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values | qwen_low_correct
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bool 1
class | deepseek_low_correct
bool 2
classes | deepseek_high_correct
bool 2
classes | gptoss_low_correct
bool 2
classes | gptoss_high_correct
bool 2
classes |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: We now define an algorithm: The definition of a(n) is numbers where iterated sum of digits of square settles down to a cyclic pattern of 13, 16, 13, 16, ... after sufficient iterations. Given the input x_list (a series of values): [54, 55, 56, 57, 58, 59, 60, 61, 62, 63], determine the corresponding output sequence y_list.
(A) [122, 123, 125, 129, 131, 132, 134, 138, 140, 141]
(B) [122, 123, 126, 127, 130, 132, 135, 137, 139, 141]
(C) [122, 124, 126, 129, 130, 133, 135, 138, 139, 142]
(D) [121, 122, 124, 128, 130, 131, 133, 137, 139, 140]
(E) [121, 123, 125, 128, 129, 132, 134, 136, 138, 141]
(F) [121, 122, 123, 127, 130, 131, 133, 135, 138, 142]
(G) [120, 122, 125, 128, 129, 132, 134, 136, 139, 142]
(H) [120, 122, 123, 127, 129, 131, 132, 136, 138, 139]
(I) [121, 123, 125, 127, 130, 133, 135, 137, 139, 141]
(J) [120, 121, 124, 127, 129, 131, 134, 136, 138, 140]
|
D
|
supergpqa_Science:cot
|
383
|
57ee3670dc1b4daab63dd87519aad8f2
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| false
| false
| true
| true
|
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: The spin of the helium-3 atom $\mathrm{He}^{3}$ is $1/2$ and it is a fermion. The density of liquid $\mathrm{He}^{3}$ near absolute zero is $0.081 \mathrm{g/cm}^{3}$. The Fermi energy $e_{F}$ is () and the Fermi temperature $T_{F}$ is ().
(A) $$
\approx4. 0 \times1 0^{-4} \mathrm{e v},
$$
$$
\approx6. \mathrm{0 K}
$$
(B) $$
\approx6. 1 \times1 0^{-4} \mathrm{e v},
$$
$$
\approx5. \mathrm{2 K}
$$
(C) $$
\approx6. 5 \times1 0^{-4} \mathrm{e v},
$$
$$
\approx5. \mathrm{7 K}
$$
(D) $$
\approx3. 6 \times1 0^{-4} \mathrm{e v},
$$
$$
\approx4. \mathrm{3 K}
$$
(E) $$
\approx4. 8 \times1 0^{-4} \mathrm{e v},
$$
$$
\approx2. \mathrm{9 K}
$$
(F) $$
\approx5. 7 \times1 0^{-4} \mathrm{e v},
$$
$$
\approx3. \mathrm{4 K}
$$
(G) $$
\approx7. 5 \times1 0^{-4} \mathrm{e v},
$$
$$
\approx3. \mathrm{8 K}
$$
(H) $$
\approx4. 3 \times1 0^{-4} \mathrm{e v},
$$
$$
\approx4. \mathrm{9 K}
$$
(I) $$
\approx3. 9 \times1 0^{-4} \mathrm{e v},
$$
$$
\approx6. \mathrm{1 K}
$$
(J) $$
\approx8. 2 \times1 0^{-4} \mathrm{e v},
$$
$$
\approx4. \mathrm{5 K}
$$
|
H
|
supergpqa_Science:cot
|
1196
|
395c8d0d96774c7bb225e4d3de2cc7e0
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| true
| true
| true
| true
|
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: Determine the maximum value of the sum \[ S = \sum_{n=1}^\infty \frac{n}{2^n} (a_1 a_2 \cdots a_n)^{1/n} \] over all sequences $a_1, a_2, a_3, \cdots$ of nonnegative real numbers satisfying \[ \sum_{k=1}^\infty a_k = 1. \]
(A) \frac{2}{3}
(B) \frac{1}{3}
(C) \frac{1}{2}
(D) \frac{7}{12}
(E) \frac{3}{5}
(F) \frac{3}{4}
(G) \frac{1}{4}
(H) \frac{5}{8}
(I) \frac{5}{6}
(J) \frac{4}{5}
|
A
|
supergpqa_Science:cot
|
106
|
e596ee43e0c34b53b09452797b80fbfa
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| true
| true
| true
| true
|
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: Consider the reaction of extraction of gold from its ore:
$$ Au(s)+2CN^{-}(aq)+\dfrac{1}{4}O_{2}(g)+\dfrac{1}{2}$$
$$ H_{2} O(1)\rightarrow Au(CN)_{2}^{-}(aq)+OH^{-}(aq)$$
Use the following data to calculate $ \triangle G^{0}$ for the above reaction.
$$ K_{f}[Au(CN)_{2}^{-}] = X $$
$$ O_{2}+2H_{2}O+4e^{-}\rightarrow 4oH^{-}; E^{0} = + 0.41 V$$
$$ Au^{3+}+ 3e^{-} \rightarrow Au; E^{0} = +1.50 V $$
$$ Au^{3+}+ 2e^{-} \rightarrow Au^{+}; E^{0} = +1.40 V$$
(A) -RT ln X + 1.31 F
(B) $$ +RT ln X + 2.11 F $$
(C) -RT ln X + 2.11 F
(D) $$ -RT ln X-1.29 F $$
(E) -RT ln X - 1.41 F
(F) -RT ln X + 1.39 F
(G) -RT ln X - 1.39 F
(H) $$ -RT ln X+ 1.29 F $$
(I) -RT ln X + 1.41 F
(J) $$ -RT ln X - 2.11 F$$
|
H
|
supergpqa_Science:cot
|
956
|
19a453d8b3ab471684268017ff86683d
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| false
| true
| true
| true
|
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: Given that the $K_{Steady state}^{\ominus}$ for $[\mathrm{Ag}(\mathrm{NH}_3)_2]^*$ is $1.1 \times 10^{7}$, and the $K_{\mathrm{sp}}^{\ominus}$ for $\mathrm{AgCl}$ is $1.8 \times 10^{-10}$. How many grams of $\mathrm{AgCl}$ solid can dissolve in $100 \ \mathrm{cm}^{3}$ of ammonia solution with a concentration of $10 \ \mathrm{mol} \cdot \mathrm{dm}^{-3}$?
(A) $$
9. 7 \mathrm{\ g}
$$
(B) $$
4. 8 \mathrm{\ g}
$$
(C) $$
6. 2 \mathrm{\ g}
$$
(D) $$
8. 0 \mathrm{\ g}
$$
(E) $$
7. 4 \mathrm{\ g}
$$
(F) $$
5. 9 \mathrm{\ g}
$$
(G) $$
1. 3 \mathrm{\ g}
$$
(H) $$
5. 0 \mathrm{\ g}
$$
(I) $$
3. 5 \mathrm{\ g}
$$
(J) $$
2. 1 \mathrm{\ g}
$$
|
F
|
supergpqa_Science:cot
|
1546
|
2e8352de221b41efbabd36039bdcf3df
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| false
| true
| null | null |
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: Compute \[ \log_2 \left( \prod_{a=1}^{2015} \prod_{b=1}^{2015} (1+e^{2\pi i a b/2015}) \right) \] Here $i$ is the imaginary unit (that is, $i^2=-1$).
(A) 13721
(B) 13720
(C) 13725
(D) 13727
(E) 13723
(F) 13726
(G) 13724
(H) 13722
(I) 13728
(J) 13729
|
C
|
supergpqa_Science:cot
|
178
|
c9820d674f5e49148ebf9a5211482572
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| true
| true
| true
| true
|
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: A neutron with a rest mass of $940 \mathrm{MeV}$ and a half-life of 13 minutes is 5,000 light-years away from Earth. How much energy is required for this neutron to reach the earth at the end of its first half-life?
(A) $$
8. 2 \times1 0^{1 1} \mathrm{M e V}
$$
(B) $$
2. 4 \times1 0^{1 1} \mathrm{M e V}
$$
(C) $$
5. 4 \times1 0^{1 1} \mathrm{M e V}
$$
(D) $$
3. 6 \times1 0^{1 1} \mathrm{M e V}
$$
(E) $$
2. 3 \times1 0^{1 1} \mathrm{M e V}
$$
(F) $$
6. 5 \times1 0^{1 1} \mathrm{M e V}
$$
(G) $$
1. 9 \times1 0^{1 1} \mathrm{M e V}
$$
(H) $$
4. 7 \times1 0^{1 1} \mathrm{M e V}
$$
(I) $$
1. 8 \times1 0^{1 1} \mathrm{M e V}
$$
(J) $$
7. 1 \times1 0^{1 1} \mathrm{M e V}
$$
|
G
|
supergpqa_Science:cot
|
588
|
d14d338a85264ca7ac2ad2fb159e78cd
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| false
| true
| true
| true
|
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: Let the function $f(x) = mathop {lim }limits_{n to infty } frac{{{x^{2n + 1}} + 1}}{{{x^{2n + 1}} - {x^{n + 1}} + x}}$, then which of the following statements about its discontinuities is correct?
(A) $x=1$ is a removable discontinuity
(B) $x=0$ is a removable discontinuity
(C) $x=-1$ is a removable discontinuity
(D) $x=1, x=-1$ are both jump discontinuities
(E) $x=1$ is a jump discontinuity
(F) $x=-1$ is a jump discontinuity
(G) $x=0$ is a jump discontinuity
|
A
|
supergpqa_Science:cot
|
987
|
fab27a77c7a04024bdf2a0e07f36df23
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| true
| false
| true
| true
|
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: 0.2 mol of $O_{2} (g)$ and 0.5 mol of $N_{2} (g)$ form an ideal mixed gas with a temperature of 298 K and a pressure of 101.325 kPa. What are the partial molar volumes of $O_{2} (g)$ and $N_{2} (g)$, as well as the volume of the mixed gas?
(A) $$
\begin{array} {r c l} {{{}}} & {{}} & {{{V \,=1 7. 7 1 \, d m^{3}}}} \\ \end{array}
$$
$$
V ( \mathrm{O}_{2} ) \;= 2 5 \,. 1 0 \, \mathrm{d m}^{3} \, \cdot\, \mathrm{m o l}^{-1} \; \;.
$$
$$
V ( \mathrm{N}_{2} ) \,=2 5 \,. 1 0 \, \mathrm{d m}^{3} \, \cdot\, \mathrm{m o l}^{-1}
$$
(B) $$V =17. 12 dm^{3}$$
$$V (O_{2} )= 2445{dm}^{3}\cdot mol^{-1}$$
$$V (N_{2} )=2445{dm}^{3} \cdot {mol}^{-1}$$
(C) $$
\begin{array} {r c l} {{{}}} & {{}} & {{{V \,=1 6. 9 5 \, d m^{3}}}} \\ \end{array}
$$
$$
V ( \mathrm{O}_{2} ) \;= 2 4 \,. 6 0 \, \mathrm{d m}^{3} \, \cdot\, \mathrm{m o l}^{-1} \; \;.
$$
$$
V ( \mathrm{N}_{2} ) \,=2 4 \,. 6 0 \, \mathrm{d m}^{3} \, \cdot\, \mathrm{m o l}^{-1}
$$
(D) $$
\begin{array} {r c l} {{{}}} & {{}} & {{{V \,=2 1. 1 8 \, d m^{3}}}} \\ \end{array}
$$
$$
V ( \mathrm{O}_{2} ) \;= 2 4 \,. 4 0 \, \mathrm{d m}^{3} \, \cdot\, \mathrm{m o l}^{-1} \; \;.
$$
$$
V ( \mathrm{N}_{2} ) \,=2 4 \,. 4 0 \, \mathrm{d m}^{3} \, \cdot\, \mathrm{m o l}^{-1}
$$
(E) $$
\begin{array} {r c l} {{{}}} & {{}} & {{{V \,=1 9. 0 5 \, d m^{3}}}} \\ \end{array}
$$
$$
V ( \mathrm{O}_{2} ) \;= 2 5 \,. 0 0 \, \mathrm{d m}^{3} \, \cdot\, \mathrm{m o l}^{-1} \; \;.
$$
$$
V ( \mathrm{N}_{2} ) \,=2 5 \,. 0 0 \, \mathrm{d m}^{3} \, \cdot\, \mathrm{m o l}^{-1}
$$
(F) $$
\begin{array} {r c l} {{{}}} & {{}} & {{{V \,=1 7. 2 5 \, d m^{3}}}} \\ \end{array}
$$
$$
V ( \mathrm{O}_{2} ) \;= 2 4 \,. 5 5 \, \mathrm{d m}^{3} \, \cdot\, \mathrm{m o l}^{-1} \; \;.
$$
$$
V ( \mathrm{N}_{2} ) \,=2 4 \,. 5 5 \, \mathrm{d m}^{3} \, \cdot\, \mathrm{m o l}^{-1}
$$
(G) $$
\begin{array} {r c l} {{{}}} & {{}} & {{{V \,=1 8. 0 8 \, d m^{3}}}} \\ \end{array}
$$
$$
V ( \mathrm{O}_{2} ) \;= 2 4 \,. 8 0 \, \mathrm{d m}^{3} \, \cdot\, \mathrm{m o l}^{-1} \; \;.
$$
$$
V ( \mathrm{N}_{2} ) \,=2 4 \,. 8 0 \, \mathrm{d m}^{3} \, \cdot\, \mathrm{m o l}^{-1}
$$
(H) $$
\begin{array} {r c l} {{{}}} & {{}} & {{{V \,=2 0. 3 0 \, d m^{3}}}} \\ \end{array}
$$
$$
V ( \mathrm{O}_{2} ) \;= 2 3 \,. 5 0 \, \mathrm{d m}^{3} \, \cdot\, \mathrm{m o l}^{-1} \; \;.
$$
$$
V ( \mathrm{N}_{2} ) \,=2 3 \,. 5 0 \, \mathrm{d m}^{3} \, \cdot\, \mathrm{m o l}^{-1}
$$
(I) $$
\begin{array} {r c l} {{{}}} & {{}} & {{{V \,=1 6. 8 9 \, d m^{3}}}} \\ \end{array}
$$
$$
V ( \mathrm{O}_{2} ) \;= 2 3 \,. 9 5 \, \mathrm{d m}^{3} \, \cdot\, \mathrm{m o l}^{-1} \; \;.
$$
$$
V ( \mathrm{N}_{2} ) \,=2 3 \,. 9 5 \, \mathrm{d m}^{3} \, \cdot\, \mathrm{m o l}^{-1}
$$
(J) $$
\begin{array} {r c l} {{{}}} & {{}} & {{{V \,=1 8. 5 1 \, d m^{3}}}} \\ \end{array}
$$
$$
V ( \mathrm{O}_{2} ) \;= 2 4 \,. 7 5 \, \mathrm{d m}^{3} \, \cdot\, \mathrm{m o l}^{-1} \; \;.
$$
$$
V ( \mathrm{N}_{2} ) \,=2 4 \,. 7 5 \, \mathrm{d m}^{3} \, \cdot\, \mathrm{m o l}^{-1}
$$
|
B
|
supergpqa_Science:cot
|
298
|
d2cbedfc62434946ba7d8acdc8f99d04
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| true
| false
| true
| true
|
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: Solve the equation
$$
(x_{1} \wedge 0.6) \lor (x_{2} \wedge 0.7) \lor (x_{3} \wedge 0.5) \lor (x_{4} \wedge 0.3) = 0.5.
