Q. For the equation x^3 - 6x^2 + 11x - 6 = 0, which of the following is a root?
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Solution
By substituting x = 2 into the equation, we find that 2 is a root since 2^3 - 6(2^2) + 11(2) - 6 = 0.
Correct Answer:
B
— 2
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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
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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 equilibrium 2A(g) ⇌ B(g) + C(g), how many moles of gas are on the reactant side?
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Solution
There are 2 moles of gas on the reactant side (2A).
Correct Answer:
B
— 2
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Q. For the equilibrium 2A(g) ⇌ B(g) + C(g), if the concentration of B is increased, what will happen to the concentration of A?
A.
Increase
B.
Decrease
C.
Remain the same
D.
Cannot be determined
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Solution
According to Le Chatelier's principle, increasing the concentration of B will shift the equilibrium to the left, resulting in a decrease in the concentration of A.
Correct Answer:
B
— Decrease
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Q. For the equilibrium 2A(g) ⇌ B(g) + C(g), if the volume of the container is decreased, what will happen?
A.
Shift to the left
B.
Shift to the right
C.
No change
D.
Depends on the temperature
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Solution
Decreasing the volume increases the pressure, and the equilibrium will shift towards the side with fewer moles of gas, which is the right side in this case.
Correct Answer:
B
— Shift to the right
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Q. For the equilibrium 2A(g) ⇌ B(g) + C(g), if the volume of the container is decreased, what will happen to the equilibrium? (2020)
A.
Shift to the left
B.
Shift to the right
C.
No change
D.
Depends on temperature
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Solution
Decreasing the volume increases the pressure, and the equilibrium will shift towards the side with fewer moles of gas, which is the right side in this case.
Correct Answer:
B
— Shift to the right
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Q. For the equilibrium 2A(g) ⇌ B(g) + C(g), if the volume of the container is doubled, what will happen to the equilibrium position?
A.
Shift to the left
B.
Shift to the right
C.
No change
D.
Depends on the temperature
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Solution
Doubling the volume decreases the pressure, which shifts the equilibrium towards the side with more moles of gas, in this case, the right side.
Correct Answer:
B
— Shift to the right
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Q. For the equilibrium 2A(g) ⇌ B(g) + C(g), if the volume of the container is doubled, what will happen to the equilibrium concentrations?
A.
Concentrations of A will increase
B.
Concentrations of B and C will increase
C.
Equilibrium will not change
D.
Concentrations of A will decrease
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Solution
Doubling the volume decreases the pressure, which shifts the equilibrium towards the side with more moles of gas (B and C).
Correct Answer:
B
— Concentrations of B and C will increase
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Q. For the equilibrium reaction 2SO2(g) + O2(g) ⇌ 2SO3(g), what is the effect of decreasing the volume of the container? (2021)
A.
Shift to the left
B.
Shift to the right
C.
No effect
D.
Increase in temperature
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Solution
Decreasing the volume increases the pressure, and according to Le Chatelier's principle, the equilibrium will shift towards the side with fewer moles of gas, which is the right side in this case.
Correct Answer:
B
— Shift to the right
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Q. For the equilibrium reaction 4HCl(g) ⇌ 2H2(g) + 2Cl2(g), how will the equilibrium shift if the temperature is decreased?
A.
Shift to the right
B.
Shift to the left
C.
No change
D.
Increase in pressure
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Solution
Decreasing the temperature for an endothermic reaction will shift the equilibrium to the left, favoring the formation of HCl.
Correct Answer:
B
— Shift to the left
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Q. For the equilibrium reaction A ⇌ B, if the concentration of B is increased, what will be the effect on the equilibrium? (2023)
A.
Shift to the right
B.
Shift to the left
C.
No change
D.
Increase in temperature
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Solution
Increasing the concentration of B will shift the equilibrium to the left to produce more A, according to Le Chatelier's principle.
Correct Answer:
B
— Shift to the left
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Q. For the equilibrium reaction CO(g) + 2H2(g) ⇌ CH3OH(g), if the temperature is decreased, what will be the effect?
A.
Shift to the right
B.
Shift to the left
C.
No change
D.
Increase in pressure
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Solution
Decreasing the temperature for an exothermic reaction will shift the equilibrium to the right, favoring the formation of products.
Correct Answer:
A
— Shift to the right
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Q. For the equilibrium reaction N2(g) + 3H2(g) ⇌ 2NH3(g), how will the equilibrium shift if H2 is added?
A.
Shift to the right
B.
Shift to the left
C.
No change
D.
Increase in temperature
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Solution
Adding H2 will shift the equilibrium to the right to produce more NH3.
Correct Answer:
A
— Shift to the right
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Q. For the equilibrium reaction N2(g) + 3H2(g) ⇌ 2NH3(g), if the pressure is increased, what will be the effect on the equilibrium?
A.
Shift to the right
B.
Shift to the left
C.
No change
D.
Increase in temperature
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Solution
Increasing the pressure will shift the equilibrium to the right, favoring the formation of NH3, which has fewer moles of gas.
Correct Answer:
A
— Shift to the right
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Q. For the equilibrium reaction N2(g) + 3H2(g) ⇌ 2NH3(g), what happens if the volume of the container is decreased?
A.
Shift to the right
B.
Shift to the left
C.
No change
D.
Increase in temperature
Show solution
Solution
Decreasing the volume increases the pressure, which shifts the equilibrium to the side with fewer moles of gas, favoring the production of NH3.
Correct Answer:
A
— Shift to the right
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Q. For the equilibrium reaction: 2NO2(g) ⇌ N2O4(g), what is the effect of increasing the pressure? (2021)
A.
Shift to the left
B.
Shift to the right
C.
No effect
D.
Increase Kp
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Solution
Increasing the pressure will shift the equilibrium to the side with fewer moles of gas, which is the right side (N2O4) in this case.
Correct Answer:
A
— Shift to the left
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Q. For the equilibrium reaction: H2(g) + I2(g) ⇌ 2HI(g), what is the effect of adding more HI?
A.
Shifts to the right
B.
Shifts to the left
C.
No change
D.
Increases the rate of reaction
Show solution
Solution
Adding more HI increases the concentration of products, which shifts the equilibrium to the left to form more reactants.
Correct Answer:
B
— Shifts to the left
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Q. For the equilibrium reaction: H2(g) + I2(g) ⇌ 2HI(g), what is the effect of adding more I2?
A.
Equilibrium shifts to the right
B.
Equilibrium shifts to the left
C.
No effect
D.
Equilibrium constant changes
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Solution
Adding more I2 increases its concentration, which shifts the equilibrium to the right to produce more HI.
Correct Answer:
A
— Equilibrium shifts to the right
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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
Show solution
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
Show solution
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 maximum value. (2023)
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Solution
The maximum occurs at x = -b/(2a) = -4/(-2) = 2. f(2) = -2^2 + 4(2) + 1 = 5.
Correct Answer:
B
— 5
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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 inflection point.
A.
(1, 1)
B.
(2, 2)
C.
(3, 3)
D.
(4, 4)
Show solution
Solution
f''(x) = 12x - 18. Setting f''(x) = 0 gives x = 1.5. The inflection point is (1.5, f(1.5)).
Correct Answer:
B
— (2, 2)
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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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