Q. What is the solution to the equation y' + 3y = 0?
A.
y = Ce^(-3x)
B.
y = Ce^(3x)
C.
y = 3Ce^(-x)
D.
y = Ce^(-x/3)
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Solution
This is a first-order linear differential equation. The solution is y = Ce^(-3x).
Correct Answer: A — y = Ce^(-3x)
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Q. What is the solution to the equation y' = 3y + 6?
A.
y = Ce^(3x) - 2
B.
y = Ce^(3x) + 2
C.
y = 2e^(3x)
D.
y = 3Ce^(x)
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Solution
This is a first-order linear equation. The integrating factor is e^(3x), leading to the solution y = Ce^(3x) + 2.
Correct Answer: B — y = Ce^(3x) + 2
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Q. What is the solution to the equation y'' + 4y = 0?
A.
y = C1 cos(2x) + C2 sin(2x)
B.
y = C1 e^(2x) + C2 e^(-2x)
C.
y = C1 e^(4x) + C2 e^(-4x)
D.
y = C1 sin(4x) + C2 cos(4x)
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Solution
The characteristic equation is r^2 + 4 = 0, giving complex roots. The general solution is y = C1 cos(2x) + C2 sin(2x).
Correct Answer: A — y = C1 cos(2x) + C2 sin(2x)
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Q. What is the solution to the equation y'' - 3y' + 2y = 0?
A.
y = C1 e^(2x) + C2 e^(x)
B.
y = C1 e^(x) + C2 e^(2x)
C.
y = C1 e^(-x) + C2 e^(-2x)
D.
y = C1 + C2x
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Solution
The characteristic equation r^2 - 3r + 2 = 0 has roots 1 and 2, leading to y = C1 e^(x) + C2 e^(2x).
Correct Answer: B — y = C1 e^(x) + C2 e^(2x)
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Q. What is the specific heat at constant volume (Cv) for a monatomic ideal gas? (2019)
A.
3R/2
B.
5R/2
C.
R
D.
2R
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Solution
For a monatomic ideal gas, the specific heat at constant volume (Cv) is 3R/2.
Correct Answer: A — 3R/2
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Q. What is the specific heat capacity of a substance if it requires 500 J to raise the temperature of 2 kg of the substance by 10°C? (2021)
A.
25 J/kg°C
B.
50 J/kg°C
C.
100 J/kg°C
D.
200 J/kg°C
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Solution
Specific heat capacity (c) = Q / (m * ΔT) = 500 J / (2 kg * 10°C) = 25 J/kg°C.
Correct Answer: B — 50 J/kg°C
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Q. What is the specific heat capacity of a substance if it requires 500 J to raise the temperature of 2 kg of the substance by 5°C? (2021)
A.
20 J/kg°C
B.
50 J/kg°C
C.
100 J/kg°C
D.
200 J/kg°C
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Solution
Specific heat capacity (c) = Q / (m * ΔT) = 500 J / (2 kg * 5°C) = 50 J/kg°C.
Correct Answer: B — 50 J/kg°C
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Q. What is the specific heat capacity of a substance if it takes 500 J to raise the temperature of 2 kg of the substance by 5°C? (2021)
A.
20 J/kg°C
B.
50 J/kg°C
C.
100 J/kg°C
D.
200 J/kg°C
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Solution
Specific heat capacity (c) = Q / (m * ΔT) = 500 J / (2 kg * 5°C) = 50 J/kg°C
Correct Answer: B — 50 J/kg°C
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Q. What is the specific heat capacity of water at constant pressure (C_p)? (2020)
A.
4.18 J/g°C
B.
2.09 J/g°C
C.
1.00 J/g°C
D.
3.00 J/g°C
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Solution
The specific heat capacity of water at constant pressure is approximately 4.18 J/g°C.
Correct Answer: A — 4.18 J/g°C
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Q. What is the specific heat capacity of water if it takes 4200 J to raise the temperature of 1 kg of water by 1°C? (2021)
A.
2100 J/kg°C
B.
4200 J/kg°C
C.
8400 J/kg°C
D.
1000 J/kg°C
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Solution
The specific heat capacity of water is defined as the amount of heat required to raise the temperature of 1 kg of water by 1°C, which is given as 4200 J/kg°C.
Correct Answer: B — 4200 J/kg°C
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Q. What is the specific heat capacity of water? (2021) 2021
A.
1 J/g°C
B.
4.18 J/g°C
C.
2 J/g°C
D.
0.9 J/g°C
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Solution
The specific heat capacity of water is approximately 4.18 J/g°C.
Correct Answer: B — 4.18 J/g°C
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Q. What is the speed of light in vacuum? (2020)
A.
3 x 10^8 m/s
B.
3 x 10^6 m/s
C.
