Q. If the true value of a measurement is 50.0 units and the measured value is 48.0 units, what is the percentage error?
A.
4.0%
B.
2.0%
C.
5.0%
D.
3.0%
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Solution
Percentage error = |(True value - Measured value) / True value| * 100 = |(50.0 - 48.0) / 50.0| * 100 = 4.0%.
Correct Answer: A — 4.0%
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Q. If the true value of a quantity is 100 and the measured value is 98, what is the percentage error?
A.
2%
B.
1%
C.
3%
D.
0.5%
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Solution
Percentage error = (|True value - Measured value| / True value) * 100 = (|100 - 98| / 100) * 100 = 2%.
Correct Answer: A — 2%
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Q. If the true value of a quantity is 200 and the measured value is 198, what is the percentage error?
A.
1%
B.
0.5%
C.
2%
D.
0.25%
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Solution
Percentage error = (|True value - Measured value| / True value) * 100 = (|200 - 198| / 200) * 100 = 1%.
Correct Answer: A — 1%
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Q. If the universal set U = {1, 2, 3, 4, 5} and A = {1, 2}, what is A'?
A.
{3, 4, 5}
B.
{1, 2}
C.
{1, 2, 3}
D.
{2, 3, 4, 5}
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Solution
The complement of A, denoted A', includes all elements in U that are not in A. Thus, A' = {3, 4, 5}.
Correct Answer: A — {3, 4, 5}
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Q. If the value of one of the resistances in a Wheatstone bridge is doubled, what effect does it have on the balance condition?
A.
It remains balanced
B.
It becomes unbalanced
C.
It depends on other resistances
D.
It cannot be determined
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Solution
Doubling one resistance will change the ratio, thus making the bridge unbalanced.
Correct Answer: B — It becomes unbalanced
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Q. If the value of one of the resistances in a Wheatstone bridge is increased, what effect does it have on the balance of the bridge?
A.
It remains balanced
B.
It becomes unbalanced
C.
It depends on the other resistances
D.
It becomes short-circuited
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Solution
Increasing one resistance will generally cause the bridge to become unbalanced.
Correct Answer: B — It becomes unbalanced
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Q. If the vapor pressure of a pure solvent is 80 mmHg and a non-volatile solute is added, resulting in a vapor pressure of 60 mmHg, what is the mole fraction of the solvent in the solution?
A.
0.75
B.
0.50
C.
0.25
D.
0.80
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Solution
Using Raoult's Law, P_solution = X_solvent * P_pure_solvent; 60 mmHg = X_solvent * 80 mmHg; X_solvent = 60/80 = 0.75.
Correct Answer: A — 0.75
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Q. If the vapor pressure of a solution is 80 mmHg and the vapor pressure of the pure solvent is 120 mmHg, what is the mole fraction of the solvent in the solution?
A.
0.67
B.
0.33
C.
0.50
D.
0.75
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Solution
Using Raoult's Law, P_solution = X_solvent * P_solvent^0; 80 = X_solvent * 120; X_solvent = 80/120 = 0.67.
Correct Answer: A — 0.67
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Q. If the vapor pressure of pure solvent A is 100 mmHg, and the mole fraction of A in the solution is 0.6, what is the vapor pressure of the solution according to Raoult's Law?
A.
60 mmHg
B.
100 mmHg
C.
40 mmHg
D.
80 mmHg
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Solution
Using Raoult's Law, P_solution = X_A * P_A^0 = 0.6 * 100 mmHg = 60 mmHg.
Correct Answer: A — 60 mmHg
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Q. If the vapor pressure of pure solvent A is 100 mmHg, and the mole fraction of A in the solution is 0.5, what is the vapor pressure of the solution according to Raoult's Law?
A.
50 mmHg
B.
100 mmHg
C.
150 mmHg
D.
200 mmHg
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Solution
According to Raoult's Law, P_solution = X_A * P_A^0 = 0.5 * 100 mmHg = 50 mmHg.
Correct Answer: A — 50 mmHg
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Q. If the vapor pressure of pure solvent A is 100 mmHg, what is the vapor pressure of a solution containing 0.5 mole fraction of A?
A.
50 mmHg
B.
100 mmHg
C.
75 mmHg
D.
25 mmHg
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Solution
According to Raoult's Law, the vapor pressure of the solution is 0.5 * 100 mmHg = 50 mmHg.
Correct Answer: A — 50 mmHg
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Q. If the vapor pressure of pure solvent is 120 mmHg, what will be the vapor pressure of a solution containing 0.2 moles of solute in 1 mole of solvent?
A.
100 mmHg
B.
110 mmHg
C.
120 mmHg
D.
80 mmHg
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Solution
Using Raoult's Law, the vapor pressure of the solution is P_solution = P_solvent * X_solvent = 120 * (1/(1+0.2)) = 120 * (1/1.2) = 100 mmHg.
Correct Answer: B — 110 mmHg
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Q. If the vapor pressure of pure solvent is 80 mmHg and the mole fraction of the solvent in the solution is 0.75, what is the vapor pressure of the solution?
A.
60 mmHg
B.
80 mmHg
C.
75 mmHg
D.
20 mmHg
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Solution
Using Raoult's Law, P_solution = X_solvent * P_solvent^0 = 0.75 * 80 mmHg = 60 mmHg.
Correct Answer: A — 60 mmHg
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Q. If the vapor pressure of pure water is 23.8 mmHg at 25°C, what is the vapor pressure of a solution containing 1 mole of NaCl in 1 kg of water?