$$
(A) $$X=([0,0.5],[0.1,1],[0.5,0.6],[0,1])U([0.5,0.6],[0,1],[0.7,1],[0.4,1])U([0,0.6],[0.4,0.5],[0.5,1],[0,1])$$
(B) $$X=([0.5,0.7],[0,1],[0.5,1],[0.2,1])U([0.7,1],[0.5,0.7],[0,1],[0.1,0.5])U([0.6,1],[0.7,1],[0,1],[0.5,1])$$
(C) $$X=([0.5,1],[0.6,1],[0,0.5],[0.3,1])U([0,0.5],[0.4,1],[0.5,1],[0.6,1])U([0,0.5],[0.6,1],[0.5,1],[0,0.3])$$
(D) $$X=([0.4,0.5],[0,1],[0.5,0.7],[0.8,1])U([0.6,1],[0.3,0.5],[0,1],[0,0.5])U([0.5,0.7],[0,1],[0,0.4],[0.5,1])$$
(E) $$X=([0.5,1],[0,0.7],[0.3,1],[0.6,1])U([0.3,0.5],[0.5,1],[0,1],[0,0.6])U([0.5,0.6],[0,1],[0,0.5],[0,0.4])$$
(F) $$X=([0.5,0.6],[0,0.3],[0.6,1],[0,0.5])U([0.3,0.5],[0.4,1],[0.5,0.6],[0.7,1])U([0,1],[0.6,1],[0.5,0.7],[0,1])$$
(G) $$X=([0.7,1],[0,1],[0.3,0.5],[0.1,1])U([0.3,0.5],[0.5,1],[0,1],[0.4,0.6])U([0.5,0.6],[0,1],[0,0.4],[0.5,1])$$
(H) $$X=([0.5,1],[0.5,0.7],[0,0.5],[0,1])U([0.5,1],[0.5,0.6],[0,1],[0.3,1])U([0.5,1],[0.3,0.5],[0,0.4],[0.5,1])$$
(I) $$
X=(0.5,[0,0.5],[0,1],[0,1])U([0,0.5],0.5,[0,1],[0,1])U([0,0.5],[0,0.5],[0.5,1],[0,1])
$$
(J) $$X=([0.6,1],[0,0.5],[0,1],[0.5,1])U([0.5,0.6],[0,0.5],[0.7,1],[0,1])U([0.5,0.6],[0,1],[0.7,1],[0.5,1])$$
|
I
|
supergpqa_Science:cot
|
303
|
0d36544692ae451b9bbad3551a7d02ec
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| false
| true
| true
| true
|
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: White light with a wavelength range of 4000Å-7000Å is incident perpendicularly on a grating. In its diffraction spectrum, what is the wavelength range of the overlap between the second-order spectrum and the third-order spectrum?
(A) 4000Å - 6000Å
(B) 5000Å - 6500Å
(C) 4000Å - 5000Å
(D) 3000Å - 4500Å
(E) 4500Å - 7000Å
(F) 3500Å - 5000Å
(G) 4667Å - 7000Å
(H) 6000Å - 7000Å
(I) 6000Å - 6667Å
(J) 4000Å - 4667Å
|
J
|
supergpqa_Science:cot
|
1354
|
42b7b89703f447b88edb78adf84f0ecd
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| false
| true
| true
| true
|
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: A large metal sphere with a radius of $R=1~\mathrm{m}$ is charged to an electric potential of $U$. Then, $n=10$ small, initially uncharged metal spheres with a radius of $r=\frac{1}{9}~\mathrm{m}$ are sequentially brought into contact with the large sphere and then removed. These 10 charged small spheres are then placed apart from each other along the circumference of a circle with a radius of $R_{0}=10~\mathrm{m}$, with the large metal sphere removed. What is the electric potential at the center of the circle? (Assuming there is no leakage of the total charge in the system throughout this process)
(A) $$
0. 1 0 4 U
$$
(B) $$
0. 0 8 3 U
$$
(C) $$
0. 0 4 9 U
$$
(D) $$
0. 0 5 6 U
$$
(E) $$
0. 0 7 2 U
$$
(F) $$
0. 1 1 0 U
$$
(G) $$
0. 0 6 5 U
$$
(H) $$
0. 0 6 1 U
$$
(I) $$
0. 0 9 9 U
$$
(J) $$
0. 0 5 8 U
$$
|
G
|
supergpqa_Science:cot
|
558
|
e95057c83a594203a4325a2d1b12c563
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| false
| true
| true
| true
|
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: The length of the latus rectum of the parabola $169\left \{ \left ( x-1 \right )^2+\left ( y-3 \right )^2 \right \}=\left ( 5x-12y+17 \right )^2$ is
(A) \displaystyle \frac{30}{13}
(B) \displaystyle \frac{14}{13}
(C) \displaystyle \frac{20}{13}
(D) \frac{26}{13}
(E) \frac{16}{13}
(F) $$\displaystyle \frac{28}{13}$$
(G) \frac{24}{13}
(H) none of these
(I) $$\displaystyle \frac{12}{13}$$
|
F
|
supergpqa_Science:cot
|
1470
|
98940d1fdfa5422696f2b7656dde78d8
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| false
| true
| false
| false
|
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: Evaluate the integral:
$$
I = \int 3 \cdot x \cdot \ln\left(4 + \frac{ 1 }{ x } \right) \, dx
$$
(A) 3/2 * x^2 * ln(4 * x + 1) - 3/4 * x^2 + 3/8 * x - 3/32 * ln(x + 1/4) - 3/2 * x^2 * ln(x) + 3/2 * x^2 + C
(B) 3/2 * x^2 * ln(4 * x + 1) - 3/4 * x^2 + 3/8 * x - 3/32 * ln(x + 1/4) - 3/2 * x^2 * ln(x) + 3/6 * x^2 + C
(C) 3/2 * x^2 * ln(4 * x + 1) - 3/4 * x^2 + 3/8 * x - 3/32 * ln(x + 1/4) - 3/2 * x^2 * ln(x) + 3/10 * x^2 + C
(D) 3/2 * x^2 * ln(4 * x + 1) - 3/4 * x^2 + 3/8 * x - 3/32 * ln(x + 1/4) - 3/2 * x^2 * ln(x) + 3/9 * x^2 + C
(E) 3/2 * x^2 * ln(4 * x + 1) - 3/4 * x^2 + 3/8 * x - 3/32 * ln(x + 1/4) - 3/2 * x^2 * ln(x) + 3/7 * x^2 + C
(F) 3/2 * x^2 * ln(4 * x + 1) - 3/4 * x^2 + 3/8 * x - 3/32 * ln(x + 1/4) - 3/2 * x^2 * ln(x) + 3/4 * x^2 + C
(G) 3/2 * x^2 * ln(4 * x + 1) - 3/4 * x^2 + 3/8 * x - 3/32 * ln(x + 1/4) - 3/2 * x^2 * ln(x) + 3/16 * x^2 + C
(H) 3/2 * x^2 * ln(4 * x + 1) - 3/4 * x^2 + 3/8 * x - 3/32 * ln(x + 1/4) - 3/2 * x^2 * ln(x) + 3/8 * x^2 + C
(I) 3/2 * x^2 * ln(4 * x + 1) - 3/4 * x^2 + 3/8 * x - 3/32 * ln(x + 1/4) - 3/2 * x^2 * ln(x) + 3/5 * x^2 + C
(J) 3/2 * x^2 * ln(4 * x + 1) - 3/4 * x^2 + 3/8 * x - 3/32 * ln(x + 1/4) - 3/2 * x^2 * ln(x) + 3/3 * x^2 + C
|
F
|
supergpqa_Science:cot
|
2141
|
1453a7800e0345038eaed00b29410ccf
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| false
| true
| null | null |
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: A bug is on a vertex of a regular icosahedron (a polyhedron with 20 triangular faces.) Every second, it can either move to one of the adjacent vertices, or teleport to the opposite vertex (i.e. the unique vertex of the icosahedron such that the distance traveled is greatest.) However, he can teleport at most twice before exhausting himself. If $M$ is the amount of ways he can move, such that he is at the original vertex after exactly $7$ seconds, compute the last $3$ nonzero digits of $M$ (Your answer should not contain any 0s.)
(A) 268
(B) 261
(C) 253
(D) 262
(E) 259
(F) 256
(G) 267
(H) 258
(I) 265
(J) 264
|
I
|
supergpqa_Science:cot
|
90
|
e751de6766964f5e85660b8271890b34
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| false
| false
| true
| true
|
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: Given $u(x, y) = x^3 + 6x^2y - 3xy^2 - 2y^3$, find the analytic function $f(z) = u + \mathrm{i} v$ that satisfies the condition $f(0) = 0$.
(A) $$
( 6+4 \mathrm{i} ) z^{5}
$$
(B) $$
( 4-5 \mathrm{i} ) z^{6}
$$
(C) $$
( 1-2 \mathrm{i} ) z^{3}
$$
(D) $$
( 2+ \mathrm{i} ) z^{3}
$$
(E) $$
( 5+3 \mathrm{i} ) z^{7}
$$
(F) $$
( 1+2 \mathrm{i} ) z^{2}
$$
(G) $$
( 2- \mathrm{i} ) z^{8}
$$
(H) $$
( 3-2 \mathrm{i} ) z^{2}
$$
(I) $$
( 2-3 \mathrm{i} ) z^{4}
$$
(J) $$
( 3+4 \mathrm{i} ) z^{5}
$$
|
C
|
supergpqa_Science:cot
|
286
|
f47fc12d61684b6e940b616840dea6bc
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| true
| true
| false
| false
|
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: Find $\frac{ d y }{d x}$, given $y=\tan(2 \cdot v)$ and $v=\arctan(2 \cdot x-1)$.
(A) 2\cdot x^2-2\cdot x+1 / 4\cdot\left(x-x^2\right)^2
(B) 2\cdot x^2-2\cdot x+1 / 4\cdot\left(x-x^2-1\right)^2
(C) 2\cdot x^2-2\cdot x+1 / 2\cdot\left(2x-2x^2+1\right)^2
(D) 2\cdot x^2-2\cdot x+1 / 4\cdot\left(2x-2x^2\right)^2
(E) 2\cdot x^2-2\cdot x+1 / 2\cdot\left(x^2-x+1\right)^2
(F) 2\cdot x^2-2\cdot x+1 / 2\cdot\left(2x-2x^2\right)^2
(G) 2\cdot x^2-2\cdot x+1 / 4\cdot\left(x-x^2+1\right)^2
(H) 2\cdot x^2-2\cdot x+1 / 4\cdot\left(x^2-x\right)^2
(I) 2\cdot x^2-2\cdot x+1 / 2\cdot\left(x-x^2\right)^2
(J) 2\cdot x^2-2\cdot x+1 / 8\cdot\left(x-x^2\right)^2
|
I
|
supergpqa_Science:cot
|
2064
|
f6e1523e6f3148a48d1a77c85ffece8e
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| false
| true
| null | null |
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: Eumerica makes bottled air at three plants in Vienna, Athens, and Moscow, and ships crates of their products to distributors in Venice, Frankfurt, and Paris. Each day the Athens plant produces 25 thousand crates, while Vienna can produce up to 18 thousand, and Moscow can produce up to 15 thousand. In addition, Venice must receive 14 thousand and Paris must receive 22 thousand crates, while Frankfurt can receive up to 19 thousand. The company pays Arope Trucking to transport their products at the following per-crate Eurodollar costs: Vienna to Frankfurt (120), Frankfurt to Athens (100), Athens to Frankfurt (120), Frankfurt to Paris (150), Paris to Athens (130), Athens to Vienna (160), Venice to Paris (240), Paris to Venice (250), Frankfurt to Venice (270), Venice to Moscow (280), Moscow to Frankfurt (290). Eumerica would like to tell Arope which shipments to make between cities so as to minimize cost. What is the minimum cost?
(A) 10,380,000
(B) 10,360,000
(C) 10,390,000
(D) 10,000,000
(E) 10,370,000
(F) 10,400,000
(G) 10,450,000
(H) 10,350,000
(I) 11,000,000
(J) 10,500,000
|
B
|
supergpqa_Science:cot
|
512
|
342a812e2bab48828937e2b882d8bfbd
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| true
| false
| true
| true
|
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: $int_{-infty}^{+infty} frac{a^{3/2}}{s^2+a^2} ds = $
(A) $0$
(B) $frac{1}{a}$
(C) $frac{3}{a}$
(D) $frac{1}{2a^2}$
(E) $frac{1}{2a}$
(F) $frac{2}{a}$
(G) $2a$
(H) $frac{3}{2a}$
(I) $frac{3}{2a^2}$
(J) $frac{1}{a^2}$
|
F
|
supergpqa_Science:cot
|
38
|
8e58552045574cd082ae3ea220ee55d9
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| false
| true
| true
| true
|
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: Which of the following sequences is not a degree sequence of a graph?
I. (3, 3, 3, 1, 0, 0)
II. (3, 2, 1, 1, 1, 0)
III. (1, 1, 1, 2, 1, 1)
IV. (2, 2, 2, 2, 2, 2)
V. (3, 2, 2, 3, 1, 1)
VI. (1, 0, 0, 3, 2, 2)
VII. (2, 2, 2, 2, 1, 7)
VIII. (1, 2, 2, 4, 3, 3)
(A) I,VI
(B) II,VII
(C) I,IV
(D) II,VI
(E) II,V
(F) IV,VII
(G) I,VII
(H) IV,V
(I) III,VIII
(J) V,VI
|
I
|
supergpqa_Science:cot
|
979
|
7d58c02c5b734876b0549134fa1202a0
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| false
| false
| true
| true
|
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: Let $ABC$ be a triangle with $AB = 13$ , $BC = 14$ , and $AC = 15$ . Let $D$ be the foot of the altitude from $A$ to $BC$ and $E$ be the point on $BC$ between $D$ and $C$ such that $BD = CE$ . Extend $AE$ to meet the circumcircle of $ABC$ at $F$ . If the area of triangle $FAC$ is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers, find $m + n$ .
(A) 633
(B) 635
(C) 632
(D) 638
(E) 634
(F) 630
(G) 631
(H) 639
(I) 637
(J) 636
|
G
|
supergpqa_Science:cot
|
517
|
b29006a0fcbc41d2a1fad1497949eb9e
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| false
| true
| true
| true
|
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: Which of these ions $Cu^+, Co^{3+}, Fe^{2+}$ is stable in the aqueous medium?Given: $E_{Cu^{2+}/Cu^+}^0 = 0.15 V;\ E_{Cu^+/Cu}^0 = 0.53 V\ E_{Co^{3+}/Co^{2+}}^0 = 1.82 V; \ E_{Fe^{3+}/Fe^{2+}}^0 = 0.77 V \ E_{Fe^{2+}/Fe}^0 = - 0.44 V; \ E_{O_2, H^+/H_2O}^0 = 1.23 V$
(A) $$Fe^{3+}$$
(B) $$Co^{2+}, Fe^{3+}$$
(C) $$Cu^+, Co^{3+}$$
(D) $$Co^{3+}, Cu^+$$
(E) $$Co^{3+}, Cu^+, Fe^{2+}$$
(F) $$Fe^{3+}, Co^{3+}$$
(G) $$Co^{2+}, Fe^{2+}$$
(H) $$Co^{2+}$$
(I) $$Co^{3+}$$
(J) $$Cu^{2+}$$
|
I
|
supergpqa_Science:cot
|
3713
|
2789bb6861f94992954c4df00ec8e656
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| false
| false
| null | null |
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: For real numbers $a,b,$ and $c,$ suppose that $\tfrac{a+b}{c}$ and $\tfrac{b+c}{a}$ are the roots of the polynomial $x^2 - 574 x + 17.$ Find $\tfrac{a+c}{b}$ .
