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1
Algebra
Wrong
\(4x - 9y = 9y + 5\)
\(hy = 2 + 4x\)
In the given system of equations, \(h\) is a constant. If the system has no solution, what is the value of \(h\)?
A
-9
B
0
C
9
18
Correct Answer
No explanation
2
Words in Context
Wrong
The general store was essential to daily life in the rural United States during the 1800s because it provided the supplies that the people living in nearby communities needed. Also, the store was a ______ of information. People socializing at the general store would share news and help spread it throughout their communities.
Which choice completes the text with the most logical and precise word or phrase?
source
Correct Answer
B
rival
C
condition
D
waste
No explanation
3
MCQ
Wrong
Length of \(x = \left(\dfrac{1}{3}\right) t^3\), \(y = \left(\dfrac{1}{2}\right) t^2\) for \(0 \leq t \leq 1\)
A
\(\int \sqrt{t^2 + 1} dt\)
B
\(\int \sqrt{t^2 + t} dt\)
\(\int \sqrt{t^4 + t^2} dt\)
Correct Answer
D
\(\left(\dfrac{1}{2}\right) \int \sqrt{4 + t^4} dt\)
E
\(\left(\dfrac{1}{6}\right) \int t^2 \sqrt{4 t^2 + 9} dt\)
Explanation
\(\dfrac{dx}{dt} = t^2\), \(\dfrac{dy}{dt} = t\). \(L = \int \sqrt{t^4 + t^2} dt\).
4
MCQ
Wrong
What is the x-coordinate of the point of inflection on \(y = \left(\dfrac{1}{3}\right) x^3 + 5 x^2 + 24\)?
A
\(5\)
B
\(0\)
C
\(-\dfrac{10}{3}\)
\(-5\)
Correct Answer
E
\(-10\)
Explanation
\(y'' = 2 x + 10 = 0 \rightarrow x = -5\).
5
Probability > Basic probability rules
Wrong
Nine-tenths of adults change their jobs at least once. The reasons for changing and what they change to are as shown in the following tables.
Assuming independence of all variables, what is the probability that an adult decides to change a job, based on higher salary, and
move into a new field?
| Reason | Career Interests | Higher Salary | Fired | Other Reason |
|---|---|---|---|---|
| Probability | 0.55 | 0.10 | 0.20 | 0.15 |
| Change To | Related Field | New Field | Unemployed | |
| Probability | 0.40 | 0.55 | 0.05 |
0.0495
Correct Answer
B
0.055
C
0.585
D
0.595
E
0.65
Explanation
When events are independent, the probability of their intersection is the product of their probabilities. In this case, \(\binom{9}{10}(0.10)(0.55) = 0.0495\).
6
Derivatives - Differentiability
Wrong
Graph the function \(f(x) = x + |x|\). Zoom in repeatedly, first toward the point \((-1, 0)\) and then toward the origin. What is different about the behavior of \(f\) in the vicinity of these two points? What do you conclude about the differentiability of \(f\)?
My Answer
(No answer)
Correct Answer
No explanation
7
Differentiation - Applied
Wrong
The biomass of a guppy population in a small aquarium is modeled using the Product Rule. Let \(N(t)\) be the number of guppies and \(w(t)\) be the average weight of each guppy at time \(t\), and \(B(t) = N(t) w(t)\) is the total biomass. Use the Product Rule to find \(B'(t)\) and interpret each term.
My Answer
(No answer)
Correct Answer
\(B'(t) = N'(t) w(t) + N(t) w'(t)\). First term: biomass increase from population growth. Second term: biomass increase from individual weight gain.
No explanation
8
Solution Sets of Linear Systems
Wrong
Describe all solutions of \(A \mathbf{x} = \mathbf{0}\) in parametric vector form, where \(A\) is row equivalent to the given matrix.
\(\begin{pmatrix} 1 & -2 & -8 & 3 & 0 & 6 \\ 0 & 0 & 1 & -5 & 0 & 4 \\ 0 & 0 & 0 & 0 & 1 & -7 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix}\)
My Answer
(No answer)
Correct Answer
Write the general solution in parametric vector form identifying free variables
No explanation
9
Implicit Diff - Orthogonal Trajectories
Wrong
Show that the given families of curves are orthogonal trajectories of each other.
