A.
e^x + C
B.
e^x
C.
x e^x + C
D.
ln(e^x) + C
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Solution
The integral of e^x is e^x + C.
Correct Answer: A — e^x + C
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Q. For a monatomic ideal gas, the ratio of specific heats (γ) is approximately: (2019)
A.
1.5
B.
1.67
C.
1.4
D.
2
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Solution
For a monatomic ideal gas, γ = C_p/C_v = 5/3 = 1.67.
Correct Answer: B — 1.67
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Q. For a process at constant volume, which of the following is true? (2023)
A.
Work done is zero
B.
Heat added equals change in internal energy
C.
Both A and B
D.
None of the above
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Solution
At constant volume, no work is done (W=0), and the heat added equals the change in internal energy (Q=ΔU).
Correct Answer: C — Both A and B
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Q. For a reaction with a rate constant of 0.02 M⁻¹s⁻¹ and initial concentration of 0.5 M, what is the time taken to reach 0.25 M in a second-order reaction? (2023)
A.
25 s
B.
50 s
C.
10 s
D.
20 s
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Solution
Using t = 1 / (k[A₀]) * (1/[A] - 1/[A₀]), t = 1 / (0.02 * 0.5) * (1/0.25 - 1/0.5) = 25 s.
Correct Answer: A — 25 s
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Q. For a reaction with an activation energy of 50 kJ/mol, what is the rate constant at 300 K if R = 8.314 J/(mol·K)? (2022)
A.
0.001 M/s
B.
0.01 M/s
C.
0.1 M/s
D.
1 M/s
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Solution
Using the Arrhenius equation, k = Ae^(-Ea/RT). Calculate k using the given values.
Correct Answer: C — 0.1 M/s
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Q. For a reaction with an activation energy of 50 kJ/mol, what will happen to the rate if the temperature is increased by 20°C? (2022)
A.
Rate decreases
B.
Rate remains the same
C.
Rate increases significantly
D.
Rate increases slightly
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Solution
Increasing the temperature generally increases the rate of reaction significantly due to higher kinetic energy.
Correct Answer: C — Rate increases significantly
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Q. For a zero-order reaction, if the initial concentration is 0.5 M and the rate constant is 0.1 M/s, how long will it take to reach 0.2 M? (2021)
A.
3 s
B.
5 s
C.
2 s
D.
4 s
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Solution
For a zero-order reaction, t = [A₀ - A] / k. Here, t = (0.5 - 0.2) / 0.1 = 3 s.
Correct Answer: D — 4 s
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Q. For a zero-order reaction, if the initial concentration is 0.5 M and the rate constant is 0.1 M/s, how long will it take to reach 0 M? (2019)
A.
5 s
B.
10 s
C.
15 s
D.
20 s
Show solution
Solution
For a zero-order reaction, t = [A₀] / k. Here, t = 0.5 M / 0.1 M/s = 5 s.
Correct Answer: B — 10 s
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Q. For an ideal gas, the work done during an isobaric process is given by which formula? (2022)
A.
W = PΔV
B.
W = nRT
C.
W = ΔU + Q
D.
W = 0
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Solution
In an isobaric process, the work done is calculated using the formula W = PΔV, where P is pressure and ΔV is the change in volume.
Correct Answer: A — W = PΔV
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Q. For an ideal gas, which law relates pressure, volume, and temperature? (2023) 2023
A.
Boyle's Law
B.
Charles's Law
C.
Ideal Gas Law
D.
Avogadro's Law
Show solution
Solution
The Ideal Gas Law (PV = nRT) relates pressure (P), volume (V), and temperature (T) for an ideal gas.
Correct Answer: C — Ideal Gas Law
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Q. For the equation x² + 4x + k = 0 to have real roots, what must be the minimum value of k? (2023)
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Solution
The discriminant must be non-negative: 4² - 4*1*k ≥ 0, thus k ≤ 4.
Correct Answer: A — -4
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Q. For the equation x² + 6x + k = 0 to have real roots, what is the minimum value of k? (2021)
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Solution
The discriminant must be non-negative: 6² - 4*1*k ≥ 0, which gives k ≤ 9, so the minimum value is -9.
Correct Answer: A — -9
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Q. For the equation x² + 6x + k = 0 to have real roots, what must be the minimum value of k? (2023)
A.
-9
B.
-6
C.
-12
D.
-15
Show solution
Solution
The discriminant must be non-negative: 6² - 4*1*k ≥ 0, thus k ≤ 9.
Correct Answer: A — -9
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Q. For the exothermic reaction A + B ⇌ C + D, what effect does increasing the temperature have on the equilibrium? (2020)
A.
Shifts to the left
B.
Shifts to the right
C.
No change
D.
Increases Kc
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Solution
For an exothermic reaction, increasing the temperature shifts the equilibrium to the left, favoring the reactants.
Correct Answer: A — Shifts to the left
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Q. For the exothermic reaction A + B ⇌ C + heat, what will happen if the temperature is increased? (2020)
A.
Equilibrium will shift to the right
B.
Equilibrium will shift to the left
C.