(A) 35
(B) 38
(C) 36
(D) 35.7
(E) 34
(F) 35.8
(G) 35.2
(H) 37
(I) 35.9
(J) 35.5
|
C
|
supergpqa_Science:cot
|
2157
|
98273697cfe64c44b258fcf337adbd0e
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| true
| true
| null | null |
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: If the given two identical charged rings lie in $xy$ plane both having linear charge density $\lambda$ varies as per $\lambda = \lambda_0 \cos \theta$ ($\lambda_0$ = constant) where $\theta$ is measured from +x-axis. Radius for both the rings is $R$. Electric force between the two rings is $\dfrac{x K \lambda_{0}^{2} \pi^{2} R^{4}}{d^4}$ then $x$ is.
(A) $$5$$
(B) $$7$$
(C) $$2$$
(D) $$1$$
(E) $$8$$
(F) $$6$$
(G) $$3$$
(H) $$9$$
(I) $$4$$
|
F
|
supergpqa_Science:cot
|
1703
|
d86004033e6248e6af8277b009841251
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| true
| true
| null | null |
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: A particle is projected with speed 100 m/s at angle $\theta ={ 60 }^{ \circ }$ with the horizontal at time t = 0. At time t the velocity vector of the particle becomes perpendicular to the direction of velocity of projection
Its tangential acceleration at time t is :
(A) $$5\sqrt{3} m/s^{2}$$
(B) $$10m/s^{ 2 }$$
(C) $$15m/s^{2}$$
(D) $$10\sqrt{3} m/s^{2}$$
(E) $$1cm/s^{ 2 }$$
(F) zero
(G) $$5m/s^{ 2 }$$
(H) $$5\sqrt{2} m/s^{2}$$
|
G
|
supergpqa_Science:cot
|
922
|
60fce05d0409453a951e1c0f2338ea4c
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| true
| true
| false
| false
|
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: Evaluate the integral:
$$
\int \int \int_{R} 3 \cdot y \, dV
$$
where $R$ is bounded by:
1. $0 \le x \le 1$
2. $0 \le y \le x$
3. $0 \le z \le \sqrt{9-y^2}$
(A) 216-86*sqrt(2)-243*arcsin(1/3)/30
(B) 216-86*sqrt(2)-243*arcsin(1/3)/24
(C) 216-86*sqrt(2)-243*arcsin(1/3)/10
(D) 216-86*sqrt(2)-243*arcsin(1/3)/32
(E) 216-86*sqrt(2)-243*arcsin(1/3)/4
(F) 216-86*sqrt(2)-243*arcsin(1/3)/6
(G) 216-86*sqrt(2)-243*arcsin(1/3)/16
(H) 216-86*sqrt(2)-243*arcsin(1/3)/8
(I) 216-86*sqrt(2)-243*arcsin(1/3)/12
(J) 216-86*sqrt(2)-243*arcsin(1/3)/20
|
H
|
supergpqa_Science:cot
|
2060
|
4f9333d59bac4d6daa38605306d8b2dc
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| false
| true
| null | null |
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: Two block of mass $2\ kg$ and $5\ kg$ are at rest on the ground. The masses are connected by a string passing over a frictionless Pulley which is under the influence of a constant upward force $F=50N$. The acceleration of $5\ kg$ and $2\ kg$ masses are
(A) $$0,0$$
(B) $2.5m/{ s }^{ 2 }$,$2.5m/{ s }^{ 2 }$
(C) 1.0m/{ s }^{ 2 },2.5m/{ s }^{ 2 }
(D) $1m/{ s }^{ 2 }$,$2.5m/{ s }^{ 2 }$
(E) 1.5m/{ s }^{ 2 },2.0m/{ s }^{ 2 }
(F) 1.5m/{ s }^{ 2 },2.5m/{ s }^{ 2 }
(G) 1.0m/{ s }^{ 2 },1.5m/{ s }^{ 2 }
(H) 1.5m/{ s }^{ 2 },0
(I) 2.0m/{ s }^{ 2 },1.5m/{ s }^{ 2 }
(J) $$0,2.5m/{ s }^{ 2 }$$
|
J
|
supergpqa_Science:cot
|
2432
|
94fc9dd8bbbf45f0bb991c4d2c3d50a7
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| true
| true
| null | null |
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: The infinite sequence of 2's and 3's \begin{align*} &2,3,3,2,3,3,3,2,3,3,3,2,3,3,2,3,3, \\ &3,2,3,3,3,2,3,3,3,2,3,3,2,3,3,3,2,\dots \end{align*} has the property that, if one forms a second sequence that records the number of 3's between successive 2's, the result is identical to the given sequence. Find the real number $r$ such that, for any $n$, the $n$th term of the sequence is 2 if and only if $n = 1 + \lfloor rm \rfloor$ for some nonnegative integer $m$. (Note: $\lfloor x \rfloor$ denotes the largest integer less than or equal to $x$.)
(A) 2 + \sqrt{7}
(B) 2 + \sqrt{8}
(C) 2 + \sqrt{3}
(D) 2 + \sqrt{5}
(E) 2 + \sqrt{11}
(F) 2 + \sqrt{1}
(G) 2 + \sqrt{4}
(H) 2 + \sqrt{10}
(I) 2 + \sqrt{2}
(J) 2 + \sqrt{6}
|
C
|
supergpqa_Science:cot
|
1043
|
0edd56b224fb472c9363ec164ef570b2
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| true
| true
| true
| true
|
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: Let $R$ be the region consisting of the points $(x,y)$ of the cartesian plane satisfying both $|x|-|y| \leq 1$ and $|y| \leq 1$. Find the area of $R$.
(A) 4
(B) 5.8
(C) 5.9
(D) 6
(E) 7
(F) 5.5
(G) 5
(H) 8
(I) 5.7
(J) 5.2
|
D
|
supergpqa_Science:cot
|
2136
|
792298d098804369927def02d340bd1a
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| true
| true
| null | null |
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: A secret spy organization needs to spread some secret knowledge to all of its members. In the beginning, only $1$ member is informed. Every informed spy will call an uninformed spy such that every informed spy is calling a different uninformed spy. After being called, an uninformed spy becomes informed. The call takes $1$ minute, but since the spies are running low on time, they call the next spy directly afterward. However, to avoid being caught, after the third call an informed spy makes, the spy stops calling. How many minutes will it take for every spy to be informed, provided that the organization has $600$ spies?
(A) 17
(B) 10
(C) 13
(D) 12
(E) 8
(F) 9
(G) 11
(H) 16
(I) 15
(J) 14
|
B
|
supergpqa_Science:cot
|
1140
|
04f76364cc5e4a00a2ae66468ee6888b
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| true
| true
| true
| true
|
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: With benzoyl peroxide as initiator and ethyl acetate as solvent, methyl methacrylate undergoes polymerization at 60°C. Assume that the total volume of polymer in the reactor is 1 L, the density is $0.878 \mathrm{~g \cdot mL}^{-1}$, the mass of the monomer is 300 g, the initiator dose is 0.6 $\gamma_{\mathrm{6}}$ of the monomer dose, $k_{\mathrm{d}} = 2.0 \times 10^{-6} \, \mathrm{s}^{-1}$, $k_{\mathrm{p}} = 367 \, \mathrm{~L} \cdot \mathrm{mol}^{-1} \cdot \mathrm{s}^{-1}$, $k_{\mathrm{t}} = 0.93 \times 10^{7} \, \mathrm{~L} \cdot \mathrm{mol}^{-1} \cdot \mathrm{s}^{-1}$, $f = 0.8$, $C_{\mathrm{I}} = 0.02$, $C_{\mathrm{M}} = 0.18 \times 10^{-4}$, $C_{\mathrm{S}} = 0.46 \times 10^{-6}$, kinetic chain termination is mainly through disproportionation, about 85%. Calculate the number average degree of polymerization of the product when the reaction is stopped at low conversion?
(A) $$
\overline{{{X_{\mathrm{n}}}}} \!=\! 5. 9 7 \! \times\! 1 0^{3}
$$
(B) $$
\overline{{{X_{\mathrm{n}}}}} \!=\! 3. 5 9 \! \times\! 1 0^{5}
$$
(C) $$
\overline{{{X_{\mathrm{n}}}}} \!=\! 7. 4 1 \! \times\! 1 0^{2}
$$
(D) $$
\overline{{{X_{\mathrm{n}}}}} \!=\! 2. 8 7 \! \times\! 1 0^{3}
$$
(E) $$
\overline{{{X_{\mathrm{n}}}}} \!=\! 6. 8 1 \! \times\! 1 0^{1}
$$
(F) $$
\overline{{{X_{\mathrm{n}}}}} \!=\! 1. 5 0 \! \times\! 1 0^{2}
$$
(G) $$
\overline{{{X_{\mathrm{n}}}}} \!=\! 8. 3 3 \! \times\! 1 0^{6}
$$
(H) $$
\overline{{{X_{\mathrm{n}}}}} \!=\! 4. 2 3 \! \times\! 1 0^{4}
$$
(I) $$
\overline{{{X_{\mathrm{n}}}}} \!=\! 4. 6 5 \! \times\! 1 0^{7}
$$
(J) $$
\overline{{{X_{\mathrm{n}}}}} \!=\! 9. 2 0 \! \times\! 1 0^{4}
$$
|
D
|
supergpqa_Science:cot
|
3271
|
0c2df859ea2c42d6ad86e582cdc81feb
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| false
| true
| null | null |
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: Given \( A=\left[ \begin{matrix} 0 & 1 \\ -2 & -3 \end{matrix} \right] \), then \( e^{At} \) equals
(A) \( \left[ egin{matrix} 2e^{-t}-e^{-2t} & e^{-t}-2e^{-2t} \ -2e^{-t}+2e^{-2t} & -2e^{-t}+2e^{-2t} \end{matrix}
ight] \)
(B) \left[ \begin{matrix} 2e^{-t}-e^{-2t} & e^{-t}-2e^{-2t} \\ -2e^{-t}+e^{-2t} & -e^{-t}+e^{-2t} \end{matrix} \right]
(C) \left[ \begin{matrix} 2e^{-t}-e^{-2t} & e^{-t}-2e^{-2t} \\ -2e^{-t}+e^{-2t} & -e^{-t}+2e^{-2t} \end{matrix} \right]
(D) \( \left[ \begin{matrix} 2e^{-t}-e^{-2t} & e^{-t}-e^{-2t} \\ -2e^{-t}+2e^{-2t} & -e^{-t}+2e^{-2t} \end{matrix} \right] \)
(E) \left[ \begin{matrix} 2e^{-t}-2e^{-2t} & e^{-t}-e^{-2t} \\ -2e^{-t}+2e^{-2t} & -e^{-t}+e^{-2t} \end{matrix} \right]
(F) \left[ \begin{matrix} 2e^{-t}-2e^{-2t} & e^{-t}-2e^{-2t} \\ -2e^{-t}+e^{-2t} & -2e^{-t}+2e^{-2t} \end{matrix} \right]
(G) \left[ \begin{matrix} 2e^{-t}-2e^{-2t} & e^{-t}-e^{-2t} \\ -2e^{-t}+e^{-2t} & -e^{-t}+2e^{-2t} \end{matrix} \right]
(H) \( \left[ egin{matrix} 2e^{-t}-e^{-2t} & 2e^{-t}-2e^{-2t} \ -2e^{-t}+2e^{-2t} & -e^{-t}+2e^{-2t} \end{matrix}
ight] \)
(I) \left[ \begin{matrix} 2e^{-t}-e^{-2t} & e^{-t}-e^{-2t} \\ -2e^{-t}+e^{-2t} & -e^{-t}+e^{-2t} \end{matrix} \right]
(J) \( \left[ egin{matrix} 2e^{-t}-2e^{-2t} & e^{-t}-2e^{-2t} \ -2e^{-t}+2e^{-2t} & -e^{-t}+2e^{-2t} \end{matrix}
ight] \)
|
D
|
supergpqa_Science:cot
|
2015
|
41402a3aeb374ef3a883cb09fe6bac83
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| true
| true
| null | null |
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: The fraction of atoms in a sample of argon gas at 400 K that has an energy of 10.0 kJ or greater is() .
(A) $$
3. 8 \times1 0^{-2}
$$
(B) $$
5. 7 \times1 0^{-2}
$$
(C) $$
7. 1 \times1 0^{-2}
$$
(D) $$
9. 0 \times1 0^{-2}
$$
(E) $$
1. 5 \times1 0^{-2}
$$
(F) $$
4. 9 \times1 0^{-2}
$$
(G) $$
2. 3 \times1 0^{-2}
$$
(H) $$
8. 4 \times1 0^{-2}
$$
(I) $$
6. 2 \times1 0^{-2}
$$
(J) $$
0. 6 \times1 0^{-2}
$$
|
F
|
supergpqa_Science:cot
|
1623
|
f4d9108ba75b40888359eeaf02df0ad8
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| true
| false
| null | null |
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: What is the value of the integral $\int_{-1}^{1} \sqrt{\frac{2+x}{1-x^{2}}} \mathrm{d} x \,$?
(A) $\int_{-1}^{1} \sqrt{\frac{2+x}{1-x^{2}}} \mathrm{d} x \,=4.368939556$
(B) $\int_{-1}^{1} \sqrt{\frac{2+x}{1-x^{2}}} \mathrm{d} x \,=5.652138492$
(C) $\int_{-1}^{1} \sqrt{\frac{2+x}{1-x^{2}}} \mathrm{d} x \,=6.239854372$
(D) $\int_{-1}^{1} \sqrt{\frac{2+x}{1-x^{2}}} \mathrm{d} x \,=5.890234109$
(E) $\int_{-1}^{1} \sqrt{\frac{2+x}{1-x^{2}}} \mathrm{d} x \,=3.785201294$
(F) $\int_{-1}^{1} \sqrt{\frac{2+x}{1-x^{2}}} \mathrm{d} x \,=5.417896562$
(G) $\int_{-1}^{1} \sqrt{\frac{2+x}{1-x^{2}}} \mathrm{d} x \,=3.462139875$
(H) $\int_{-1}^{1} \sqrt{\frac{2+x}{1-x^{2}}} \mathrm{d} x \,=4.110984263$
(I) $\int_{-1}^{1} \sqrt{\frac{2+x}{1-x^{2}}} \mathrm{d} x \,=2.948212134$
(J) $\int_{-1}^{1} \sqrt{\frac{2+x}{1-x^{2}}} \mathrm{d} x \,=6.023745921$
|
A
|
supergpqa_Science:cot
|
2332
|
ee235d078927400abeeb50bdf9a7ee39
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| true
| true
| null | null |
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: Calculate the following curve integral using the residue theorem $ointlimits_{|z|=3}frac{z^{13}}{(z^2+5)^3(z^4+1)^2}dz$
(A) $4pi i$
(B) $2pi$
(C) -1
(D) $4pi$
(E) $-2pi$
(F) $-2pi i$
(G) $2pi i$
(H) $-4pi i$
(I) 0
|
G
|
supergpqa_Science:cot
|
1006
|
c8178a367bb9432db19da985f5276626
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| false
| false
| true
| true
|
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: A 100 ml mixture of $ Na_{2}CO_{3} $ and $ NaHCO_{3} $ is titrated against 1 M - HCl. If $ V_{1} L $ and $ V_{2} L $ are consumed when phenolphthalein and methyl orange are used as indicators, respectively, in two separate titrations, which of the following is true for molarities in the original solution?