\(y = c x^2\), \(\quad x^2 + 2y^2 = k\)
My Answer
(No answer)
Correct Answer
First family: \(y' = 2c x\). Since \(c = \dfrac{y}{x^2}\), \(y'_1 = \dfrac{2y}{x}\). Second family: \(2x + 4y y' = 0\), \(y'_2 = -\dfrac{x}{2y}\). Product: \(\dfrac{2y}{x} \cdot \left(-\dfrac{x}{2y}\right) = -1\).
No explanation
10
Probability > Random variables
Wrong
Suppose \(X\) and \(Y\) are random variables with \(E(X) = 29\), \(\text{var}(X) = 7\), \(E(Y) = 35\), and \(\text{var}(Y) = 9\). What are the expected value and variance of the random variable \(X + Y\)?
A
\(E(X + Y) = 32\), \(\text{var}(X + Y) = 8\)
B
\(E(X + Y) = 64\), \(\text{var}(X + Y) = 8\)
C
\(E(X + Y) = 64\), \(\text{var}(X + Y) = 16\)
D
\(E(X + Y) = 64\), \(\text{var}(X + Y) = \sqrt{7^2 + 9^2}\)
There is insufficient information to answer this question.
Correct Answer
Explanation
\(E(X + Y) = 29 + 35 = 64\). However, without independence we cannot determine \(\text{var}(X + Y)\) from the information given.
11
MCQ
Wrong
If \(f(x) = \dfrac{x^2}{e^x}\), then \(f'(1) =\)
A
\(0\)
\(\dfrac{1}{e}\)
Correct Answer
C
\(\dfrac{2}{e}\)
D
\(2\)
E
\(2 e\)
Explanation
\(f(x) = x^2 e^{-x}\). \(f'(x) = 2 x e^{-x} - x^2 e^{-x} = e^{-x}(2 x - x^2)\). At \(x = 1\): \(e^{-1}(2 - 1) = \dfrac{1}{e}\).
12
Derivatives - Differentiability
Wrong
(a) Sketch the graph of the function \(g(x) = x + |x|\).
(b) For what values of \(x\) is \(g\) differentiable?
(c) Find a formula for \(g'\).
My Answer
(No answer)
Correct Answer
No explanation
13
Probability > Normal probabilities
Wrong
The starting national average salary for a computer security specialist is \$58,760. Assuming a roughly normal distribution and a standard deviation of \$6,500, what is the probability that a randomly chosen computer security specialist will start with a salary between \$50,000 and \$60,000?
A
\(P\left(\dfrac{50000}{6500} < z < \dfrac{60000}{6500}\right)\)
B
\(2P\left(z > \dfrac{60000-50000}{6500}\right)\)
C
\(P\left(\dfrac{58760-50000}{6500} < z < \dfrac{58760-60000}{6500}\right)\)
D
\(P\left(\dfrac{58760-50000}{\dfrac{6500}{\sqrt{n}}} < z < \dfrac{58760-60000}{\dfrac{6500}{\sqrt{n}}}\right)\)
\(P\left(\dfrac{50000-58760}{6500} < z < \dfrac{60000-58760}{6500}\right)\)
Correct Answer
Explanation
The two critical z-scores are \(\dfrac{50000-58760}{6500}\) and \(\dfrac{60000-58760}{6500}\).
14
Statistical Inference > CI for proportions
Wrong
An online news magazine periodically gives viewers an opportunity to record electronically their agreement or disagreement with some viewpoint or commentary. On one such occasion, 250 out of 800 respondents agreed with a statement that the most practical way of meeting a potential spouse is through online dating sites. The immediate online calculation concluded that 31.25% of the viewers, with a margin of error of ±3.2%, agreed with the statement. The fine print below stated that the calculation was made with 95% confidence. What is the proper conclusion?
A
We are 95% confident that the proportion of viewers who believe that online dating sites are the most practical way of meeting a potential spouse is between 0.280 and 0.345.
B
Without knowing whether both \(n p\) and \(n(1 - p)\) are ≥ 10, the calculation is inappropriate.
C
Without knowing whether or not the 800 respondents represent less than 10% of the entire population of viewers, the calculation is inappropriate.
