Find the domain of the expression $\frac{\sqrt{x-2}}{\sqrt{5-x}}$.}
Solution:
The expressions inside each square root must be non-negative. Therefore, $x-2 \ge 0$, so $x\ge2$, and $5 - x \ge 0$, so $x \le 5$. Also, the denominator cannot be equal to zero, so $5-x>0$, which gives $x<5$. Therefore, the domain of the expression is $\boxed{[2,5)}$.
Final Answer: The final answer is $[2,5)$. I hope it is correct.
Problem:
If $\det \mathbf{A} = 2$ and $\det \mathbf{B} = 12,$ then find $\det (\mathbf{A} \mathbf{B}).$
Solution:
We have that $\det (\mathbf{A} \mathbf{B}) = (\det \mathbf{A})(\det \mathbf{B}) = (2)(12) = \boxed{24}.$
Final Answer: The final answer is $24$. I hope it is correct.
Problem:
Terrell usually lifts two 20-pound weights 12 times. If he uses two 15-pound weights instead, how many times must Terrell lift them in order to lift the same total weight?
Solution:
If Terrell lifts two 20-pound weights 12 times, he lifts a total of $2\cdot 12\cdot20=480$ pounds of weight. If he lifts two 15-pound weights instead for $n$ times, he will lift a total of $2\cdot15\cdot n=30n$ pounds of weight. Equating this to 480 pounds, we can solve for $n$:
\begin{align*}
30n&=480\\
\Rightarrow\qquad n&=480/30=\boxed{16}
\end{align*}
Final Answer: The final answer is $16$. I hope it is correct.
Problem:
If the system of equations
\begin{align*}
6x-4y&=a,\\
6y-9x &=b.
\end{align*}has a solution $(x, y)$ where $x$ and $y$ are both nonzero,
find $\frac{a}{b},$ assuming $b$ is nonzero.
Solution:
If we multiply the first equation by $-\frac{3}{2}$, we obtain
$$6y-9x=-\frac{3}{2}a.$$Since we also know that $6y-9x=b$, we have
MuTual is a retrieval-based dataset for multi-turn dialogue reasoning, which is
modified from Chinese high school English listening comprehension test data.
Homepage: https://github.com/Nealcly/MuTual
### Citation
```
@inproceedings{mutual,
title = "MuTual: A Dataset for Multi-Turn Dialogue Reasoning",
author = "Cui, Leyang and Wu, Yu and Liu, Shujie and Zhang, Yue and Zhou, Ming" ,
booktitle = "Proceedings of the 58th Conference of the Association for Computational Linguistics",
year = "2020",
publisher = "Association for Computational Linguistics",
}
```
### Groups and Tasks
#### Groups
* Not part of a group yet.
#### Tasks
*`mutual`
*`mutual_plus`
### Checklist
For adding novel benchmarks/datasets to the library:
* [ ] Is the task an existing benchmark in the literature?
* [ ] Have you referenced the original paper that introduced the task?
* [ ] If yes, does the original paper provide a reference implementation? If so, have you checked against the reference implementation and documented how to run such a test?
If other tasks on this dataset are already supported:
* [ ] Is the "Main" variant of this task clearly denoted?
* [ ] Have you provided a short sentence in a README on what each new variant adds / evaluates?
* [ ] Have you noted which, if any, published evaluation setups are matched by this variant?
*`qasper`: executes both `qasper_bool` and `qasper_freeform`
#### Tasks
*`qasper_bool`: Multiple choice task that evaluates the task with `answer_type="bool"`
*`qasper_freeform`: Greedy generation task that evaluates the samples from the task with `answer_type="free form answer"`
### Checklist
For adding novel benchmarks/datasets to the library:
* [ ] Is the task an existing benchmark in the literature?
* [ ] Have you referenced the original paper that introduced the task?
* [ ] If yes, does the original paper provide a reference implementation? If so, have you checked against the reference implementation and documented how to run such a test?
If other tasks on this dataset are already supported:
* [ ] Is the "Main" variant of this task clearly denoted?
* [ ] Have you provided a short sentence in a README on what each new variant adds / evaluates?
* [ ] Have you noted which, if any, published evaluation setups are matched by this variant?
Abstract: `link to paper PDF or arXiv abstract goes here`
`Short description of paper / benchmark goes here:`
Homepage: `homepage to the benchmark's website goes here, if applicable`
### Citation
```
BibTeX-formatted citation goes here
```
### Subtasks
List or describe tasks defined in this folder, and their names here:
*`task_name`: `1-sentence description of what this particular task does`
*`task_name2`: .....
### Checklist
For adding novel benchmarks/datasets to the library:
* [ ] Is the task an existing benchmark in the literature?
* [ ] Have you referenced the original paper that introduced the task?
* [ ] If yes, does the original paper provide a reference implementation? If so, have you checked against the reference implementation and documented how to run such a test?
If other tasks on this dataset are already supported:
* [ ] Is the "Main" variant of this task clearly denoted?
* [ ] Have you provided a short sentence in a README on what each new variant adds / evaluates?
* [ ] Have you noted which, if any, published evaluation setups are matched by this variant?