(A) molarity of $ NaHCO_{3} = 10 (2V_{1}-2V_{2}) $
(B) molarity of $ NaHCO_{3} = 10 (2V_{1}-V_{2}) $
(C) molarity of $ NaHCO_{3} = 10 (V_{2}-3V_{1}) $
(D) molarity of $ Na_{2}CO_{3} = 20V_{1} $
(E) molarity of $ Na_{2}CO_{3} = 10(V_{2}+V_{1}) $
(F) molarity of $ NaHCO_{3} = 10 (2V_{2}-V_{1}) $
(G) molarity of $ NaHCO_{3} = 10 (2V_{2}-2V_{1}) $
(H) molarity of $ NaHCO_{3} = 10 (V_{2}-V_{1}) $
(I) molarity of $ NaHCO_{3} = 10 (V_{2}-2V_{1}) $
|
I
|
supergpqa_Science:cot
|
3933
|
ed67f9aece4b4375826a90c191e917c3
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| true
| true
| null | null |
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: Given that $\mathbf{y}=f( \mathbf{x} )$ satisfies $\mathbf{y}^{\prime\prime}+2 \mathbf{y}^{\prime}+5 f \ \ \mathbf{( x )} \ =0$, and $f \ \ ( 0 ) \ \ =1 \,, \ \ f^{\prime} \ \ ( 0 ) \ \ =\ -1 \,$.
We set $a_{n}=\int_{n \pi}^{+\infty} f \left( x \right) \! \mathrm{d} x$ , so what is the value of $\sum_{n=1}^{\infty} a_{n}$?
(A) $$
\frac{1} {5 \left( e^{\pi}+1 \right)}
$$
(B) $$
\frac{1} {4 \left( e^{\pi}-1 \right)}
$$
(C) $$
\frac{1} { \left( e^{\pi}-1 \right)}
$$
(D) $$
\frac{1} {3 \left( e^{\pi}-1 \right)}
$$
(E) $$
\frac{1} {3 \left( e^{\pi}+1 \right)}
$$
(F) $$
\frac{1} {2 \left( e^{\pi}+1 \right)}
$$
(G) $$
\frac{1} {1 \left( e^{\pi}+1 \right)}
$$
(H) $$
\frac{1} {4 \left( e^{\pi}+1 \right)}
$$
(I) $$
\frac{1} {5 \left( e^{\pi}-1 \right)}
$$
(J) $$
\frac{1} {2\left( e^{\pi}-1 \right)}
$$
|
I
|
supergpqa_Science:cot
|
1338
|
ebe7a35d7a10447e85f872b9d18143f2
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| true
| false
| true
| true
|
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: Time period of oscillation of magnet of magnetic moment M and moment of inertia I in a vertical plane perpendicular to the magnetic meridian at a place where earth's horizontal and vertical component of magnetic field are $B_h$ and $B_v$ respectively is:
(A) $$T= 2 \pi \sqrt {\dfrac{1}{M(Bv^2 + Bh^2)^{1/2}}}$$
(B) $$T= 2 \pi \sqrt{\dfrac{1}{MBh^2}}$$
(C) $$T= 2 \pi \sqrt{\dfrac{1}{MBv^2}}$$
(D) $$T= 2 \pi \sqrt {\dfrac{1}{M(Bv + Bh)^2}}$$
(E) $$T= 2 \pi \sqrt{\dfrac{1}{MBv}}$$
(F) $$T= 2 \pi \sqrt {\dfrac{1}{M(Bv + Bh)}}$$
(G) infinite
(H) $$T= 2 \pi \sqrt {\dfrac{1}{MBv^3}}$$
(I) $$T= 2 \pi \sqrt {\dfrac{1}{M(Bv - Bh)}}$$
(J) $$T= 2 \pi \sqrt {\dfrac{1}{MBh}}$$
|
E
|
supergpqa_Science:cot
|
1752
|
089603ad154f4b82a1e1f24560f629c2
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| false
| false
| null | null |
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: An ideal diatomic gas is expanded so that the amount of heat transferred to the gas is equal to the decrease in its internal energy.
The process can be represented by the equation $TV^n$= constant where the value of n is:
(A) $$n=\frac {1}{5}$$
(B) $$n=\frac {3}{5}$$
(C) $$n=\frac {7}{5}$$
(D) $$n=\frac{4}{5}$$
(E) $$n=\frac{1}{3}$$
(F) $$n=\frac {2}{5}$$
(G) $$n=\frac{1}{4}$$
(H) $$n=\frac{2}{3}$$
(I) $$n=\frac {3}{2}$$
(J) $$n=\frac{3}{4}$$
|
A
|
supergpqa_Science:cot
|
1821
|
28b8e9105db04b8b9d3a7e2f6aedff5c
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| true
| true
| null | null |
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: Two infinitely long parallel wires carry currents of magnitude $I_1$ and $I_2$ and are at a distance $4$ cm apart. The magnitude of the net magnetic field is found to reach a non-zero minimum value between the two wires and $1$ cm away from the first wire. The ratio of the two currents and their mutual direction is?
(A) $\displaystyle\dfrac{I_2}{I_1}=3$, parallel
(B) \(\displaystyle\dfrac{I_2}{I_1}=24\), antiparallel
(C) \(\displaystyle\dfrac{I_2}{I_1}=25\), antiparallel
(D) $\displaystyle\dfrac{I_2}{I_1}=9$, parallel
(E) $\displaystyle\dfrac{I_2}{I_1}=3$, antiparallel
(F) \(\displaystyle\dfrac{I_2}{I_1}=27\), antiparallel
(G) $\displaystyle\dfrac{I_2}{I_1}=27$, parallel
(H) \(\displaystyle\dfrac{I_2}{I_1}=27\), parallel
(I) \(\displaystyle\dfrac{I_2}{I_1}=8\), antiparallel
(J) $\displaystyle\dfrac{I_2}{I_1}=9$, antiparallel
|
J
|
supergpqa_Science:cot
|
804
|
8760f984946a4005a7fdaf292cbdf95e
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| true
| true
| true
| true
|
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: Reduce the following big-O natations: $O [ \mathrm{~ e}^{\Omega}+\mathrm{a n}^{\mathrm{~ 1 0}} ]=$ ____.
(A) $O[n^{10}]$
(B) $O[e^{\ln}]$
(C) $O[e^{\eta}]$
(D) $O[e^{10+}]$
(E) $O[ an^{10}]$
(F) $O[e^{10}]$
(G) $O[e^{\alpha}]$
(H) $O[e^n]$
(I) $O[e^{\omega}]$
(J) $O[e^n+n^{10}]$
|
H
|
supergpqa_Science:cot
|
2196
|
e093ece1af544bcc845a69d8e81d0ac6
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| true
| true
| null | null |
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: In a set of experiments on a hypothetical one-electron atom, the wavelengths of the photons emitted from transitions ending in the ground state $(n=1)$ are shown in the energy diagram above. The possible energy of the atom in $n=3$ cannot be
(A) -0.5125\space eV
(B) $$-1.95\space eV$$
(C) -0.5425\space eV
(D) -1.55\space eV
(E) -2.15\space eV
(F) -1.75\space eV
(G) $$-7.8\space eV$$
(H) $$-0.121\space eV$$
(I) $$-0.4875\space eV$$
(J) -0.5625\space eV
|
B
|
supergpqa_Science:cot
|
2740
|
cf0a8834963c480f925330c19b40ad4b
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| false
| false
| null | null |
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: We now define an algorithm: The definition of a(n) is: Let r+i*s be the sum of the distinct first-quadrant Gaussian integers dividing n; sequence gives r values. Note that a Gaussian integer is a complex number of the form z = a+bi, where both a and b are integers, and the first-quadrant Gaussian integers have both a and b non-negative. Given the input x_list (a series of values): [56, 57, 58, 59, 60, 61, 62, 63, 64, 65], determine the corresponding output sequence y_list.
(A) [184, 88, 159, 68, 380, 81, 136, 112, 198, 184]
(B) [180, 84, 155, 64, 376, 77, 132, 108, 194, 180]
(C) [175, 79, 150, 59, 371, 72, 127, 103, 189, 175]
(D) [178, 82, 153, 62, 374, 75, 130, 106, 192, 178]
(E) [182, 86, 157, 66, 378, 79, 134, 110, 196, 182]
(F) [179, 83, 154, 63, 375, 76, 131, 107, 193, 179]
(G) [177, 81, 152, 61, 373, 74, 129, 105, 191, 177]
(H) [181, 85, 156, 65, 377, 78, 133, 109, 195, 181]
(I) [176, 80, 151, 60, 372, 73, 128, 104, 190, 176]
(J) [183, 87, 158, 67, 379, 80, 135, 111, 197, 183]
|
I
|
supergpqa_Science:cot
|
2387
|
a2b58a4f7d2b48e1bdb27af38030c4e4
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| false
| true
| null | null |
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: Given a positive integer $x$, define the function $p_x(o) = \prod_{{k=1}}^x \cos(ko)$.Determine the smallest $x$ such that the absolute value of the second derivative at zero satisfies $|p_x''(0)| > 89688$.
(A) 60
(B) 63
(C) 62
(D) 66
(E) 68
(F) 61
(G) 64
(H) 65
(I) 67
(J) 69
|
H
|
supergpqa_Science:cot
|
1090
|
4b3480d85cd0470aa9d628df35121c08
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| true
| true
| true
| true
|
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: A circular membrane with a radius of $0.015 \mathrm{~m}$ is fixed at the perimeter. Assuming its surface density is $\rho_{\mathrm{s}} = 2 \, \mathrm{kg/m}^{2}$, what is the minimum tension required for its fundamental frequency to be below 5000 Hz?
(A) $$
T \,=\, 6. 3 8 \, \times\, 1 0^{4} \mathrm{~ N / m}
$$
(B) $$
T \,=\, 5. 5 0 \, \times\, 1 0^{4} \mathrm{~ N / m}
$$
(C) $$
T \,=\, 4. 0 4 \, \times\, 1 0^{4} \mathrm{~ N / m}
$$
(D) $$
T \,=\, 3. 7 2 \, \times\, 1 0^{4} \mathrm{~ N / m}
$$
(E) $$
T \,=\, 7. 2 5 \, \times\, 1 0^{4} \mathrm{~ N / m}
$$
(F) $$
T \,=\, 8. 3 2 \, \times\, 1 0^{4} \mathrm{~ N / m}
$$
(G) $$
T \,=\, 9. 1 5 \, \times\, 1 0^{4} \mathrm{~ N / m}
$$
(H) $$
T \,=\, 8. 0 6 \, \times\, 1 0^{4} \mathrm{~ N / m}
$$
(I) $$
T \,=\, 6. 9 0 \, \times\, 1 0^{4} \mathrm{~ N / m}
$$
(J) $$
T \,=\, 7. 6 7 \, \times\, 1 0^{4} \mathrm{~ N / m}
$$
|
J
|
supergpqa_Science:cot
|
243
|
c9f9c9b6cb984b06a607754755ee0a0d
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| false
| true
| true
| true
|
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: Find the sum of all positive integers $n$ that are within 250 of exactly 15 perfect squares.
(A) 10003
(B) 9996
(C) 10000
(D) 9998
(E) 9999
(F) 9997
(G) 9995
(H) 10002
(I) 10001
(J) 10004
|
E
|
supergpqa_Science:cot
|
1124
|
749da5e79ed7454fb7d9722bc33fd4f3
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| false
| true
| true
| true
|
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: Distilled water is a dielectric having the constants\epsilon_r= 81, \mu_r= 1. If a wave is incident from water onto a water-air interface, calculate the critical angle. If the incident E_1 = 1 V m^-1 and the incident angle is 45°, calculate the magnitude of the field strength in the air (a) at the interface and (b) \lambda/4 away from the interface.
(A) at the interface: 1.52 Vm^-1, away from the interface: 80.2 \mu Vm^-1
(B) at the interface: 2.00 Vm^-1, away from the interface: 90.0 \mu Vm^-1
(C) at the interface: 0.72 Vm^-1, away from the interface: 37.2 \mu Vm^-1
(D) at the interface: 1.42 Vm^-1, away from the interface: 73.2 \mu Vm^-1
(E) at the interface: 1.82 Vm^-1, away from the interface: 63.2 \mu Vm^-1
(F) at the interface: 0.92 Vm^-1, away from the interface: 30.4 \mu Vm^-1
(G) at the interface: 1.22 Vm^-1, away from the interface: 60.5 \mu Vm^-1
(H) at the interface: 1.62 Vm^-1, away from the interface: 40.8 \mu Vm^-1
(I) at the interface: 2.42 Vm^-1, away from the interface: 83.2 \mu Vm^-1
(J) at the interface: 1.00 Vm^-1, away from the interface: 50.0 \mu Vm^-1
|
D
|
supergpqa_Science:cot
|
672
|
66bbc4f83bce42188ec373be48d0039c
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| false
| true
| true
| true
|
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: Two $5$ -digit numbers are called "responsible" if they are:
$$\begin{align*} &\text {i. In form of abcde and fghij such that fghij = 2(abcde)}\\ &\text {ii. all ten digits, a through j are all distinct.}\\ &\text {iii.} a + b + c + d + e + f + g + h + i + j = 45\end{align*}$$
If two "responsible" numbers are small as possible, what is the sum of the three middle digits of $\text {abcde}$ and last two digits on the $\text {fghij}$ ? That is, $b + c + d + i + j$ .
(A) 22
(B) 21
(C) 20
(D) 23
(E) 24
(F) 26
(G) 25
(H) 27
|
A
|
supergpqa_Science:cot
|
79
|
d16499eb98a643b0a414050a50467596
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| false
| true
| true
| true
|
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: Let $( M^{2}, d s^{2} )$ be a minimal surface in $\mathbb{R}^{3}$ , where $d s^{2}$ igtereticig of the Euclidean metric. Assume that the Gaussian curvature $K$ of $( M^{2}, d s^{2} )$ Hatin Daote v $\widetilde{K}$ the Gaussian curvature of the metric $\widetilde{d s^{2}}=-K d s^{2}$ . So $\widetilde{K}=$ _______ .