D
The z-distribution was used when the t-distribution should have been used, so the calculation is inappropriate.
The data set was not a simple random sample, so the calculation is inappropriate.
Correct Answer
Explanation
This is an example of a voluntary response survey, which is a typically strongly biased survey that completely violates the simple random sample assumption.
15
Riemann Zeta Function
Wrong
Euler also found the sum of the \(p\)-series with \(p = 4\):
\(\zeta(4) = \displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^4} = \dfrac{\pi^4}{90}\)
Use Euler's result to find the sum of the series.
(a) \(\displaystyle\sum_{n=1}^{\infty} \left(\dfrac{3}{n}\right)^4\)
(b) \(\displaystyle\sum_{k=5}^{\infty} \dfrac{1}{(k - 2)^4}\)
My Answer
(No answer)
Correct Answer
No explanation
16
Geometry
Wrong
Triangle \(\triangle A B C\) has side lengths \(A B = 80\), \(B C = 45\), and \(A C = 75\). The bisector \(\angle B\) and the altitude to side \(\overline{A B}\) intersect at point \(P\). What is \(B P\)?
A
18
B
19
C
20
21
Correct Answer
E
22
No explanation
17
Geometry and Trigonometry
Wrong
The floor of a ballroom has an area of 600 square meters. An architect creates a scale model of the floor of the ballroom, where the length of each side of the model is \(\dfrac{1}{10}\) times the length of the corresponding side of the actual floor of the ballroom. What is the area, in square meters, of the scale model?
6
Correct Answer
B
10
C
60
D
150
Explanation
Area scales by (1/10)^2 = 1/100. 600/100 = 6.
18
Craft and Structure
Wrong
The following text is adapted from Aphra Behn's 1689 novel The Lucky Mistake. Atlante and Rinaldo are neighbors who have been secretly exchanging letters through Charlot, Atlante's sister.
[Atlante] gave this letter to Charlot; who immediately ran into the balcony with it, where she still found Rinaldo in a melancholy posture, leaning his head on his hand: She showed him the letter, but was afraid to toss it to him, for fear it might fall to the ground; so he ran and fetched a long cane, which he cleft at one end, and held it while she put the letter into the cleft, and stayed not to hear what he said to it. But never was man so transported with joy, as he was at the reading of this letter; it gives him new wounds; for to the generous, nothing obliges love so much as love.
Which choice best describes the overall structure of the text?
It describes the delivery of a letter, and then portrays a character's happiness at reading that letter.
Correct Answer
B
It establishes that a character is desperate to receive a letter, and then explains why another character has not yet written that letter.
C
It presents a character's concerns about delivering a letter, and then details the contents of that letter.
D
It reveals the inspiration behind a character's letter, and then emphasizes the excitement that another character feels upon receiving that letter.
No explanation
19
Polar Coordinates - Polar Equation
Wrong
Find a polar equation for the curve represented by the given Cartesian equation.
\(x^2 + y^2 = 2c x\)
My Answer
(No answer)
Correct Answer
No explanation
20
Average Value
Wrong
\(h(u) = \dfrac{\ln u}{u}\), \([1, 5]\)
My Answer
(No answer)
Correct Answer
No explanation
21
Command of Evidence (Quantitative)
Wrong
Maximum Height of Maple Trees When Fully Grown
For a school project, a forestry student needs to recommend a maple tree that is native to North America and won't grow more than 60 feet in height. Based on the characteristics of five common maple trees, she has decided to select a _______
Which choice most effectively uses data from the table to complete the text?
| Tree type | Maximum height (feet) | Native to North America |
|---|---|---|
| Sugar maple | 75 | yes |
| Silver maple | 70 | yes |
| Red maple | 60 | yes |
| Japanese maple | 25 | no |
| Norway maple | 50 | no |
A
silver maple.
B
sugar maple.
red maple.
Correct Answer
D
Norway maple.
No explanation
22
Separable Equations - IVP
Wrong
Find the solution of the differential equation that satisfies the given initial condition.
