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 $$-\frac{{3}}{{2}}a=b\Rightarrow\frac{{a}}{{b}}=\boxed{{-\frac{{2}}{{3}}}}.$$
Final Answer: The final answer is $-\frac{{2}}{{3}}$. I hope it is correct.
"You are an expert Python programmer, and here is your task: Write a function to find the similar elements from the given two tuple lists. Your code should pass these tests:\n\n assert similar_elements((3, 4, 5, 6),(5, 7, 4, 10)) == (4, 5)\n assert similar_elements((1, 2, 3, 4),(5, 4, 3, 7)) == (3, 4) \n assert similar_elements((11, 12, 14, 13),(17, 15, 14, 13)) == (13, 14) \n"
"You are an expert Python programmer, and here is your task: Write a python function to identify non-prime numbers. Your code should pass these tests:\n\n assert is_not_prime(2) == False \n assert is_not_prime(10) == True \n assert is_not_prime(35) == True \n"
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role="BOT",
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"[BEGIN]\n 'import math\r\ndef is_not_prime(n):\r\n result = False\r\n for i in range(2,int(math.sqrt(n)) + 1):\r\n if n % i == 0:\r\n result = True\r\n return result' \n[DONE] \n\n "
),
dict(
role="HUMAN",
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"You are an expert Python programmer, and here is your task: Write a function to find the largest integers from a given list of numbers using heap queue algorithm. Your code should pass these tests:\n\n assert heap_queue_largest( [25, 35, 22, 85, 14, 65, 75, 22, 58],3)==[85, 75, 65] \n assert heap_queue_largest( [25, 35, 22, 85, 14, 65, 75, 22, 58],2)==[85, 75] \n assert heap_queue_largest( [25, 35, 22, 85, 14, 65, 75, 22, 58],5)==[85, 75, 65, 58, 35] \n"
template='''Yes or no: Q: Do hamsters provide food for any animals?
A: Hamsters are prey animals. Prey are food for predators. Thus, hamsters provide food for some animals.
So the answer is yes.
Q: Yes or no: Could Brooke Shields succeed at University of Pennsylvania?
A: Brooke Shields went to Princeton University. Princeton University is about as academically rigorous as the University of Pennsylvania. Thus, Brooke Shields could also succeed at the University of Pennsylvania.
So the answer is yes.
Q: Yes or no: Hydrogen's atomic number squared exceeds number of Spice Girls?
A: Hydrogen has an atomic number of 1. 1 squared is 1. There are 5 Spice Girls. Thus, Hydrogen's atomic number squared is less than 5.
So the answer is no.
Q: Yes or no: Is it common to see frost during some college commencements?
A: College commencement ceremonies can happen in December, May, and June. December is in the winter, so there can be frost. Thus, there could be frost at some commencements.
So the answer is yes.
Q: Yes or no: Could a llama birth twice during War in Vietnam (1945-46)?
A: The War in Vietnam was 6 months. The gestation period for a llama is 11 months, which is more than 6 months. Thus, a llama could not give birth twice during the War in Vietnam.
So the answer is no.
Q: Yes or no: Would a pear sink in water?
A: The density of a pear is about 0.6g/cm3, which is less than water. Objects less dense than water float. Thus, a pear would float.
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'Hamsters are prey animals. Prey are food for predators. Thus, hamsters provide food for some animals.\nSo the answer is yes'
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'Question: Could Brooke Shields succeed at University of Pennsylvania?\nAnswer:'
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'Question: Hydrogen\'s atomic number squared exceeds number of Spice Girls?\nAnswer:'
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'Hydrogen has an atomic number of 1. 1 squared is 1. There are 5 Spice Girls. Thus, Hydrogen\'s atomic number squared is less than 5.\nSo the answer is no'
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'Question: Is it common to see frost during some college commencements?\nAnswer:'
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'College commencement ceremonies can happen in December, May, and June. December is in the winter, so there can be frost. Thus, there could be frost at some commencements.\nSo the answer is yes'
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role='HUMAN',
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'Question: Yes or no: Could a llama birth twice during War in Vietnam (1945-46)?\nAnswer:'
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'The War in Vietnam was 6 months. The gestation period for a llama is 11 months, which is more than 6 months. Thus, a llama could not give birth twice during the War in Vietnam.\nSo the answer is no'
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role='BOT',
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'The density of a pear is about 0.6g/cm3, which is less than water. Objects less dense than water float. Thus, a pear would float.\nSo the answer is no'