(A) $$-3$$
(B) $$\pi$$
(C) $$\frac{1}{2}$$
(D) $$\frac{-1}{2}$$
(E) $$1$$
(F) $$0$$
(G) $$-2$$
(H) $$3$$
(I) $$-1$$
(J) $$2$$
|
E
|
supergpqa_Science:cot
|
324
|
1ab34e22c8ef45f08ce4e2d79ca4ee36
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| false
| true
| true
| true
|
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: A charged spherical drop of mercury is in equilibrium in a plane horizontal air capacitor and the intensity of the electric field is $6\times 10^4 \;Vm^{-1}$. If the charge on the drop is $8\times 10^{-18} \;C$, the radius of the drop is :$\left [\rho_{air} = 1.29 \;kg/m^3 ; \rho_{Hg} = 13.6\times 10^3 \;kg/m^3 \right ]$
(A) $$1.90 \times 10^{-8} \;m$$
(B) $$2.7 \times 10^{-10} \;m$$
(C) $$1.90 \times 10^{-10} \;m$$
(D) $$0.95\times 10^{-8} \;m$$
(E) $$0.95\times 10^{-6} \;m$$
(F) $$3.80 \times 10^{-8} \;m$$
(G) $$1.90 \times 10^{-6} \;m$$
(H) $$2.7\times 10^{-8} \;m$$
(I) $$1.35 \times 10^{-10} \;m$$
|
E
|
supergpqa_Science:cot
|
771
|
4b460c2d40974304b34a313768f1a9af
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| false
| true
| true
| true
|
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: The experiment measured that the heat capacity of iron is $c_{V}^{( 1 )}=0.054 \ \text{cal/mol}\cdot K$ at $T_{1} = 20 \ \text{K}$, and $c_{V}^{( 2 )}=0.18 \ \text{cal/mol}\cdot K$ at $T_{2} = 30 \ \text{K}$. What is the Debye temperature of iron ?
(A) $$
5 9 1 ( \mathrm{K} )
$$
(B) $$
2 7 8 ( \mathrm{K} )
$$
(C) $$
4 1 3 ( \mathrm{K} )
$$
(D) $$
5 0 7 ( \mathrm{K} )
$$
(E) $$
4 2 5 ( \mathrm{K} )
$$
(F) $$
3 8 4 ( \mathrm{K} )
$$
(G) $$
3 2 9 ( \mathrm{K} )
$$
(H) $$
4 7 6 ( \mathrm{K} )
$$
(I) $$
6 3 0 ( \mathrm{K} )
$$
(J) $$
4 8 2 ( \mathrm{K} )
$$
|
C
|
supergpqa_Science:cot
|
3281
|
ddd9b681a2764e84a5950cb359481b61
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| false
| true
| null | null |
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: Let $p(x)$ be a nonzero polynomial of degree less than 1992 having no nonconstant factor in common with $x^3 - x$. Let \[ \frac{d^{1992}}{dx^{1992}} \left( \frac{p(x)}{x^3 - x} \right) = \frac{f(x)}{g(x)} \] for polynomials $f(x)$ and $g(x)$. Find the smallest possible degree of $f(x)$.
(A) 3986
(B) 3983
(C) 3985
(D) 3989
(E) 3987
(F) 3988
(G) 3984
(H) 3982
(I) 3980
(J) 3981
|
G
|
supergpqa_Science:cot
|
58
|
3f2aceb643cd49dcbbfeb538d52f3c38
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| false
| true
| true
| true
|
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: $aK_2Cr_2O_7+bKCl+cH_2SO_4\rightarrow xCrO_2Cl_2+yKHSO_4+zH_2O$Find the value of $a,b,c,x,y,\ and\ z$.
(A) a=4,b=2,c=6,x=2,y=6,z=3
(B) a=2,b=4,c=6,x=2,y=6,z=3
(C) a=4,b=2,c=6,x=6,y=2,z=3
(D) a=4,b=1,c=6,x=2,y=6,z=3
(E) a=3,b=4,c=6,x=2,y=6,z=3
(F) a=2,b=4,c=6,x=2,y=6,z=5
(G) a=4,b=2,c=6,x=3,y=6,z=2
(H) a=6,b=4,c=2,x=6,y=3,z=w
(I) a=2,b=4,c=6,x=2,y=6,z=4
(J) a=1,b=4,c=6,x=2,y=6,z=3
|
J
|
supergpqa_Science:cot
|
1882
|
4f8d7205da624a7dabfde77528c8578a
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| false
| true
| null | null |
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: A spherical soap bubble, which is filled with air, the air quality is not counted, the bubble is a vacuum, its radius is $r_0$ when equilibrium, due to the disturbance, the soap bubble does a small radial expansion, contraction vibration. What is its vibration period ?
( The quality and surface strength coefficient of the soap bubble are known to be $m$ and $\sigma$ respectively, and the air temperature in the bubble remains unchanged during the vibration process. )
(A) $$
\frac{\sigma}{\sqrt{8 \pi m}}$$
(B) $$
\sqrt{\frac{8 m} {\pi \sigma}}$$
(C) $$
\sqrt{\frac{\pi m} {8 \sigma}}
$$
(D) $$
\sqrt{\frac{8 \sigma} {\pi m}}$$
(E) $$
\frac{8 \sigma}{\sqrt{\pi m}}$$
(F) $$
\sqrt{\frac{\pi \sigma} {8 m}}$$
(G) $$
\sqrt{\frac{m^2 \pi} {8 \sigma}}$$
(H) $$
\sqrt{\frac{8 \pi \sigma }{m}}$$
(I) $$
\frac{m}{\sqrt{8 \pi \sigma}}$$
(J) $$
\pi \sqrt{\frac{8 m}{ \sigma}}$$
|
C
|
supergpqa_Science:cot
|
2606
|
ee71cc2ee26f41c1b1435bce1caf707e
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| true
| true
| null | null |
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: A car travelling towards a hill at 10 m/s sounds its horn which has a frequency 500 Hz. This is heard in a second ear travelling behind the first car in the same direction with speed 20 m/s. The sound can also be heard in the second car will be : (speed of sound in air = 340 m/s)
(A) 27 Hz
(B) 34 Hz
(C) 24 Hz
(D) 25 Hz
(E) 23 Hz
(F) 21 Hz
(G) 33 Hz
(H) 29 Hz
(I) 31 Hz
(J) 32 Hz
|
I
|
supergpqa_Science:cot
|
3950
|
36b6e7cf3610481b8bf41fc6525de512
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| false
| true
| null | null |
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: The molar volume of mercury at P = 0 and T = 273°K is 14.72 cm^3 mol^-1 , and the compressibility is \beta = 3.88 × 10^-11 m^2 N^-1. If \beta is assumed to be constant over the pressure range, calculate the free energy change for the compression of mercury from 0 to 3000 kg cm^-2.
(A) 6100 Nm/mole
(B) 5100 Nm/mole
(C) 3200 Nm/mole
(D) 3900 Nm/mole
(E) 4700 Nm/mole
(F) 4500 Nm/mole
(G) 5500 Nm/mole
(H) 3700 Nm/mole
(I) 2900 Nm/mole
(J) 4300 Nm/mole
|
J
|
supergpqa_Science:cot
|
1686
|
215a16d072ea431bb743ad40c757b413
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| false
| true
| null | null |
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: A cube of coefficient of linear expansion $a_s$ is floating in a bath containing a liquid of coefficient of volume exertion $y_1$. When the temperature is raised by $\Delta T$, the depth d upto which the cube is submerged in the liquid remains the same. Then the relation between $a_s$ and $y _1$ is
(A) $$y_1 = 5 a_s / 2$$
(B) $$y_1 = a_s /2$$
(C) $$y_1 = 2 a_s$$
(D) $$y_1 = 3 a_s / 4$$
(E) $$y_1 = 3 a_s$$
(F) $$y_1 = 3 a_s /2$$
(G) $$y_1 = \frac{5 a_s}{3}$$
(H) $$y_1 = 7 a_s / 3$$
(I) $$y_1 = 4 a_s / 3$$
(J) $$y_1 = \frac{3 a_s}{4}$$
|
C
|
supergpqa_Science:cot
|
2823
|
2c629fc99d574bad84a8a59432afcecb
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| false
| true
| null | null |
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: If $\cos^4(1^{\circ}) + \cos^4(2^{\circ}) + \cdots + \cos^4(179^{\circ}) = \dfrac{m}{n}$ where $m,n$ are relatively prime positive integers, find $m+n$ . (Note: $\cos^4(\theta) = (\cos \theta)^4$ )
(A) 138
(B) 132
(C) 137
(D) 133
(E) 130
(F) 131
(G) 139
(H) 136
(I) 134
(J) 135
|
J
|
supergpqa_Science:cot
|
2546
|
859df26ea5bd4120a308e38f3e8f19bb
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| false
| true
| null | null |
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: Given one-dimensional composite lattice $m=5 \times 1.67 \times 10^{-24} \, \mathrm{g}, \, \frac{M}{m}=4, \beta=1.5 \times 10^{1} \, \mathrm{N/m}$, $(1.5 \times 10^{4} \, \mathrm{dyn/cm})$, find the average number of phonons at 300K, which is ().
(A) 0.123,0.398,0.765
(B) 0.367,0.433,0.897
(C) 0.234,0.309,0.765
(D) 0.415,0.358,0.912
(E) 0.231,0.291,0.782
(F) 0.341,0.274,0.863
(G) 0.452,0.219,0.813
(H) 0.222,0.279,0.876
(I) 0.289,0.194,0.654
(J) 0.345,0.265,0.678
|
H
|
supergpqa_Science:cot
|
2211
|
5e806a42f9f4451eb863390167af3b03
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| false
| false
| null | null |
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: The vapour pressure of a certain liquid is given by the equation:$log_{10}P=3.54595+\dfrac {313.7}{T}+1.40655 log_{10}T$, where, P is the vapour pressure in mm and T is temperature in K. The molar latent heat of vaporisation as a function of temperature and its value in cal at 80 K is :
(A) 1024.56
(B) 822.84
(C) $$922.84$$
(D) $$1056.24$$
(E) 922.48
(F) $$1194$$
(G) $$597$$
(H) $$778.56$$
|
F
|
supergpqa_Science:cot
|
2455
|
20dfe9252f674f55a5a434b315f8540d
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| false
| true
| null | null |
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: We now define an algorithm: The definition of a(n) is: Fundamental discriminants of real quadratic fields; indices of primitive positive Dirichlet L-series. Fundamental discriminants are discriminants of quadratic fields that are square-free and congruent to 0 or 1 mod 4. The discriminant of a quadratic field \\(\\mathbb{Q}(\\sqrt{d})\\) for a square-free integer \\(d\\) is \\(d\\) if \\(d \\equiv 1 (mod~4)\\) or \\(4d\\) if \\(d \\equiv 2, 3 (mod~4)\\). Given the input x_list (a series of values): [52, 53, 54, 55, 56, 57, 58, 59, 60, 61], determine the corresponding output sequence y_list.
(A) [167, 172, 176, 180, 182, 185, 188, 193, 196, 198]
(B) [166, 170, 173, 178, 181, 183, 187, 189, 193, 196]
(C) [168, 172, 173, 177, 181, 184, 185, 188, 193, 197]
(D) [170, 174, 176, 179, 183, 186, 188, 191, 195, 199]
(E) [166, 171, 173, 176, 181, 184, 186, 189, 192, 194]
(F) [165, 170, 172, 175, 178, 182, 185, 187, 191, 195]
(G) [164, 168, 172, 177, 179, 183, 185, 188, 192, 196]
(H) [169, 173, 175, 178, 182, 185, 187, 189, 194, 198]
(I) [167, 171, 174, 176, 180, 183, 186, 190, 192, 196]
(J) [163, 169, 175, 178, 181, 184, 186, 191, 194, 197]
|
C
|
supergpqa_Science:cot
|
1662
|
50c875453bbb4858b2bbf952a9604c94
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| false
| false
| null | null |
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: We now define an algorithm: The definition of a(n) is: Triangle with subscripts (1,1),(2,1),(1,2),(3,1),(2,2),(1,3), etc. in which entry (i,j) is the floor function of the ratio i/j. Given the input x_list (a series of values): [93, 94, 95, 96, 97, 98, 99, 100, 101, 102], determine the corresponding output sequence y_list.
(A) [5, 5, 3, 3, 2, 2, 1, 1, 1, 1]
(B) [6, 4, 2, 2, 1, 1, 0, 0, 0, 0]
(C) [5, 4, 3, 2, 2, 1, 1, 1, 0, 0]
(D) [7, 3, 3, 3, 2, 1, 1, 1, 0, 0]
(E) [6, 4, 2, 2, 1, 1, 1, 0, 0, 0]
(F) [6, 4, 2, 3, 1, 0, 0, 0, 0, 0]
(G) [6, 4, 3, 1, 1, 1, 0, 0, 0, 0]
(H) [6, 5, 2, 2, 1, 1, 1, 0, 0, 0]
(I) [4, 4, 4, 2, 2, 2, 1, 0, 0, 0]
(J) [5, 4, 2, 2, 1, 1, 0, 0, 0, 0]
|
B
|
supergpqa_Science:cot
|
1370
|
bacbc45b1e2a4169aef2af6009d2bb44
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| false
| true
| true
| true
|
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: For the substances $\mathrm{C_{2}H_{5}OH (l)}$, $\mathrm{CO_{2} (g)}$, and $\mathrm{H_{2}O (l)}$ at 298 K, the standard enthalpies of formation per mole are $-276.1 \, \text{kJ} \cdot \text{mol}^{-1}$, $-393.3 \, \text{kJ} \cdot \text{mol}^{-1}$, and $-285.8 \, \text{kJ} \cdot \text{mol}^{-1}$ respectively. The combustion enthalpies of $\mathrm{CO (g)}$ and $\mathrm{CH_{4} (g)}$ at 298 K are $-284.5 \, \text{kJ} \cdot \text{mol}^{-1}$ and $-887 \, \text{kJ} \cdot \text{mol}^{-1}$ respectively; the molar heat capacities at constant pressure, $G_{p, m}$, for $\mathrm{CH_{4} (g)}$, $\mathrm{CO_{2} (g)}$, and $\mathrm{C_{2}H_{5}OH (l)}$ are $20.92 \, \text{J} \cdot \text{mol}^{-1} \cdot \text{K}^{-1}$, $29.29 \, \text{J} \cdot \text{mol}^{-1} \cdot \text{K}^{-1}$, and $133.9 \, \text{J} \cdot \text{mol}^{-1} \cdot \text{K}^{-1}$ respectively. Calculate the standard enthalpy change, $\Delta_{t} H_{m}^{\Theta}$, for the following reaction at $298 \, \text{K}$:
$$ 3 \, \mathrm{CH_{4} \ (g)} + \mathrm{CO_{2} \ (g)} \Longrightarrow 2 \, \mathrm{C_{2}H_{5}OH}. $$.
