Q. At what temperature does a reaction become spontaneous if ΔH = 50 kJ and ΔS = 0.1 kJ/K?
-
A.
500 K
-
B.
250 K
-
C.
1000 K
-
D.
200 K
Solution
Set ΔG = 0: 0 = ΔH - TΔS; T = ΔH/ΔS = 50 kJ / 0.1 kJ/K = 500 K.
Correct Answer: A — 500 K
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Q. At what temperature does the Gibbs Free Energy change from negative to positive?
-
A.
At absolute zero
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B.
At the melting point
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C.
At the boiling point
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D.
At the transition temperature
Solution
The Gibbs Free Energy changes from negative to positive at the transition temperature, where the system shifts from one phase to another.
Correct Answer: D — At the transition temperature
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Q. At what temperature does the volume of a gas become zero according to Charles's Law?
-
A.
0 K
-
B.
-273.15 °C
-
C.
273.15 K
-
D.
None of the above
Solution
According to Charles's Law, the volume of a gas approaches zero at absolute zero, which is -273.15 °C.
Correct Answer: B — -273.15 °C
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Q. At what temperature does the volume of a gas theoretically become zero?
-
A.
0°C
-
B.
0 K
-
C.
273 K
-
D.
100 K
Solution
According to Charles's Law, the volume of a gas approaches zero at absolute zero, which is 0 K.
Correct Answer: B — 0 K
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Q. At what temperature will the RMS speed of a gas be 1000 m/s if its molar mass is 0.044 kg/mol?
-
A.
300 K
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B.
400 K
-
C.
500 K
-
D.
600 K
Solution
Using v_rms = sqrt(3RT/M), we rearrange to find T = (v_rms^2 * M) / (3R). Plugging in values gives T approximately 500 K.
Correct Answer: C — 500 K
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Q. At what temperature will the RMS speed of a gas be 1000 m/s if its molar mass is 0.044 kg/mol? (R = 8.314 J/(mol K))
-
A.
500 K
-
B.
600 K
-
C.
700 K
-
D.
800 K
Solution
Using v_rms = sqrt(3RT/M), we solve for T: T = (v_rms^2 * M) / (3R) = (1000^2 * 0.044) / (3 * 8.314) = 700 K.
Correct Answer: C — 700 K
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Q. At what temperature will the RMS speed of a gas be 300 m/s if its molar mass is 28 g/mol?
-
A.
300 K
-
B.
600 K
-
C.
900 K
-
D.
1200 K
Solution
Using the formula v_rms = sqrt((3RT)/M), we can rearrange to find T. Setting v_rms = 300 m/s and M = 28 g/mol, we find T = (M * v_rms^2)/(3R) = 600 K.
Correct Answer: B — 600 K
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Q. At what temperature will the RMS speed of a gas be 500 m/s if its molar mass is 0.02 kg/mol? (2000)
-
A.
250 K
-
B.
500 K
-
C.
1000 K
-
D.
2000 K
Solution
Using v_rms = sqrt(3RT/M), rearranging gives T = (v_rms^2 * M) / (3R). Substituting values gives T = 500 K.
Correct Answer: B — 500 K
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Q. At what temperature will the RMS speed of a gas be 600 m/s if its molar mass is 0.02 kg/mol?
-
A.
300 K
-
B.
600 K
-
C.
900 K
-
D.
1200 K
Solution
Using v_rms = sqrt(3RT/M), we can rearrange to find T = (v_rms^2 * M) / (3R). Plugging in values gives T = (600^2 * 0.02) / (3 * 8.314) = 900 K.
Correct Answer: C — 900 K
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Q. Calculate the area between the curves y = x and y = x^2 from x = 0 to x = 1.
-
A.
0.25
-
B.
0.5
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C.
0.75
-
D.
1
Solution
The area is given by the integral from 0 to 1 of (x - x^2) dx. This evaluates to [x^2/2 - x^3/3] from 0 to 1 = (1/2 - 1/3) = 1/6 = 0.5.
Correct Answer: B — 0.5
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Q. Calculate the area between the curves y = x^2 and y = 2x from x = 0 to x = 2.
Solution
The area is given by the integral from 0 to 2 of (2x - x^2) dx. This evaluates to [x^2 - x^3/3] from 0 to 2 = (4 - 8/3) = 4/3.
Correct Answer: A — 2
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Q. Calculate the area between the curves y = x^2 and y = 4 from x = 0 to x = 2.
Solution
The area is given by the integral from 0 to 2 of (4 - x^2) dx. This evaluates to [4x - x^3/3] from 0 to 2 = (8 - 8/3) = 16/3.
Correct Answer: A — 4
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Q. Calculate the area under the curve y = cos(x) from x = 0 to x = π/2.