No change in equilibrium
D.
Equilibrium will shift to the center
Show solution
Solution
For an exothermic reaction, increasing the temperature shifts the equilibrium to the left, favoring the reactants, as the system tries to absorb the added heat.
Correct Answer: B — Equilibrium will shift to the left
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Q. For the exothermic reaction: N2(g) + 3H2(g) ⇌ 2NH3(g), what will happen if the temperature is increased? (2023)
A.
The equilibrium will shift to the right
B.
The equilibrium will shift to the left
C.
No change in equilibrium
D.
The reaction will stop
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Solution
According to Le Chatelier's principle, increasing the temperature of an exothermic reaction shifts the equilibrium to the left, favoring the reactants.
Correct Answer: B — The equilibrium will shift to the left
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Q. For the function f(x) = -x^2 + 4x + 1, find the x-coordinate of the vertex. (2023)
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Solution
The vertex x-coordinate is given by -b/(2a) = -4/(2*-1) = 2.
Correct Answer: A — 2
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Q. For the function f(x) = -x^2 + 6x, find the x-coordinate of the vertex. (2022)
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Solution
The vertex x-coordinate is found using x = -b/(2a) = -6/(-2) = 3.
Correct Answer: A — 3
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Q. For the function f(x) = 2x^2 - 8x + 10, find the minimum value. (2022)
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Solution
The minimum occurs at x = 2. f(2) = 2(2^2) - 8(2) + 10 = 2.
Correct Answer: C — 4
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Q. For the function f(x) = 2x^3 - 9x^2 + 12x, find the critical points. (2022)
A.
(0, 0)
B.
(1, 5)
C.
(2, 0)
D.
(3, 3)
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Solution
Set f'(x) = 0. f'(x) = 6x^2 - 18x + 12 = 0. Critical points are x = 2.
Correct Answer: C — (2, 0)
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Q. For the function f(x) = 2x^3 - 9x^2 + 12x, find the intervals of increase. (2022)
A.
(-∞, 0)
B.
(0, 3)
C.
(3, ∞)
D.
(0, 2)
Show solution
Solution
f'(x) = 6x^2 - 18x + 12. The critical points are x = 1 and x = 2. Test intervals show increase in (0, 3).
Correct Answer: B — (0, 3)
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Q. For the function f(x) = 2x^3 - 9x^2 + 12x, find the local maxima. (2023) 2023
A.
(1, 5)
B.
(2, 6)
C.
(3, 3)
D.
(0, 0)
Show solution
Solution
Find f'(x) = 6x^2 - 18x + 12, set to 0. Local maxima at x = 2 gives f(2) = 6.
Correct Answer: B — (2, 6)
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Q. For the function f(x) = 3x^2 - 12x + 7, find the minimum value. (2022)
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Solution
The vertex occurs at x = 2. f(2) = 3(2^2) - 12(2) + 7 = -5.
Correct Answer: B — -4
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Q. For the function f(x) = 3x^2 - 12x + 7, find the x-coordinate of the vertex. (2022)
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Solution
The x-coordinate of the vertex is given by x = -b/(2a) = 12/(2*3) = 2.
Correct Answer: B — 2
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Q. For the function f(x) = 3x^2 - 12x + 9, find the vertex. (2021)
A.
(2, 3)
B.
(3, 0)
C.
(0, 9)
D.
(1, 6)
Show solution
Solution
The vertex occurs at x = -b/(2a) = 12/(2*3) = 2. f(2) = 3(2^2) - 12(2) + 9 = 3.
Correct Answer: A — (2, 3)
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Q. For the function f(x) = 3x^2 - 12x + 9, find the x-coordinate of the vertex. (2021)
Show solution
Solution
The vertex x-coordinate is found using -b/(2a) = 12/(2*3) = 2.
Correct Answer: B — 2
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Q. For the function f(x) = x^2 + 2x, find the local maximum. (2022)
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Solution
f'(x) = 2x + 2. Setting f'(x) = 0 gives x = -1. f(-1) = 1.
Correct Answer: A — -1
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Q. For the matrix E = [[1, 2, 3], [0, 1, 4], [5, 6, 0]], find det(E). (2021)
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Solution
Using the determinant formula, det(E) = 1*(1*0 - 4*6) - 2*(0*0 - 4*5) + 3*(0*6 - 1*5) = 1*(-24) - 2*(-20) - 15 = -24 + 40 - 15 = 1.
Correct Answer: A — -24
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Q. For the matrix E = [[1, 2, 3], [0, 1, 4], [5, 6, 0]], find the determinant. (2023)
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Solution
Determinant of E = 1(1*0 - 4*6) - 2(0*0 - 4*5) + 3(0*6 - 1*5) = 1(0 - 24) - 2(0 - 20) + 3(0 - 5) = -24 + 40 - 15 = 1.
Correct Answer: A — -24
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Q. For the matrix E = [[1, 2], [2, 4]], what is the determinant? (2021)
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Solution
Determinant of E = (1*4) - (2*2) = 4 - 4 = 0.
Correct Answer: A — 0
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