(A) $$
1 1 2. 6 \mathrm{~ k J ~ \cdot~ m o l}^{-1}
$$
(B) $$
8 5. 2 \mathrm{~ k J ~ \cdot~ m o l}^{-1}
$$
(C) $$
6 5. 9 \mathrm{~ k J ~ \cdot~ m o l}^{-1}
$$
(D) $$
1 0 0. 8 \mathrm{~ k J ~ \cdot~ m o l}^{-1}
$$
(E) $$
7 9. 4 \mathrm{~ k J ~ \cdot~ m o l}^{-1}
$$
(F) $$
6 0. 7 \mathrm{~ k J ~ \cdot~ m o l}^{-1}
$$
(G) $$
5 0. 3 \mathrm{~ k J ~ \cdot~ m o l}^{-1}
$$
(H) $$
9 0. 1 \mathrm{~ k J ~ \cdot~ m o l}^{-1}
$$
(I) $$
7 4. 8 \mathrm{~ k J ~ \cdot~ m o l}^{-1}
$$
(J) $$
8 0. 3 \mathrm{~ k J ~ \cdot~ m o l}^{-1}
$$
|
I
|
supergpqa_Science:cot
|
3532
|
081b00459e7c4799b0d62e54263990eb
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| true
| true
| null | null |
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: Twenty red points are equally spaced on the circumference of a circle. How many right triangles are there whose vertices are all red points?
(A) 188
(B) 190
(C) 168
(D) 176
(E) 170
(F) 172
(G) 182
(H) 180
(I) 184
(J) 160
|
H
|
supergpqa_Science:cot
|
3045
|
6bcdfaf70ef64941b23ca8cda118c8e2
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| true
| true
| null | null |
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: Find the volume of the region bounded by \((\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2})^2=ax (a,b,c>0)\).
(A) \(\frac{1}{3}\pi a^2b^2c^2 \)
(B) \( rac{1}{3}\pi ab^2c^2\)
(C) \( rac{1}{3}\pi ab^3c^3\)
(D) \(\frac{1}{3}\pi a^3bc\)
(E) \(\frac{1}{3}\pi a^2b^2c \)
(F) \(\frac{1}{3}\pi a^3b^2c \)
(G) \(\frac{1}{3}\pi a^2b^3c \)
(H) \(\frac{1}{3}\pi ab^2c^3\)
(I) \( rac{1}{3}\pi abc\)
(J) \(\frac{1}{3}\pi a^2bc^2\)
|
D
|
supergpqa_Science:cot
|
993
|
2260ef96858d4055ae069b4f34c1877a
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| false
| true
| true
| true
|
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: The degradation of $CF_{3}CH_{2}F$ (an HFC) by OH radicals in the troposphere is first order in each reactant and has a rate constant of $k=1. 6 \times1 0^{8} \, M^{-1} \, \mathrm{s}^{-1}$ at 4 °C. If the tropospheric concentrations of $OH$ and $CF_{3}CH_{2}F$ are 8.1 $\times1 0^{5}$ and 63 $\times1 0^{8}$ molecules $\mathrm{c m}^{-3},$ respectively, what is the rate of reaction at this temperature in $M/s$?
(A) $$
2. 3 \times1 0^{-1 9} \, M / {\mathrm{s}}
$$
(B) $$
2. 3 \times1 0^{-1 1} \, M / {\mathrm{s}}
$$
(C) $$
4. 8 \times1 0^{-1 9} \, M / {\mathrm{s}}
$$
(D) $$
9. 4 \times1 0^{-1 6} \, M / {\mathrm{s}}
$$
(E) $$
8. 5 \times1 0^{-1 9} \, M / {\mathrm{s}}
$$
(F) $$
6. 9 \times1 0^{-2 0} \, M / {\mathrm{s}}
$$
(G) $$
7. 2 \times1 0^{-1 8} \, M / {\mathrm{s}}
$$
(H) $$
5. 7 \times1 0^{-1 8} \, M / {\mathrm{s}}
$$
(I) $$
3. 2 \times1 0^{-1 9} \, M / {\mathrm{s}}
$$
(J) $$
1. 9 \times1 0^{-1 7} \, M / {\mathrm{s}}
$$
|
A
|
supergpqa_Science:cot
|
311
|
a058e159f637424a9710f2dc58fe6bd2
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| false
| false
| true
| true
|
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: In certain localized regions of some semiconductor devices, there exists an extremely strong electric field, causing the electron temperature $T$ to differ from the lattice temperature $T_{\mathrm{L}}$ (magnetic field effect). Let $\mu_{\mathrm{n}}$ and $\mathrm{n}_{\mathrm{e}}$ be the electron mobility and energy relaxation time, respectively. Suppose $\tau_{\mathrm{e}} = 10^{-11} \, \mathrm{s}$, $\mu_{\mathrm{n}} = 10^{3} \, \mathrm{cm}^{2} / \mathrm{V} \cdot\, \mathrm{s}$, and $\mathcal{E} = 10^{3} \, \mathrm{V} / \mathrm{cm}$; what is the difference $\Delta T$ between the electron temperature $T_{\mathrm{e}}$ and the lattice temperature?
(A) $$
6 6. 1 \mathrm{K}
$$
(B) $$
4 3. 7 \mathrm{K}
$$
(C) $$
7 0. 0 \mathrm{K}
$$
(D) $$
8 1. 6 \mathrm{K}
$$
(E) $$
7 7. 3 \mathrm{K}
$$
(F) $$
9 8. 5 \mathrm{K}
$$
(G) $$
9 9. 0 \mathrm{K}
$$
(H) $$
8 8. 2 \mathrm{K}
$$
(I) $$
6 4. 4 \mathrm{K}
$$
(J) $$
5 5. 9 \mathrm{K}
$$
|
E
|
supergpqa_Science:cot
|
215
|
211f3b0fe38b4ae993212a7398a5a2f2
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| true
| true
| true
| true
|
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: Suppose that $\Omega_1$ and $\Omega_2$ are two circles with radii $9$ and $16$ , respectively, and that they are externally tangent at $P$ . A common external tangent $\ell$ meets $\Omega_1$ at $X$ and $\Omega_2$ at $Y$ . $\overleftrightarrow{XP}$ meets $\Omega_2$ again at $W$ , and $\overleftrightarrow{YP}$ meets $\Omega_1$ again at $Z$ . If $WZ$ can be expressed as $a \sqrt b$ for positive integers $a$ and $b$ with $b$ squarefree, compute $a+ b$ .
(A) 194
(B) 195
(C) 190
(D) 198
(E) 196
(F) 191
(G) 192
(H) 199
(I) 193
(J) 197
|
B
|
supergpqa_Science:cot
|
2143
|
707e6a8314384365be4ed870d8801c1e
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| false
| true
| null | null |
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: The value of $\frac{1}{399!}\left(\sum_{i=2}^{200}\frac{199!(399-i)!}{(200-i)!}-\sum_{i=2}^{100}\frac{99!(399-i)!}{(100-i)!}\right)$ can be expressed as $\frac{m}{n}$ where $\gcd(m,n)=1$ . Find the remainder when $m+n$ is divided by $1000$
(A) 596
(B) 598
(C) 597
(D) 599
(E) 595
(F) 592
(G) 593
(H) 594
(I) 601
|
I
|
supergpqa_Science:cot
|
2547
|
8c881e2d900943ffa1fa6ea5f4935831
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| true
| true
| null | null |
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: If the concentrations of ADP and Pi inside the cell are $3mM$ and $1mM$ respectively, and the standard free energy change for ATP hydrolysis is $\Delta G^{\circ}=-7.3 \mathrm{kcal} \cdot \mathrm{mol}^{-1}$, try to calculate the concentration of ATP at equilibrium (temperature is 37°C). However, under these conditions, if the concentration of ATP is $10mM$, what would be the $\Delta G_{T\cdot P}$ of the reaction $\mathbf{ATP} \rightarrow \mathbf{ADP} + \mathbf{Pi}$?
(A) $$
- 5 1. 2 9 \mathbf{k J} \cdot\mathrm{m o l}^{-1}
$$
(B) $$
- 5 3. 6 6 \mathbf{k J} \cdot\mathrm{m o l}^{-1}
$$
(C) $$
- 4 9. 8 5 \mathbf{k J} \cdot\mathrm{m o l}^{-1}
$$
(D) $$
- 4 2. 1 0 \mathbf{k J} \cdot\mathrm{m o l}^{-1}
$$
(E) $$
- 6 2. 3 4 \mathbf{k J} \cdot\mathrm{m o l}^{-1}
$$
(F) $$
- 5 5. 4 0 \mathbf{k J} \cdot\mathrm{m o l}^{-1}
$$
(G) $$
- 5 8. 7 8 \mathbf{k J} \cdot\mathrm{m o l}^{-1}
$$
(H) $$
- 6 0. 1 9 \mathbf{k J} \cdot\mathrm{m o l}^{-1}
$$
(I) $$
- 5 0. 9 4 \mathbf{k J} \cdot\mathrm{m o l}^{-1}
$$
(J) $$
- 4 7. 3 2 \mathbf{k J} \cdot\mathrm{m o l}^{-1}
$$
|
A
|
supergpqa_Science:cot
|
1237
|
2ced4b4da0c44003b4a396f088d2c15e
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| true
| true
| false
| true
|
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: A closed rectangular vessel completely filled with a liquid of density $\rho$ moves with an acceleration $a=g$ The value of the pressure difference $\left( P _ { 1 } - P _ { 2 } \right)$ is:
(A) $$\rho g \left( b + \frac{h}{5} \right)$$
(B) $$\rho ( a b - g h )$$
(C) $$\rho g \left( b + \frac{h}{2} \right)$$
(D) $$\rho g b$$
(E) $$\rho g \left( b + \frac{h}{3} \right)$$
(F) $$\dfrac { \rho g ( b + h ) } { 2 }$$
(G) $$\rho g \left( b + \frac{h}{4} \right)$$
(H) $$\rho g \left( b - \frac{h}{2} \right)$$
(I) $$\rho gh$$
(J) $$\rho g \left( b + \frac{h}{6} \right)$$
|
I
|
supergpqa_Science:cot
|
894
|
17b6b7a5588d41d4934564ab3fae25ba
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| false
| false
| true
| true
|
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: In order to calculate the cationic and anionic mobilities of the potassium (K^+) and chloride ions (Cl^-) in a .05 normal solution at 25°C, the moving boundary technique was used to first determine t_+, the cationic transport number. In a cell having a cross-sectional area of 1 cm^2, a current of .007A was applied for 1 hr. The observed boundary moved 2.56 cm. Given that ^ = 136.0 cm^2 mol^-1 \Omega^-1 for the solution, find \mu_+ and \mu_-, the cationic and anionic mobilities, respectively.
(A) \mu_+ = 5.91 × 10^-8m^2s^-1v^-1, \mu_- = 7.00 × 10^-8m^2s^-1v^-1
(B) \mu_+ = 7.19 × 10^-9m^2s^-1v^-1, \mu_- = 6.91 × 10^-9m^2s^-1v^-1
(C) \mu_+ = 6.91 × 10^-9m^2s^-1v^-1, \mu_- = 7.19 × 10^-9m^2s^-1v^-1
(D) \mu_+ = 6.91 × 10^-8m^2s^-1v^-1, \mu_- = 8.19 × 10^-8m^2s^-1v^-1
(E) \mu_+ = 6.91 × 10^-8m^2s^-1v^-1, \mu_- = 7.19 × 10^-8m^2s^-1v^-1
(F) \mu_+ = 5.00 × 10^-8m^2s^-1v^-1, \mu_- = 5.30 × 10^-8m^2s^-1v^-1
(G) \mu_+ = 8.00 × 10^-8m^2s^-1v^-1, \mu_- = 8.50 × 10^-8m^2s^-1v^-1
(H) \mu_+ = 7.50 × 10^-8m^2s^-1v^-1, \mu_- = 6.50 × 10^-8m^2s^-1v^-1
(I) \mu_+ = 7.19 × 10^-8m^2s^-1v^-1, \mu_- = 6.91 × 10^-8m^2s^-1v^-1
(J) \mu_+ = 6.91 × 10^-7m^2s^-1v^-1, \mu_- = 7.19 × 10^-7m^2s^-1v^-1
|
E
|
supergpqa_Science:cot
|
2410
|
eb9943bcd11449f291e7fb0458b04c0c
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| true
| true
| null | null |
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: A gaseous mixture enclosed in a vessel of volume V consists of one mole of a gas A with $\gamma$ = $\dfrac { 5 }{ 3 }$ and another gas B with $\gamma$ = $\dfrac { 7 }{ 5 }$ at a certain temperature T. the molar masses of the gases A and B are 4 and 32, respectively. The gases A and B do not reach with each other and are assumed to be ideal. The gaseous mixture follows the equation P${ V }^{ \dfrac { 19 }{ 13 } }$ = constant, in adiabatic processes. The number of moles of the gas B in the gaseous mixture.
(A) 3.2
(B) 3
(C) 5
(D) 1
(E) 1.8
(F) 1.5
(G) 4
(H) 2
(I) 6
(J) 2.5
|
H
|
supergpqa_Science:cot
|
687
|
9ba351c0e8fa484285fddf34d530139b
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| true
| true
| true
| true
|
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: Let $A=(11,4), B=(15,19),$ and $C=(3,18)$ be points in the coordinate plane. A point $D$ is located inside $\triangle ABC$ such that the areas of $\triangle ABD, \triangle BCD, \triangle CAD$ are in a $3:3:2$ ratio. Find the product of the coordinates of $D$ .
(A) 117.25
(B) 116.5
(C) 115
(D) 117.5
(E) 116
(F) 114
(G) 117
(H) 119
(I) 116.25
(J) 118
|
G
|
supergpqa_Science:cot
|
1028
|
c181036d1e724096b6ba0ade0556b714
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| false
| true
| false
| false
|
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: Solve the integral:
$$
\int 3 \cdot \sin(2 \cdot x)^4 \cdot \cos(2 \cdot x)^2 \, dx
$$
(A) \frac{3}{16}\cdot\left(x-\frac{\sin(8\cdot x)}{8}+\frac{\sin(4\cdot x)}{8}+\frac{\sin(12\cdot x)}{24}\right)+C
(B) \frac{3}{16}\cdot\left(x-\frac{\sin(8\cdot x)}{8}-\frac{\sin(4\cdot x)}{8}+\frac{\sin(12\cdot x)}{24}\right)+C
(C) \frac{3}{16}\cdot\left(x+\frac{\sin(8\cdot x)}{8}-\frac{\sin(4\cdot x)}{8}+\frac{\sin(12\cdot x)}{24}\right)+C
(D) \frac{3}{16}\cdot\left(x-\frac{\sin(8\cdot x)}{8}+\frac{\sin(4\cdot x)}{8}-\frac{\sin(12\cdot x)}{24}\right)+C
|
B
|
supergpqa_Science:cot
|
2184
|
0e6375bdf7fb4b5eb235fbdae21b6318
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| false
| true
| null | null |
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: Let $S_n$ denote the set $\{1, 2,..., n\}$ , and define $f(S)$ , where $S$ is a subset of the positive integers, to output the greatest common divisor of all elements in $S$ , unless $S$ is empty, in which case it will output $0$ . Find the last three digits of $\sum_{S \subseteq S_{10}}f(S)$ , where $S$ ranges over all subsets of $S_{10}$ .
(A) 111
(B) 107
(C) 103
(D) 102
(E) 105
(F) 108
(G) 106
(H) 104
(I) 110
(J) 109
|
F
|
supergpqa_Science:cot
|
3159
|
5f1d14b7898843078179131808d8b102
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| false
| true
| null | null |
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: What is the aqueous ammonia concentration of a solution prepared by dissolving $0.15$ mole of $NH_{4}^{+}CH3COO^{-}$ in $1 L$ of water? Given: $K_{a} (CH_{3}COOH) = 1.8 \times 10^{-5}$; $K_{b} (NH_{4}OH) = 1.8 \times 10^{-5}$.
