dfrac{k}{n}=0.57,\ n=120$

Q19. If $\lim_{x\to c}(f(x)+g(x))=3$ and $\lim_{x\to c}(f(x)-g(x))=-1$, find $\lim_{x\to c} f(x)g(x)$.

-2

Q20. 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 healthy, wealthy, nor wise.

$\neg h \land (\neg w \lor \neg s)$

$\neg(h \land w \land s)$

$\neg h \land \neg w \land \neg s$

$h \land w \land s$

$\neg h \lor \neg w \lor \neg s$

Q21. Find the volume of the region cut from the solid cylinder $x^2+y^2\le 1$ by the sphere $x^2+y^2+z^2=4$.

$\frac{4\pi}{3}(4-\sqrt{3})$

$\frac{4\pi}{3}(8-3\sqrt{3})$

$\frac{2\pi}{3}(8-3\sqrt{3})$

$\frac{8\pi}{3}(8-3\sqrt{3})$

$2\pi(8-3\sqrt{3})$

Q22. Evaluate the indefinite integral: $$\int (e^x-e^{-x})(e^x+e^{-x})^3\,dx$$

$$\frac{1}{4}(e^x-e^{-x})^4+C$$

$$\frac{1}{4}(e^x+e^{-x})^2+C$$

$$(e^x+e^{-x})^4+C$$

$$\frac{1}{3}(e^x+e^{-x})^3+C$$

$$\frac{1}{4}(e^x+e^{-x})^4+C$$

Q23. Find the sum of the series $\displaystyle \sum_{n=1}^{\infty}\left(\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}\right)$.

$1$

$\sqrt{2}-1$

$0$

Diverges

$\frac{1}{2}$

Q24. Determine the order of the differential equation and whether it is linear or nonlinear: $\displaystyle \frac{dy}{dt}+ty^2=0$.

Third-order nonlinear

First-order nonlinear

First-order linear

Second-order linear

Second-order nonlinear

Q25. Two half-cylinders of diameter $2$ intersect at right angles. Find the volume of the region common to both.

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

$\frac{8}{3}$

$\frac{16\pi}{3}$

$\frac{4}{3}$

$\frac{16}{3}$

Q26. An urn contains 10 chips. We test $H_0$: exactly half the chips are white versus $H_1$: more than half the chips are white. Three chips are drawn without replacement, and $H_0$ is rejected if at least 2 of the 3 are white. Find $\alpha$, and find $\beta$ when the urn is (a) $60\%$ white and (b) $70\%$ white.

$\alpha=\dfrac{1}{2},\ \beta(0.6)=0.352,\ \beta(0.7)=0.216$

$\alpha=\dfrac{1}{2},\ \beta(0.6)=\dfrac{3}{10},\ \beta(0.7)=\dfrac{7}{40}$

$\alpha=\dfrac{5}{12},\ \beta(0.6)=\dfrac{1}{3},\ \beta(0.7)=\dfrac{11}{60}$

$\alpha=\dfrac{7}{12},\ \beta(0.6)=\dfrac{2}{3},\ \beta(0.7)=\dfrac{49}{60}$

$\alpha=\dfrac{1}{2},\ \beta(0.6)=\dfrac{1}{3},\ \beta(0.7)=\dfrac{11}{60}$

Q27. A proof of the statement "The product of an even integer and an odd integer is even" begins: "Suppose $m$ is even and $n$ is odd. If $mn$ is even, then..." What is the main mistake?

Odd numbers cannot be written in the form $2q+1$.

Even numbers cannot be written in the form $2p$.

The product of an even and an odd integer is actually odd.

It assumes the conclusion by starting with "If $mn$ is even".

The proof should have used contradiction, because direct proof is impossible.

Q28. Which proof strategy is best for the statement: "The product of any nonzero rational number and any irrational number is irrational"?

Disprove it with the example $q=1$ and $r=\sqrt2$.

Use contraposition: if $qr$ is irrational, then $q$ is rational.

Use induction on all rational numbers.

Use contradiction: if $q\ne0$ is rational and $r$ is irrational but $qr$ were rational, then $r=\frac{qr}{q}$ would be rational.

Check only the case $q=2$ and $r=\sqrt2$.

Q29. Use reduction of order to solve $(2-t)y^{\prime\prime\prime}+(2t-3)y^{\prime\prime}-ty^{\prime}+y=0$, $t<2$, given that $y_1(t)=e^t$.

$y=C_1e^t+C_2te^{-t}+C_3t$

$y=C_1e^t+C_2te^t+C_3$

$y=C_1e^{-t}+C_2te^t+C_3t$

$y=C_1e^t+C_2te^t+C_3t^2$

$y=C_1e^t+C_2te^t+C_3t$

Q30. Find a parametrization for the circle $x^2+y^2=1$ starting at $(1,0)$ and moving counterclockwise to the point $(0,1)$, using the angle $\theta$ as the parameter.

