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1
Vectors - Applications
Wrong
The tension \(\mathbf{T}\) at each end of a chain has magnitude 25 N (see the figure). What is the weight of the chain?
My Answer
(No answer)
Correct Answer
No explanation
2
Problem-Solving and Data Analysis
Wrong
The speed of a vehicle is increasing at a rate of \(7.3\) meters per second squared. What is this rate, in miles per minute squared, rounded to the nearest tenth? (Use 1 mile = 1,609 meters.)
A
0.3
16.3
Correct Answer
C
195.8
D
220.4
No explanation
3
Advanced Math
Wrong
\(-9x^2 + 30x + c = 0\)
In the given equation, \(c\) is a constant. The equation has exactly one solution. What is the value of \(c\)?
A
3
B
0
\(-25\)
Correct Answer
D
\(-53\)
No explanation
4
Limit Laws - Algebraic
Wrong
Evaluate the limit, if it exists.
\(\operatorname*{lim}\limits_{t \rightarrow 0} \dfrac{\sqrt{1 + t} - \sqrt{1 - t}}{t}\)
My Answer
(No answer)
Correct Answer
1
Explanation
Multiply by conjugate: \(\dfrac{(1 + t) - (1 - t)}{t(\sqrt{1 + t} + \sqrt{1 - t})} = \dfrac{2 t}{t(\sqrt{1 + t} + \sqrt{1 - t})} = \dfrac{2}{\sqrt{1 + t} + \sqrt{1 - t}}\). Then \(\dfrac{2}{1 + 1} = 1\).
5
Parametric Equations - Graph Matching
Wrong
Match the graphs of the parametric equations \(x = f(t)\) and \(y = g(t)\) in (a)-(d) with the parametric curves labeled I-IV. Give reasons for your choices.
My Answer
(No answer)
Correct Answer
No explanation
6
MCQ
Wrong
What does the limit statement \(\operatorname*{lim}\limits_{x \rightarrow 1} \dfrac{\ln(x + 1) - \ln 2}{x - 1}\) represent?
A
\(0\)
B
\(\dfrac{d}{d x}[\ln(x + 1)]\)
\(f'(1)\), if \(f(x) = \ln(x + 1)\)
Correct Answer
D
\(1\)
E
The limit does not exist
Explanation
This is the definition of the derivative of \(f(x) = \ln(x + 1)\) at \(x = 1\). Numerically the value equals \(f'(1) = \dfrac{1}{2}\).
7
Trig Derivatives - Basic
Wrong
Differentiate.
\(h(\theta) = \theta^2 \sin \theta\)
My Answer
(No answer)
Correct Answer
\(h'(\theta) = 2 \theta \sin \theta + \theta^2 \cos \theta\)
No explanation
8
Geometry and Trigonometry
Wrong
A circle in the \(x y\)-plane has the equation \((x - 13)^2 + (y - k)^2 = 64\). Which of the following gives the center of the circle and its radius?
The center is at \((13, k)\) and the radius is 8.
Correct Answer
B
The center is at \((k, 13)\) and the radius is 8.
C
The center is at \((k, 13)\) and the radius is 64.
D
The center is at \((13, k)\) and the radius is 64.
No explanation
9
Number Theory
Wrong
A set of numbers is called sum-free if whenever \(x\) and \(y\) are (not necessarily distinct) elements of the set, \(x + y\) is not an element of the set. For example, \({1, 4, 6}\) and the empty set are sum-free, but \({2, 4, 5}\) is not. What is the greatest possible number of elements in a sum-free subset of \({1, 2, 3, ..., 20}\)?
A
8
B
9
10
Correct Answer
D
11
E
12
No explanation
10
Cross Product - Geometric
Wrong
Find \(|\mathbf{u} \times \mathbf{v}|\) and determine whether \(\mathbf{u} \times \mathbf{v}\) is directed into the page or out of the page.
My Answer
(No answer)
Correct Answer
No explanation