(A) $$5.52 \times 10^{-3}M $$
(B) $$7.3 \times 10^{-4}M$$
(C) $$8.3 \times 10^{-4}M $$
(D) $$9.3 \times 10^{-4}M$$
(E) $$0.15 M$$
(F) $$8.1 \times 10^{-4}M$$
(G) $$1.2 \times 10^{-4}M$$
(H) $$3.8 \times 10^{-4}M $$
(I) $$7.8 \times 10^{-4}M$$
(J) $$6.3 \times 10^{-4}M$$
|
C
|
supergpqa_Science:cot
|
3922
|
7e39ba378ed74eeeaa30d57a63aea490
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| true
| true
| null | null |
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: Let $x_{k} = x_{0} + k h, k = 0, 1, 2, 3$. What is $$\operatorname*{max}_{x_{0} \leqslant x \leqslant x_{3}} | l_{2} ( x )$$?
(A) $$
\approx0. 7 6 3 9
$$
(B) $$
\approx1. 0 3 4 8
$$
(C) $$
\approx1. 0 5 6 3
$$
(D) $$
\approx1. 0 1 4 6
$$
(E) $$
\approx0. 9 8 5 4
$$
(F) $$
\approx1. 0 6 6 0
$$
(G) $$
\approx1. 0 7 2 1
$$
(H) $$
\approx1. 1 4 9 2
$$
(I) $$
\approx1. 1 2 5 7
$$
(J) $$
\approx0. 8 9 7 2
$$
|
C
|
supergpqa_Science:cot
|
2646
|
1ba13c49703e4b848ea2a127fbdaebaf
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| false
| true
| null | null |
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: One mole of an ideal monatomic gas undergoes thermodynamic cycle $ 1 \rightarrow 2 \rightarrow 3 \rightarrow 1 $ as shown in Fig. $ 2.161 . $ Initial temperature of gas is $ T_{0}=300 \mathrm{K} $
Process $ 1 \rightarrow 2: P=a V $
Process $ 2 \rightarrow 3: P V= $ Constant
Process $ 3 \rightarrow 1: P= $ Constant
(Take $ \ln |3|=1.09) $
Find the net work done by the cycle.
(A) $$6.12 R T_{0}$$
(B) $$4.92 R T_{0}$$
(C) $$ 5.81 R T_{0} $$
(D) $$5.12 R T_{0}$$
(E) $$5.32 R T_{0}$$
(F) $$ 4.53 R T_{0} $$
(G) $$5.92 R T_{0}$$
(H) $$ 3.27 R T_{0} $$
(I) $$5.61 R T_{0}$$
(J) $$ 6.83 R T_{0} $$
|
C
|
supergpqa_Science:cot
|
3770
|
673d516d4d404d9c98c7deb2c759cd1e
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| true
| true
| null | null |
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: We now define an algorithm: The definition of a(n) is the sequence generated by the positions of bell 1 (the treble bell) in the n-th permutation of the Plain Bob Minimus change-ringing method, a traditional sequence of permutations in bell-ringing that covers all permutations of {1,2,3,4} with a period of 24. Given the input x_list (a series of values): [90, 91, 92, 93, 94, 95, 96, 97, 98, 99], determine the corresponding output sequence y_list.
(A) [1, 3, 2, 4, 4, 1, 3, 2, 4, 1]
(B) [1, 2, 3, 2, 4, 1, 3, 4, 1, 2]
(C) [3, 1, 4, 2, 2, 3, 4, 1, 3, 2]
(D) [3, 4, 1, 2, 2, 4, 3, 3, 1, 4]
(E) [4, 2, 3, 1, 4, 3, 2, 1, 4, 3]
(F) [1, 1, 2, 2, 3, 3, 4, 4, 3, 2]
(G) [4, 4, 3, 1, 1, 2, 2, 3, 2, 3]
(H) [2, 3, 4, 4, 3, 2, 1, 1, 2, 3]
(I) [4, 3, 2, 1, 1, 2, 3, 4, 2, 1]
(J) [2, 1, 4, 3, 3, 4, 2, 1, 4, 3]
|
H
|
supergpqa_Science:cot
|
2372
|
355ca841d6394ec89681ede639221c19
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| false
| true
| null | null |
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: There is uniform magnetic field B in a circular region of radius R as shown in fig. Whose magnitude changed at the rate of dB/dt. The emf induced across the ends of a circular concentric conducting arc of radius $R_1$ having an angle $\theta$. $(\angle OAO'=\theta)$ is
(A) $$\dfrac{\theta}{2 \pi}R^2 \dfrac{dB}{dt}$$
(B) $$\dfrac {\theta}{2 \pi}R_1^2 \dfrac {dB}{dt}$$
(C) $$\dfrac{\theta}{2 \pi}R_1 \dfrac{dB}{dt}$$
(D) $$\dfrac{\theta}{2 \pi}R_1 R \dfrac{dB}{dt}$$
(E) $$\dfrac {\theta}{2 }R^2 \dfrac {dB}{dt}$$
(F) none of these
(G) $$\dfrac{\theta}{\pi}R_1^2 \dfrac{dB}{dt}$$
(H) $$\dfrac{\theta}{2 \pi}R_1^3 \dfrac{dB}{dt}$$
|
E
|
supergpqa_Science:cot
|
3937
|
93fb85fa16e54fffb106e0e8b6ea0a41
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| true
| false
| null | null |
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: Solve the integral:
$$
\int \left(\frac{ x+3 }{ x-3 }\right)^{\frac{ 3 }{ 2 }} \, dx
$$
(A) C + sqrt((x+3)/(x-3)) * (x-15) - 9 * ln(abs((sqrt(x-3) + sqrt(x+3)) / (sqrt(x-3) - 3*sqrt(x+3))))
(B) C + sqrt((x+3)/(x-3)) * (x-15) - 9 * ln(abs((sqrt(x-3) + 2*sqrt(x+3)) / (sqrt(x-3) - 2*sqrt(x+3))))
(C) C + sqrt((x+3)/(x-3)) * (x-15) - 9 * ln(abs((sqrt(x-3) + 3*sqrt(x+3)) / (sqrt(x-3) - sqrt(x+3))))
(D) C + sqrt((x+3)/(x-3)) * (x-15) - 9 * ln(abs((sqrt(x-3) + 2*sqrt(x+3)) / (sqrt(x-3) - sqrt(x+3))))
(E) C + sqrt((x+3)/(x-3)) * (x-15) - 9 * ln(abs((sqrt(x-3) - 2*sqrt(x+3)) / (sqrt(x-3) + sqrt(x+3))))
(F) C + sqrt((x+3)/(x-3)) * (x-15) - 9 * ln(abs((sqrt(x-3) + sqrt(x+3)) / (sqrt(x-3) - 2*sqrt(x+3))))
(G) C + sqrt((x+3)/(x-3)) * (x-15) - 9 * ln(abs((sqrt(x-3) - 2*sqrt(x+3)) / (sqrt(x-3) + 2*sqrt(x+3))))
(H) C + sqrt((x+3)/(x-3)) * (x-15) - 9 * ln(abs((sqrt(x-3) + sqrt(x+3)) / (sqrt(x-3) + 2*sqrt(x+3))))
(I) C + sqrt((x+3)/(x-3)) * (x-15) - 9 * ln(abs((sqrt(x-3) + sqrt(x+3)) / (sqrt(x-3) - sqrt(x+3))))
(J) C + sqrt((x+3)/(x-3)) * (x-15) - 9 * ln(abs((sqrt(x-3) - sqrt(x+3)) / (sqrt(x-3) + sqrt(x+3))))
|
J
|
supergpqa_Science:cot
|
1097
|
4787cccfebd7483a988142b5f0a99e68
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| false
| true
| true
| true
|
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: If two monomers copolymerize with reactivity ratios of $r_{1} = 0.40$ and $r_{2} = 0.60$, and it is required that the ratio of the two structural units in the resulting copolymer be $F_{1} = 0.50$, try to design a reasonable feed ratio for the two monomers. If the feed concentration of monomer 1 is $2 \ \mathrm{mol} \cdot \mathrm{L}^{-1}$, what is the feed concentration of monomer 2?
(A) $$
1. 8 0 {\mathrm{~ m o l ~}} \cdot\mathrm{L}^{-1}
$$
(B) $$
1. 5 9 {\mathrm{~ m o l ~}} \cdot\mathrm{L}^{-1}
$$
(C) $$
1. 6 4 {\mathrm{~ m o l ~}} \cdot\mathrm{L}^{-1}
$$
(D) $$
2. 7 1 {\mathrm{~ m o l ~}} \cdot\mathrm{L}^{-1}
$$
(E) $$
2. 3 5 {\mathrm{~ m o l ~}} \cdot\mathrm{L}^{-1}
$$
(F) $$
2. 1 0 {\mathrm{~ m o l ~}} \cdot\mathrm{L}^{-1}
$$
(G) $$
1. 4 7 {\mathrm{~ m o l ~}} \cdot\mathrm{L}^{-1}
$$
(H) $$
1. 7 2 {\mathrm{~ m o l ~}} \cdot\mathrm{L}^{-1}
$$
(I) $$
1. 3 8 {\mathrm{~ m o l ~}} \cdot\mathrm{L}^{-1}
$$
(J) $$
1. 9 8 {\mathrm{~ m o l ~}} \cdot\mathrm{L}^{-1}
$$
|
C
|
supergpqa_Science:cot
|
2231
|
85b42cdcffa743c7981a2df1f2b5a43b
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| true
| true
| null | null |
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: Two identical coherent sources are placed on a diameter of a circle of radius $R$ at separation $x(<<R)$ symmetrical about the center of the circle. The sources emit identical wavelength $\lambda$ each. The number of points on the circle of maximum intensity is $(x=5\lambda$):
(A) $$24$$
(B) 21
(C) $$20$$
(D) $$22$$
(E) $$26$$
(F) 28
(G) 19
(H) 23
(I) 25
|
C
|
supergpqa_Science:cot
|
2912
|
4a0b44d076df4b458538e29e594ab4a8
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| true
| true
| null | null |
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: The locus of point of trisections of the focal chords of the parabola, $y^2=4x$ is?
(A) $$y^2=x-1$$
(B) $$y^2=2(1-x)$$
(C) 8y^2=4ax
(D) 9y^2=4x+2
(E) 8y^2=4x
(F) $$9y^2=4ax$$
(G) $$9y^2=4ax+1$$
(H) 9y^2=4x
(I) 9y^2=4x+1
(J) None of these
|
F
|
supergpqa_Science:cot
|
1460
|
60ec4fff3c8042b8b26bd4cfdb51ab50
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| false
| false
| false
| true
|
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: In acute triangle $ABC,$ $\ell$ is the bisector of $\angle BAC$ . $M$ is the midpoint of $BC$ . a line through $M$ parallel to $\ell$ meets $AC,AB$ at $E,F,$ respectively. Given that $AE=1,EF=\sqrt{3}, AB=21,$ the sum of all possible values of $BC$ can be expressed as $\sqrt{a}+\sqrt{b},$ where $a,b$ are positive integers. What is $a+b$ ?
(A) 890
(B) 892
(C) 894
(D) 891
(E) 893
|
A
|
supergpqa_Science:cot
|
148
|
db7b40739ad6499ebcf0c05e99fd2e3d
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| false
| false
| true
| true
|
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: A quantity of $5.08\ g$ of iodine held in suspension in water is slowly acted upon by $460\ ml$ of $H_2S$ measured at $0^oC$ and $1$ atm. What weight of sulphur will be liberated? $(I=127)$
(A) $$0.017\ g$$
(B) $$0.64\ g$$
(C) $$0.668\ g$$
(D) $$0.657\ g$$
(E) $$0.665\ g$$
(F) $$0.667\ g$$
(G) $$0.659\ g$$
(H) $$1.297\ g$$
(I) $$0.658\ g$$
(J) $$0.660\ g$$
|
D
|
supergpqa_Science:cot
|
3846
|
ab8979c2a8a2403cbfed885d845ed9e5
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| false
| false
| null | null |
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: Calculate the integral:
$$
\int_{-\sqrt{2}}^{\sqrt{2}} \frac{ 2 \cdot x^7+3 \cdot x^6-10 \cdot x^5-7 \cdot x^3-12 \cdot x^2+x+1 }{ x^2+2 } \, dx
$$
(A) $$
\frac{5\pi - 64}{14\sqrt{2}}
$$
(B) $$
\frac{5\pi - 64}{10\sqrt{2}}
$$
(C) $$
\frac{5\pi - 64}{7\sqrt{2}}
$$
(D) $$
\frac{5\pi - 64}{9\sqrt{2}}
$$
(E) $$
\frac{5\pi - 64}{8\sqrt{2}}
$$
(F) $$
\frac{5\pi - 64}{15\sqrt{2}}
$$
(G) $$
\frac{5\pi - 64}{11\sqrt{2}}
$$
(H) $$
\frac{5\pi - 64}{16\sqrt{2}}
$$
(I) $$
\frac{5\pi - 64}{12\sqrt{2}}
$$
(J) $$
\frac{5\pi - 64}{13\sqrt{2}}
$$
|
B
|
supergpqa_Science:cot
|
1536
|
329ec98bffc0498aa62becacd7e4e108
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| true
| true
| null | null |
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: Let $N$ denote the number of $8$ -tuples $(a_1,a_2,... a_8)$ of real numbers such that $a_1 = 10$ and
$$$\left|a^2_1 - a^2_2 \right|= 10$$$
$$$\left|a^2_2 - a^2_3 \right|= 20$$$
$$$...$$$
$$$\left|a^2_7 - a^2_8 \right|= 70$$$
$$$\left|a^1_8 - a^2_1 \right|= 80$$$
Determine the remainder obtained when $N$ is divided by $1000$ .
(A) 477
(B) 479
(C) 470
(D) 473
(E) 474
(F) 471
(G) 478
(H) 476
(I) 475
(J) 472
|
J
|
supergpqa_Science:cot
|
1148
|
a8cfb5bd783d4a9790cf3801552cf9f3
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| true
| true
| true
| true
|
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: Dissolve 1.0 × 10^-3 mol of sodium naphthalenide in tetrahydrofuran, then quickly add 2.0 mol of styrene. The total volume of the solution is 1L. Assuming the monomers are instantly and uniformly mixed, and half of the monomers have polymerized after 2000 seconds. How much time is needed to reach a polymerization degree of 3000?