Solution
The area under the curve y = cos(x) from x = 0 to x = π/2 is given by ∫(from 0 to π/2) cos(x) dx = [sin(x)] from 0 to π/2 = 1 - 0 = 1.
Correct Answer: A — 1
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Q. Calculate the area under the curve y = x^2 + 2x from x = 0 to x = 2.
Solution
The area under the curve is given by ∫(from 0 to 2) (x^2 + 2x) dx = [x^3/3 + x^2] from 0 to 2 = (8/3 + 4) = 20/3.
Correct Answer: B — 6
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Q. Calculate the area under the curve y = x^4 from x = 0 to x = 2.
Solution
The area under the curve y = x^4 from x = 0 to x = 2 is given by ∫(from 0 to 2) x^4 dx = [x^5/5] from 0 to 2 = (32/5) - 0 = 32/5.
Correct Answer: B — 8
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Q. Calculate the derivative of f(x) = e^(2x).
-
A.
2e^(2x)
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B.
e^(2x)
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C.
2xe^(2x)
-
D.
e^(x)
Solution
Using the chain rule, f'(x) = d/dx(e^(2x)) = 2e^(2x).
Correct Answer: A — 2e^(2x)
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Q. Calculate the derivative of f(x) = x^2 * e^x.
-
A.
(2x + x^2)e^x
-
B.
2xe^x
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C.
x^2e^x
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D.
(x^2 + 2x)e^x
Solution
Using the product rule, f'(x) = d/dx(x^2 * e^x) = (x^2 + 2x)e^x.
Correct Answer: D — (x^2 + 2x)e^x
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Q. Calculate the determinant of the matrix [[1, 2], [3, 4]].
Solution
Determinant = (1*4) - (2*3) = 4 - 6 = -2.
Correct Answer: A — -2
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Q. Calculate the determinant of the matrix \( B = \begin{pmatrix} 2 & 3 \\ 5 & 7 \end{pmatrix} \).
Solution
The determinant is calculated as \( 2*7 - 3*5 = 14 - 15 = -1 \).
Correct Answer: D — 10
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Q. Calculate the determinant of the matrix \( G = \begin{pmatrix} 2 & 1 & 3 \\ 1 & 0 & 1 \\ 3 & 1 & 2 \end{pmatrix} \).
Solution
The determinant is calculated as \( 2(0*2 - 1*1) - 1(1*2 - 3*1) + 3(1*1 - 3*0) = 0 \).
Correct Answer: A — -1
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Q. Calculate the determinant of the matrix \( \begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix} \).
Solution
The determinant is calculated as \( 2*4 - 3*1 = 8 - 3 = 5 \).
Correct Answer: A — 5
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Q. Calculate the determinant of the matrix \( \begin{pmatrix} 2 & 3 \\ 5 & 7 \end{pmatrix} \).
Solution
The determinant is calculated as (2*7) - (3*5) = 14 - 15 = -1.
Correct Answer: A — 1
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Q. Calculate the determinant of the matrix: | 1 1 1 | | 2 2 2 | | 3 3 3 |
Solution
The rows are linearly dependent, hence the determinant is 0.
Correct Answer: A — 0
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Q. Calculate the determinant \( \begin{vmatrix} 1 & 2 & 1 \\ 2 & 3 & 1 \\ 3 & 4 & 1 \end{vmatrix} \).
Solution
The determinant is 0 because the rows are linearly dependent.
Correct Answer: A — 0
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Q. Calculate the determinant \( \begin{vmatrix} 2 & 3 \\ 5 & 7 \end{vmatrix} \)
Solution
The determinant is \( 2*7 - 3*5 = 14 - 15 = -1 \).
Correct Answer: A — 1
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Q. Calculate the determinant \( \begin{vmatrix} a & b \\ c & d \end{vmatrix} \)
-
A.
ad - bc
-
B.
ab + cd
-
C.
ac - bd
-
D.
bc - ad
Solution
The determinant is calculated as \( ad - bc \).
Correct Answer: A — ad - bc
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Q. Calculate the determinant | 1 0 0 | | 0 1 0 | | 0 0 1 |.
Solution
The determinant of the identity matrix is 1.
Correct Answer: B — 1
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Q. Calculate the determinant | 2 3 | | 4 5 | + | 1 1 | | 1 1 |.
Solution
The first determinant is -2 and the second is 0, so the total is -2 + 0 = -2.
Correct Answer: B — 1
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Q. Calculate the determinant: | 2 3 1 | | 1 0 2 | | 0 1 3 |.
Solution
The determinant evaluates to 0 as the rows are linearly dependent.
Correct Answer: A — -1
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Q. Calculate the determinant: | 2 3 1 | | 1 0 4 | | 0 5 2 |.
Solution
Using the determinant formula, we find that the determinant evaluates to 0.
Correct Answer: A — -1
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