A.
20.0 mmHg
B.
22.0 mmHg
C.
23.0 mmHg
D.
23.8 mmHg
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Solution
Using Raoult's law, the vapor pressure lowering is calculated considering the van 't Hoff factor for NaCl.
Correct Answer: B — 22.0 mmHg
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Q. If the variance of a data set is 16, what is the standard deviation?
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Solution
Standard deviation = √variance = √16 = 4.
Correct Answer: A — 4
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Q. If the vector A = (1, 2) and B = (2, 1), what is the angle between them?
A.
0 degrees
B.
90 degrees
C.
45 degrees
D.
180 degrees
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Solution
Cosine of angle = (A · B) / (|A| |B|) = (1*2 + 2*1) / (√5 * √5) = 4/5, angle = cos^(-1)(4/5).
Correct Answer: C — 45 degrees
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Q. If the vector a = (1, 2) and b = (3, 4), find the angle between them using the dot product.
A.
0 degrees
B.
90 degrees
C.
45 degrees
D.
60 degrees
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Solution
cos(θ) = (a · b) / (|a| |b|). a · b = 1*3 + 2*4 = 11, |a| = √(1^2 + 2^2) = √5, |b| = √(3^2 + 4^2) = 5. Thus, cos(θ) = 11 / (√5 * 5) = 11 / (5√5), θ = 60 degrees.
Correct Answer: D — 60 degrees
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Q. If the vector a = (2, -1) and b = (1, 3), what is a + b?
A.
(3, 2)
B.
(1, 2)
C.
(2, 2)
D.
(3, 1)
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Solution
a + b = (2 + 1, -1 + 3) = (3, 2)
Correct Answer: A — (3, 2)
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Q. If the vector a = (2, -1) and b = (1, 3), what is the cross product a × b?
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Solution
Cross product in 2D = a1*b2 - a2*b1 = 2*3 - (-1)*1 = 6 + 1 = 7
Correct Answer: A — 5
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Q. If the vector a = (2, 2) and b = (2, -2), what is the angle between them?
A.
90 degrees
B.
45 degrees
C.
0 degrees
D.
180 degrees
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Solution
Angle = cos⁻¹((a·b) / (|a||b|)) = cos⁻¹(0) = 90 degrees
Correct Answer: A — 90 degrees
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Q. If the vector a = (2, 2) is scaled by a factor of 3, what is the resulting vector?
A.
(6, 6)
B.
(3, 3)
C.
(2, 2)
D.
(1, 1)
Show solution
Solution
Scaled vector = 3 * a = 3 * (2, 2) = (6, 6)
Correct Answer: A — (6, 6)
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Q. If the vector a = (2, 3) and b = (4, 1), what is the cross product a × b?
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Solution
Cross product a × b = 2*1 - 3*4 = 2 - 12 = -10.
Correct Answer: A — -10
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Q. If the vector a = (2, 3) and b = (4, 1), what is the resultant vector a + b?
A.
(6, 4)
B.
(2, 4)
C.
(4, 2)
D.
(6, 2)
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Solution
Resultant vector a + b = (2+4, 3+1) = (6, 4).
Correct Answer: A — (6, 4)
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Q. If the vector A = (2, 3) is multiplied by 2, what is the resulting vector?
A.
(4, 6)
B.
(2, 3)
C.
(1, 1.5)
D.
(0, 0)
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Solution
Resulting vector = 2 * A = 2 * (2, 3) = (4, 6).
Correct Answer: A — (4, 6)
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Q. If the vector A = (2, 3) is reflected across the line y = x, what is the resulting vector?
A.
(3, 2)
B.
(2, 3)
C.
(0, 0)
D.
(1, 1)
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Solution
Reflection across y = x gives vector (3, 2).
Correct Answer: A — (3, 2)
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Q. If the vector A = (2, 3) is scaled by a factor of 2, what is the resulting vector?
A.
(4, 6)
B.
(2, 3)
C.
(1, 1.5)
D.
(0, 0)
Show solution
Solution
Scaled vector = 2 * A = 2 * (2, 3) = (4, 6).
Correct Answer: A — (4, 6)
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Q. If the vector a = (2, 3, 4) and b = (1, 0, -1), what is a + b?
A.
(3, 3, 3)
B.
(1, 3, 3)
C.
(2, 3, 3)
D.
(2, 3, 5)
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Solution
a + b = (2+1, 3+0, 4-1) = (3, 3, 3).
Correct Answer: A — (3, 3, 3)
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Q. If the vector a = (2, 3, 4) and b = (1, 0, -1), what is the scalar triple product a · (b × a)?
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Solution
The scalar triple product is 0 because a · (b × a) = 0.
Correct Answer: A — 0
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Q. If the vector a = (2, 3, 4) is scaled by a factor of 2, what is the resulting vector?
A.
(4, 6, 8)
B.
(2, 3, 4)
C.
(1, 1.5, 2)
D.
(0, 0, 0)
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Solution
Scaling the vector a by 2 gives (2*2, 2*3, 2*4) = (4, 6, 8).
Correct Answer: A — (4, 6, 8)
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Q. If the vector a = (3, 4) and b = (1, 2), find the cross product a × b.
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Solution
In 2D, a × b = a1*b2 - a2*b1 = 3*2 - 4*1 = 6 - 4 = 2.
Correct Answer: A — -2
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