$x=\cos\theta,\ y=\sin\theta,\ 0\le\theta\le\dfrac{\pi}{2}$

$x=\cos\theta,\ y=\sin\theta,\ -\dfrac{\pi}{2}\le\theta\le 0$

$x=\cos\theta,\ y=-\sin\theta,\ 0\le\theta\le\dfrac{\pi}{2}$

$x=\sin\theta,\ y=\cos\theta,\ 0\le\theta\le\dfrac{\pi}{2}$

$x=\cos\theta,\ y=\sin\theta,\ 0\le\theta\le\pi$

Q31. Find the outward flux of the field $F=2xy\mathbf{i}+2yz\mathbf{j}+2xz\mathbf{k}$ across the surface of the cube cut from the first octant by the planes $x=a$, $y=a$, and $z=a$.

$2a^4$

$3a^3$

$3a^4$

$6a^4$

$a^4$

Q32. Find the volume of the parallelepiped with one vertex at the origin and adjacent vertices at $(1,0,-3)$, $(1,2,4)$, and $(5,1,0)$.

$23$

$27$

$21$

$18$

$24$

Q33. Convert the polar equation $r\cos\theta+r\sin\theta=1$ to an equivalent Cartesian equation and identify the graph.

$x^2+y^2=1$ (a circle)

$x+y=1$ (a line)

$y=\frac{1}{x}$ (a hyperbola)

$x-y=1$ (a line)

$x+y=-1$ (a line)

Q34. How many arrangements of the letters in the word FLEEMOSTYNARY are there such that the S is immediately followed by a Y?

199584000

239500800

217728000

479001600

119750400

Q35. An urn contains twenty chips numbered 1 through 20. Two are drawn simultaneously. What is the probability that the numbers differ by more than 2?

$\dfrac{76}{95}$

$\dfrac{37}{190}$

$\dfrac{19}{20}$

$\dfrac{9}{10}$

$\dfrac{153}{190}$

Q36. Use the method of moments to estimate $\theta$ in the pdf $f_Y(y;\theta)=(\theta^2+\theta)y^{\theta-1}(1-y)$, $0\le y\le 1$, for a random sample of size $n$.

$\hat{\theta}_{\text{MM}}=\dfrac{\bar Y}{1-\bar Y}$

$\hat{\theta}_{\text{MM}}=\dfrac{1-\bar Y}{2\bar Y}$

$\hat{\theta}_{\text{MM}}=\dfrac{2\bar Y}{1-\bar Y}$

$\hat{\theta}_{\text{MM}}=\dfrac{2(1-\bar Y)}{\bar Y}$

$\hat{\theta}_{\text{MM}}=\bar Y(2-\bar Y)$

Q37. Assume that all matrices mentioned below have appropriate sizes. Which statements are true?

I. If $A$ and $B$ are $m\times n$, then both $AB^T$ and $A^TB$ are defined.
II. If $AB=C$ and $C$ has $2$ columns, then $A$ has $2$ columns.
III. Left-multiplying a matrix $B$ by a diagonal matrix $A$ with nonzero entries on the diagonal scales the rows of $B$.
IV. If $BC=BD$, then $C=D$.
V. If $AC=0$, then either $A=0$ or $C=0$.

I and III only

II and V only

III, IV, and V only

I, II, and III only

I and IV only

Q38. Evaluate the iterated integral $\int_1^4\int_0^4 \left(\frac{x}{2}+\sqrt{y}\right)dx\,dy$.

$32$

$\frac{80}{3}$

$\frac{88}{3}$

$\frac{92}{3}$

$28$

Q39. Country A launches 10 guided missiles, 6 of which have nuclear warheads. Country B fires 5 antimissile rockets, each destroying exactly one incoming missile uniformly at random. What is the probability that Country B is hit by at least one nuclear missile?

$1$

$\dfrac{63}{64}$

$\dfrac{1}{2}$

$\dfrac{5}{6}$

$\dfrac{4}{5}$

Q40. Differentiate $y=\dfrac{1}{(x^2-1)(x^2+x+1)}$.

$\dfrac{4x^3+3x^2-1}{(x^2-1)(x^2+x+1)}$

$-\dfrac{4x^3+3x^2-1}{(x^2-1)^2(x^2+x+1)^2}$

$\dfrac{4x^3+3x^2-1}{(x^2-1)^2(x^2+x+1)^2}$

$-\dfrac{(2x)(x^2+x+1)+(x^2-1)(2x+1)}{(x^2-1)^2(x^2+x+1)^2}$

$-\dfrac{4x^3+3x^2-1}{(x^2-1)(x^2+x+1)^2}$

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