3 x 10^7 m/s
D.
3 x 10^9 m/s
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Solution
The speed of light in vacuum is approximately 3 x 10^8 m/s.
Correct Answer: A — 3 x 10^8 m/s
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Q. What is the square of the modulus of the complex number 1 + 2i? (2014)
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Solution
The modulus is √(1^2 + 2^2) = √(1 + 4) = √5. The square of the modulus is 5.
Correct Answer: A — 5
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Q. What is the square of the modulus of the complex number 1 + i? (2020)
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Solution
The modulus is √(1^2 + 1^2) = √2, and the square of the modulus is 2.
Correct Answer: A — 2
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Q. What is the square root of 64? (2020)
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Solution
The square root of 64 is 8.
Correct Answer: C — 8
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Q. What is the square root of the complex number -1? (2021)
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Solution
The square root of -1 is defined as i.
Correct Answer: A — i
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Q. What is the square root of the complex number -4? (2020)
A.
2i
B.
-2i
C.
4i
D.
-4i
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Solution
The square root of -4 is √(-1) * √4 = 2i.
Correct Answer: A — 2i
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Q. What is the standard electrode potential of the half-reaction Zn²⁺ + 2e⁻ → Zn?
A.
0.76 V
B.
0.34 V
C.
-0.76 V
D.
-0.34 V
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Solution
The standard electrode potential for Zn²⁺/Zn is -0.76 V.
Correct Answer: C — -0.76 V
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Q. What is the standard electrode potential of the hydrogen electrode? (2023) 2023
A.
0 V
B.
1 V
C.
2 V
D.
0.5 V
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Solution
The standard electrode potential of the hydrogen electrode is defined as 0 V.
Correct Answer: A — 0 V
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Q. What is the structure of a nucleotide?
A.
Sugar, phosphate, and base
B.
Amino group, carboxyl group, and side chain
C.
Glycerol and fatty acids
D.
Monosaccharides
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Solution
A nucleotide consists of a sugar, a phosphate group, and a nitrogenous base.
Correct Answer: A — Sugar, phosphate, and base
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Q. What is the sum of the angles in a triangle? (2017)
A.
180 degrees
B.
360 degrees
C.
90 degrees
D.
270 degrees
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Solution
The sum of the angles in a triangle is always 180 degrees.
Correct Answer: A — 180 degrees
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Q. What is the sum of the coefficients in the expansion of (2 + 3)^4? (2022)
A.
81
B.
64
C.
100
D.
125
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Solution
The sum of the coefficients is (2 + 3)^4 = 5^4 = 625.
Correct Answer: A — 81
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Q. What is the sum of the coefficients in the expansion of (x + 2)^5? (2021)
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Solution
The sum of the coefficients is found by substituting x=1. So, (1 + 2)^5 = 3^5 = 243.
Correct Answer: B — 64
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Q. What is the sum of the complex numbers 1 + 2i and 3 - 4i? (2023)
A.
4 - 2i
B.
4 + 2i
C.
2 - 2i
D.
2 + 2i
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Solution
(1 + 2i) + (3 - 4i) = (1 + 3) + (2 - 4)i = 4 - 2i.
Correct Answer: A — 4 - 2i
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Q. What is the sum of the complex numbers 3 + 2i and 1 - 4i? (2023)
A.
4 - 2i
B.
2 - 2i
C.
4 + 2i
D.
2 + 2i
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Solution
(3 + 2i) + (1 - 4i) = (3 + 1) + (2 - 4)i = 4 - 2i.
Correct Answer: A — 4 - 2i
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Q. What is the sum of the roots of the equation 2x² - 4x + 1 = 0? (2023)
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Solution
The sum of the roots is given by -b/a = 4/2 = 2.
Correct Answer: A — 2
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Q. What is the sum of the roots of the equation 3x² + 12x + 9 = 0? (2021)
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Solution
The sum of the roots is given by -b/a = -12/3 = -4.
Correct Answer: A — -4
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Q. What is the symbol for Gold?
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Solution
The symbol for Gold is Au.
Correct Answer: A — Au
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Q. What is the temperature in Celsius if the temperature in Kelvin is 273 K?
A.
0°C
B.
100°C
C.
-273°C
D.
273°C
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Solution
To convert Kelvin to Celsius, use the formula: °C = K - 273. Therefore, 273 K - 273 = 0°C.
Correct Answer: A — 0°C
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Q. What is the temperature of a gas in Kelvin if its temperature in Celsius is 25°C? (2019)
A.
273 K
B.
298 K
C.
310 K
D.
325 K
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Solution
Temperature in Kelvin (K) = Temperature in Celsius (°C) + 273.15 = 25 + 273.15 = 298.15 K.
Correct Answer: B — 298 K
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