(A) $$
t {=} 4 5 0 0 \mathrm{~ s}
$$
(B) $$
t {=} 3 8 0 0 \mathrm{~ s}
$$
(C) $$
t {=} 4 1 0 0 \mathrm{~ s}
$$
(D) $$
t {=} 5 2 0 0 \mathrm{~ s}
$$
(E) $$
t {=} 5 0 0 0 \mathrm{~ s}
$$
(F) $$
t {=} 3 9 0 0 \mathrm{~ s}
$$
(G) $$
t {=} 3 5 0 0 \mathrm{~ s}
$$
(H) $$
t {=} 4 2 0 0 \mathrm{~ s}
$$
(I) $$
t {=} 4 0 0 0 \mathrm{~ s}
$$
(J) $$
t {=} 4 6 0 0 \mathrm{~ s}
$$
|
I
|
supergpqa_Science:cot
|
1213
|
a35ecaa015bf4cd19a3e54ff0541ae49
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| true
| false
| true
| true
|
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: Compute the integral:
$$
\int \frac{ -\sin(2 \cdot x)^4 }{ \cos(2 \cdot x) } \, dx
$$
(A) C + (1/3) * (sin(2 * x))^3 - sin(2 * x) + (1/2) * ln(|1 - sin(2 * x)| / |sin(2 * x) + 1|)
(B) C + (1/3) * (sin(2 * x))^3 + sin(2 * x) + (1/2) * ln(|1 + sin(2 * x)| / |sin(2 * x) - 1|)
(C) C + (1/3) * (sin(2 * x))^3 + sin(2 * x) + (1/2) * ln(|1 - sin(2 * x)| / |sin(2 * x) + 1|)
(D) C + (1/3) * (sin(2 * x))^3 - sin(2 * x) - (1/2) * ln(|1 - sin(2 * x)| / |sin(2 * x) - 1|)
(E) C + (1/3) * (sin(2 * x))^3 + sin(2 * x) - (1/2) * ln(|1 + sin(2 * x)| / |sin(2 * x) - 1|)
(F) C + (1/3) * (sin(2 * x))^3 - sin(2 * x) + (1/2) * ln(|1 + sin(2 * x)| / |sin(2 * x) - 1|)
(G) C + (1/3) * (sin(2 * x))^3 - sin(2 * x) - (1/2) * ln(|1 - sin(2 * x)| / |sin(2 * x) + 1|)
(H) C + (1/3) * (sin(2 * x))^3 + sin(2 * x) - (1/2) * ln(|1 - sin(2 * x)| / |sin(2 * x) + 1|)
(I) C + (1/3) * (sin(2 * x))^3 - sin(2 * x) - (1/2) * ln(|1 + sin(2 * x)| / |sin(2 * x) - 1|)
(J) C + (1/3) * (sin(2 * x))^3 - sin(2 * x) - (1/2) * ln(|1 + sin(2 * x)| / |sin(2 * x) + 1|)
|
E
|
supergpqa_Science:cot
|
108
|
d2d2c4c0864b415da77be5779a29a6c0
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| false
| false
| true
| true
|
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: What is the solubility of solid zinc hydroxide at a $pH$ of $13$? Given that$Zn(OH)_{2}(g) \leftrightharpoons Zn(OH)_{2}(aq); K_{1} = 10^{-6}M$$Zn(OH)_{2}(aq) \leftrightharpoons Zn(OH)^{+} + OH^{-}; K_{2} = 10^{-7}M$$Zn(OH)^{+} \leftrightharpoons Zn^{2+} + OH^{-}; K_{3} = 10^{-4}M$$Zn(OH)_{2}(aq) + OH^{-} \leftrightharpoons Zn(OH)_{3}^{-}; K_{4} = 10^{3}M^{-1}$$Zn(OH)_{3}^{-} + OH^{-} \leftrightharpoons Zn(OH)_{4}^{2-}; K_{5} = 10M^{-1}$
(A) $$10^{-17}$$
(B) $$4 \times 10^{-4}$$
(C) $$5 \times 10^{-4}$$
(D) $$2 \times 10^{-4}$$
(E) $$3 \times 10^{-4}$$
(F) $$10^{-6}$$
(G) $$4 \times 10^{-5}$$
(H) 3 \times 10^{-5}
(I) $$10^{-4}$$
(J) $$1.5 \times 10^{-4}$$
|
D
|
supergpqa_Science:cot
|
2966
|
9373091dca644cb79937d56b33b82ea8
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| true
| true
| null | null |
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: The only electron in the hydrogen atom resides under ordinary conditions on the first orbit. When energy is supplied, the electron moves to higher energy orbit depending on the amount of energy absorbed. When this electron returns to any of the lower orbits, it emits energy. Lyman series is formed when the electron returns to the lowest orbit while Balmer series is formed when the electron returns to second orbit. Similarly, Paschen, Brackett and Pfund series are formed when electron returns to the third, fourth and fifth orbits from higher energy orbits respectively. The maximum number of lines produced when an electron jumps from nth level to ground level is equal Maximum number of lines produced when an electron jumps from nth level to ground level is equal to. $\displaystyle \:\frac{1\left ( n-1 \right )}{2}.$ For example, in the case of $n= 4, $ number of lines produces is 6. $\displaystyle \:\left ( 4\rightarrow 3,4\rightarrow 2,4\rightarrow 1,3\rightarrow 2,3\rightarrow 1,2\rightarrow 1 \right ).$ When an electron returns from to state, the number of lines in the spectrum will be equal to $\displaystyle \:\frac{\left ( n_{2}-n_{1} \right )\left ( n_{2}-n_{1}+1 \right )}{2}$ If the electron comes back from energy level having energy to energy level having energy, then the difference may be expressed in the terms of energy of photon as $\displaystyle \:E_{2}-E_{1}\Delta E,\lambda = \frac{hc}{\Delta E}$ Since h and c are constants, $\Delta E$ corresponds to definite energy; thus each transition from one energy level to another will produce a light of definite wavelength. This is observed as a line in the spectrum of the hydrogen atom. Wavenumber of line is given by the formula $\displaystyle \:\bar{v}= \left ( \frac{1}{n_{1}^{2}} -\frac{1}{n_{2}^{2}}\right ).$ where R is a Rydberg's constant $\displaystyle \:\left ( R= 1.1\times 10^{7}m^{-1} \right )$The energy photon emitted corresponding to transition n = 3 to n=1 is:[$h=6\times10^{-34} J-sec]$
(A) $1.76 \times 10^{-20}$ J
(B) $1.76 \times 10^{-22}$ J
(C) $1.98\times 10^{-18}$ J
(D) $1.76 \times 10^{-19}$ J
(E) None of these
(F) 1.76 \times 10^{-16} J
(G) $1.76 \times 10^{-18}$ J
(H) 1.76 \times 10^{-21} J
(I) $1.76 \times 10^{-17}$ J
|
G
|
supergpqa_Science:cot
|
2452
|
388c8de03ce2427b81ae0ce91fd80137
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| false
| true
| null | null |
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: The equivalent weight of $CuSO_{4}$ (mol. wt. $=M$) in the following reaction is$5CuS + 8KMnO_4 +12 H_2SO_4 \rightarrow 5CuSO_4 +4 K_2SO_{4} + 8MnSO_4 +12H_2O $
(A) \(\dfrac{M}{7} \)
(B) $$\dfrac {M}{6}$$
(C) $$\dfrac {M}{2}$$
(D) \dfrac{M}{5}
(E) \dfrac{M}{12}
(F) $$\dfrac {M}{8}$$
(G) \dfrac{M}{9}
(H) \dfrac{M}{11}
(I) \dfrac{M}{10}
(J) $$\dfrac {M}{4}$$
|
F
|
supergpqa_Science:cot
|
944
|
b92162c2e99e42e0bb7b8204f88151df
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| false
| true
| true
| true
|
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: The equilibrium constant $(K_p)$ for the decomposition of gaseous $H_2O$
$${H_2}O(g)\leftrightharpoons {H_2}(g) + \frac{1}{2}{O_2}(g)$$
is related to degree of dissociated $\alpha $ at a total pressure $p$ is given by
(A) $$K_p = \frac{a^{3/2} p^2}{(1 + \alpha)(2 + \alpha)^{1/2}}$$
(B) $$K_p = \frac{a^3 p^{3/2}}{(1 + \alpha)(2 + \alpha)^{1/2}}$$
(C) $${K_p} = \frac{{{a^{3/2}}{p^{1/2}}}}{{(1 + 2\alpha ){{(2 + \alpha )}^{1/2}}}}$$
(D) $${K_p} = \frac{{{a^{5/2}}{p^{1/2}}}}{{(1 + \alpha ){{(2 + \alpha )}^{1/2}}}}$$
(E) $$K_p = \frac{a^{3/2} p^{1/2}}{(1 + \alpha)(2 + \alpha)^{1/2}}$$
(F) $${K_p} = \frac{{{a^{3/2}}{p^{1/3}}}}{{(1 + \alpha ){{(2 + \alpha )}^{1/2}}}}$$
(G) $${K_p} = \frac{{{a^{3/2}}{p^{1/2}}}}{{(1 - \alpha ){{(2 + \alpha )}^{1/2}}}}$$
(H) $$K_p = \frac{a^3 p^{1/2}}{(1 + \alpha)(2 + \alpha)^{1/2}}$$
(I) $${K_p} = \frac{{{a^{3/2}}{p^{3/2}}}}{{(1 + \alpha ){{(2 + \alpha )}^{1/2}}}}$$
(J) $${K_p} = \frac{{{a^{3/2}}{p^{1/2}}}}{{(1 + \alpha ){{(2 + 2\alpha )}^{1/2}}}}$$
|
G
|
supergpqa_Science:cot
|
2881
|
a3732fc48a5347cd991d8c81dc4cb3fd
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| false
| true
| null | null |
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: $int {{{1 + sin x} over {sin x(1 + cos x)}}} dx = \left( {} \right)$
(A) ${1 over 4}{tan ^3}{x over 2} + tan {x over 2} + ln left| {tan x} right| + C$
(B) ${1 over 4}{tan ^2}{x over 2} + {1 over 2}tan {x over 2} + ln left| {tan {x over 2}} right| + C$
(C) ${1 over 4}{tan ^2}{x over 2} + {1 over 2}tan {x over 2} + {1 over 2}ln left| {tan {x over 2}} right| + C$
(D) ${1 over 4}{tan ^2}{x over 2} + {1 over 2}tan {x over 2} + {1 over 2}ln left| {tan x} right| + C$
(E) ${1 over 4}{tan ^3}{x over 2} + tan {x over 2} + {1 over 2}ln left| {tan {x over 2}} right| + C$
(F) ${1 over 4}{tan ^2}{x over 2} + tan {x over 2} + {1 over 2}ln left| {tan {x over 2}} right| + C$
(G) ${1 over 4}{tan ^2}{x over 2} + {1 over 2}tan {x over 2} + {1 over 2}ln left| {tan {x over 2}} right| + {1 over 2}ln left| {cos x} right| + C$
(H) ${1 over 4}{tan ^2}{x over 2} + {1 over 2}ln left| {tan {x over 2}} right| + C$
(I) ${1 over 4}{tan ^2}{x over 2} + {1 over 2}tan {x over 2} + {1 over 2}ln left| {tan {x over 2}} right| + {1 over 2}ln left| {sin x} right| + C$
(J) ${1 over 4}{tan ^2}{x over 2} + {1 over 2}tan {x over 2} + {1 over 2}ln left| {tan {x over 2}} right| + ln left| {sin x} right| + C$
|
F
|
supergpqa_Science:cot
|
20
|
f138e09ed4b443a39613e9dd0054e080
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| false
| true
| true
| true
|
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: We now define an algorithm: The definition of a(n) is the number of digits in n! (n factorial) excluding any trailing zeros in its decimal representation. Given the input x_list (a series of values): [67, 68, 69, 70, 71, 72, 73, 74, 75, 76], determine the corresponding output sequence y_list.
(A) [82, 83, 86, 87, 87, 89, 91, 93, 93, 95]
(B) [81, 82, 85, 85, 87, 89, 90, 92, 92, 94]
(C) [79, 81, 83, 84, 85, 87, 89, 91, 91, 93]
(D) [77, 80, 82, 84, 85, 87, 89, 91, 91, 93]
(E) [80, 82, 84, 84, 86, 88, 89, 92, 93, 94]
(F) [80, 81, 83, 86, 86, 88, 90, 92, 93, 94]
(G) [80, 82, 84, 85, 86, 88, 90, 92, 92, 94]
(H) [78, 80, 82, 83, 84, 86, 88, 90, 90, 92]
(I) [79, 83, 84, 86, 87, 88, 90, 92, 92, 95]
(J) [81, 83, 85, 86, 87, 89, 91, 93, 93, 95]
|
G
|
supergpqa_Science:cot
|
1380
|
c941ad8b8dd043f0876cc2cedd62b553
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| false
| true
| true
| true
|
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: Compute the integral:
$$
\int_{0}^1 \frac{ \sqrt{x}+1 }{ \sqrt[3]{x}+1 } \, dx
$$
(A) 3*ln(2)+3*pi/2-409/77
(B) 3*ln(2)+3*pi/2-409/78
(C) 3*ln(2)+3*pi/2-409/71
(D) 3*ln(2)+3*pi/2-409/70
(E) 3*ln(2)+3*pi/2-409/69
(F) 3*ln(2)+3*pi/2-409/74
(G) 3*ln(2)+3*pi/2-409/76
(H) 3*ln(2)+3*pi/2-409/75
(I) 3*ln(2)+3*pi/2-409/73
(J) 3*ln(2)+3*pi/2-409/72
|
D
|
supergpqa_Science:cot
|
3061
|
95d255c051414b4a86c2da7fcbe409e4
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| true
| true
| null | null |
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: The average $O-H$ bond energy in $H_{2}O$ with the help of following data :(1) $H_{2}O(l) \rightarrow H_{2}O(g); \triangle H = + 40.6\ KJ\ mol^{-1}$(2) $2H(g) \rightarrow H_{2}(g); \triangle H = + 435.0\ KJ\ mol^{-1}$(3) $O_{2}(g) \rightarrow 2O(g); \triangle H = + 489.6\ KJ\ mol^{-1}$(4) $2H_{2}(g) + O_{2}(g)\rightarrow 2H_{2}O(l); \triangle H = -571.6\ KJ\ mol^{-1}$
(A) $$463.5\ KJ\ mol^{-1}$$
(B) $$461.5\ KJ\ mol^{-1}$$
(C) $$925\ KJ\ mol^{-1}$$
(D) $$445.8\ KJ\ mol^{-1}$$
(E) $$231.3\ KJ\ mol^{-1}$$
(F) $$345.6\ KJ\ mol^{-1}$$
(G) $$489.9\ KJ\ mol^{-1}$$
(H) $$584.9\ KJ\ mol^{-1}$$
(I) $$279.8\ KJ\ mol^{-1}$$
(J) $$462.5\ KJ\ mol^{-1}$$
|
J
|
supergpqa_Science:cot
|
1852
|
e8c044dba20a4a3b959ececad4fa1080
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| true
| true
| null | null |
Answer the following multiple-choice question by giving the correct answer letter in parentheses. Provide CONCISE reasoning for the answer, and make sure to finish the response with "ANSWER: X" where X is one of (A), (B), (C), (D), etc.
Question: Determine the smallest positive integer $y$ such that for any polynomial $t(x)$ with integer coefficients and any integer $k$, the value \[ t^{(y)}(k) = \left. \frac{d^y}{dx^y} t(x) \right|_{x=k} \] (the $y$-th derivative of $t(x)$ evaluated at $k$) is divisible by 2016.
(A) 11
(B) 12
(C) 8
(D) 4
(E) 9
(F) 5
(G) 10
(H) 7
(I) 14
(J) 6
|
C
|
supergpqa_Science:cot
|
1166
|
472e3ff65a29404cb9da9fe5fee7bd96
|
supergpqa
|
supergpqa_Science:cot
| false
| true
| true
| true
| true
| true
|
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