\( \dfrac{d P}{d t} = \sqrt{P t} \), \(\quad P(1) = 2\)
My Answer
(No answer)
Correct Answer
No explanation
23
Kinematics
Wrong
Vectors \(V_1\) and \(V_2\) shown above have equal magnitudes. The vectors represent the velocities of an object at times \(t_1\) and \(t_2\), respectively. The average acceleration of the object between time \(t_1\) and \(t_2\) was
A
directed north
B
directed west
C
directed north of east
directed north of west
Correct Answer
Explanation
Average acceleration = Δv/Δt = (V₂ - V₁)/Δt. The change in velocity vector points north of west.
24
Integration
Wrong
Find the integer \(n\) that allows for integration by substitution (two natural choices exist), then evaluate: \(\int \dfrac{x^n}{\sqrt{1 - x^4}} d x\)
My Answer
(No answer)
Correct Answer
\(n = 3\) gives \(-\dfrac{1}{2}\sqrt{1 - x^4} + C\); \(n = 1\) gives \(\dfrac{1}{2}\sin^{-1}(x^2) + C\)
Explanation
For \(n = 3\): \(u = 1 - x^4\) leads to direct substitution. For \(n = 1\): \(u = x^2\) transforms to \(\dfrac{1}{2}\int \dfrac{d u}{\sqrt{1 - u^2}} = \dfrac{1}{2}\sin^{-1}(u) + C\).
25
Chain Rule - Complex Nested
Wrong
Find the derivative of the function.
\(y = \cos^4(\sin^3 x)\)
My Answer
(No answer)
Correct Answer
\(y' = 4 \cos^3(\sin^3 x) \cdot (-\sin(\sin^3 x)) \cdot 3 \sin^2 x \cos x = -12 \sin^2 x \cos x \cos^3(\sin^3 x) \sin(\sin^3 x)\)
No explanation
26
Derivatives - Limit Interpretation
Wrong
Each limit represents the derivative of some function \(f\) at some number \(a\). State such an \(f\) and \(a\) in each case.
\(\operatorname*{lim}\limits_{h \rightarrow 0} \dfrac{\sqrt{9 + h} - 3}{h}\)
My Answer
(No answer)
Correct Answer
No explanation
27
Differentiation - Patterns/nth
Wrong
Find the \(n\)th derivative of each function by calculating the first few derivatives and observing the pattern that occurs.
(a) \(f(x) = x^n\)
(b) \(f(x) = \dfrac{1}{x}\)
My Answer
(No answer)
Correct Answer
(a) \(f^{(n)}(x) = n!\); (b) \(f^{(n)}(x) = \dfrac{(-1)^n n!}{x^{n+1}}\)
No explanation
28
Series
Wrong
Consider the series \(\displaystyle\sum_{n=1}^{\infty} \dfrac{n}{3^n}\).
(a) Use the Ratio Test to determine whether the series converges or diverges. Show all steps.
(b) If the series converges, explain why absolute convergence and conditional convergence are the same in this case.
(c) Using the geometric series formula \(\displaystyle\sum_{n=0}^{\infty} x^n = \dfrac{1}{1-x}\) for \(|x| < 1\), differentiate both sides and find a closed form for \(\displaystyle\sum_{n=1}^{\infty} n x^{n-1}\).
My Answer
(No answer)
Correct Answer
(a) \(L = \dfrac{1}{3} < 1\), converges. (b) All terms positive, so absolute = conditional. (c) \(\dfrac{d}{d x}[\dfrac{1}{1-x}] = \dfrac{1}{(1-x)^2}\)
Explanation
Full solution requires showing Ratio Test limit calculation, understanding of positive series, and differentiation of power series.
29
Series - Convergence
Wrong
Determine whether the series is convergent or divergent. If it is convergent, find its sum.
\(\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{1 + \left(\dfrac{2}{3}\right)^n}\)
My Answer
(No answer)
Correct Answer
Divergent
No explanation
30
MCQ
Wrong
\(\operatorname*{lim}\limits_{x \rightarrow 3} f(x) = 7\), which must be true? I. continuous at 3 II. differentiable at 3 III. \(f(3) = 7\)
None
Correct Answer
B
II only
C
III only
D
I and III only
E
I, II, and III
Explanation
Limit existing at 3 doesn't require \(f(3)\) to be defined, let alone equal 7.