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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)
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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)
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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) = 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], [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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Q. For the quadratic 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, thus k ≤ 9. The minimum value of k is -9.
Correct Answer: A — -9
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Q. For the reaction 2SO2(g) + O2(g) ⇌ 2SO3(g), what will happen if the volume of the container is increased? (2020) 2020
A.
Equilibrium shifts to the right
B.
Equilibrium shifts to the left
C.
No change in equilibrium
D.
Reaction stops
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Solution
Increasing the volume decreases the pressure, and according to Le Chatelier's principle, the equilibrium will shift to the side with more moles of gas, which is the left side in this case.
Correct Answer: B — Equilibrium shifts to the left
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Q. For the reaction N2(g) + 3H2(g) ⇌ 2NH3(g), what happens to the equilibrium if the volume of the container is decreased? (2020)
A.
Equilibrium shifts to the right
B.
Equilibrium shifts to the left
C.
No change in equilibrium
D.
Equilibrium shifts to the side with more moles
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Solution
Decreasing the volume increases the pressure, and according to Le Chatelier's principle, the equilibrium will shift to the side with fewer moles of gas, which is the right side in this case.
Correct Answer: A — Equilibrium shifts to the right
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Q. For the reaction: 2SO2(g) + O2(g) ⇌ 2SO3(g), what will happen if the volume of the container is increased? (2020)
A.
Equilibrium shifts to the right
B.
Equilibrium shifts to the left
C.
No change in equilibrium
D.
Equilibrium shifts to the center
Show solution
Solution
Increasing the volume decreases the pressure, and according to Le Chatelier's principle, the equilibrium will shift to the side with more moles of gas, which is the left side in this case.
Correct Answer: B — Equilibrium shifts to the left
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Q. For which value of m does the equation x² - mx + 9 = 0 have roots 3 and 3? (2023)
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Solution
The sum of the roots is 3 + 3 = 6, hence m = 6.
Correct Answer: A — 6
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Q. For which value of p does the equation x² + px + 4 = 0 have roots 2 and -2? (2022)
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Solution
Using the sum of roots: 2 + (-2) = -p, hence p = 0.
Correct Answer: C — -4
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Q. For which value of p does the equation x² + px + 4 = 0 have roots that are both negative? (2022)
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Solution
For both roots to be negative, p must be greater than 0 and p² > 16. Thus, p < -4.
Correct Answer: C — -4
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Q. For which value of p does the equation x² + px + 9 = 0 have roots that are both negative? (2021)
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Solution
For both roots to be negative, p must be positive and p² > 4*9. Thus, p > 6, so p = -4 is valid.
Correct Answer: B — -4
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Q. How many different 4-digit PINs can be formed using the digits 0-9 if digits cannot be repeated?
A.
5040
B.
10000
C.
9000
D.
7200
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Solution
The first digit can be any of 10 digits, the second can be any of 9, the third can be any of 8, and the fourth can be any of 7. Total = 10 * 9 * 8 * 7 = 5040.
Correct Answer: A — 5040
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Q. How many different 4-digit PINs can be formed using the digits 0-9 without repetition?
A.
5040
B.
9000
C.
10000
D.
1000
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Solution
The number of 4-digit PINs is P(10, 4) = 5040.
Correct Answer: A — 5040
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Q. How many different ways can 3 boys and 2 girls be seated in a row?
A.
30
B.
60
C.
120
D.
180
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Solution
The number of ways to arrange 5 people is 5! = 120.
Correct Answer: C — 120
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Q. How many different ways can the letters of the word 'MATH' be arranged?
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Solution
The number of arrangements of the letters in 'MATH' is 4! = 24.
Correct Answer: B — 24
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Q. How many grams are in 0.5 moles of glucose (C6H12O6)? (2023)
A.
90 g
B.
180 g
C.
45 g
D.
60 g
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Solution
Molar mass of C6H12O6 = 6*12 + 12*1 + 6*16 = 180 g/mol. Mass = 0.5 moles x 180 g/mol = 90 g.
Correct Answer: A — 90 g
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Q. How many grams are in 4 moles of potassium chloride (KCl)? (2022)
A.
74 g
B.
148 g
C.
296 g
D.
37 g
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Solution
Molar mass of KCl = 39 + 35.5 = 74.5 g/mol. Mass = 4 moles x 74.5 g/mol = 298 g.
Correct Answer: C — 296 g
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Q. How many grams are in 4 moles of sulfuric acid (H2SO4)? (2023)
A.
196 g
B.
98 g
C.
392 g
D.
294 g
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Solution
Molar mass of H2SO4 = 2*1 + 32 + 4*16 = 98 g/mol. Mass = 4 moles x 98 g/mol = 392 g.
Correct Answer: C — 392 g
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Q. How many grams of KCl are needed to prepare 0.5 M solution in 500 mL of water? (2023)
A.
14.9 g
B.
7.45 g
C.
29.8 g
D.
3.73 g
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Solution
Molarity = moles/volume(L). Moles = 0.5 mol/L * 0.5 L = 0.25 mol. Mass = moles * molar mass = 0.25 mol * 74.5 g/mol = 18.625 g.
Correct Answer: B — 7.45 g
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Q. How many grams of KCl are needed to prepare 250 mL of a 0.5 M solution? (2023) 2023
A.
7.45 g
B.
12.5 g
C.
9.25 g
D.
5.25 g
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Solution
Molar mass of KCl = 74.55 g/mol. Grams = Molarity * Volume (L) * Molar mass = 0.5 M * 0.25 L * 74.55 g/mol = 9.31 g.
Correct Answer: A — 7.45 g
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Q. How many grams of KCl are needed to prepare 250 mL of a 2 M solution? (2023)
A.
37.25 g
B.
50 g
C.
75 g
D.
25 g
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Solution
Molar mass of KCl = 74.55 g/mol. Grams = moles × molar mass = 0.5 moles × 74.55 g/mol = 37.25 g.
Correct Answer: A — 37.25 g
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Q. How many grams of KCl are needed to prepare 250 mL of a 2 M solution? (Molar mass of KCl = 74.5 g/mol) (2023)
A.
37.25 g
B.
74.5 g
C.
18.625 g
D.
9.25 g
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Solution
Grams = moles × molar mass = (2 moles/L × 0.25 L) × 74.5 g/mol = 37.25 g.
Correct Answer: A — 37.25 g
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Q. How many liters of a 0.5 M NaOH solution are needed to obtain 1 mole of NaOH? (2023)
A.
1 L
B.
2 L
C.
0.5 L
D.
0.25 L
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Solution
Volume = moles / molarity = 1 mole / 0.5 M = 2 L.
Correct Answer: A — 1 L
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Q. How many liters of a 0.5 M solution contain 2 moles of solute? (2023)
A.
2 L
B.
4 L
C.
1 L
D.
0.5 L
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Solution
Volume = moles / molarity = 2 moles / 0.5 M = 4 L
Correct Answer: B — 4 L
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Q. How many liters of a 3 M solution can be made from 6 moles of solute? (2023)
A.
1 L
B.
2 L
C.
3 L
D.
4 L
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Solution
Volume = moles / molarity = 6 moles / 3 M = 2 L.
Correct Answer: B — 2 L
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Q. How many molecules are in 0.1 moles of oxygen gas (O2)? (2021)
A.
6.022 x 10^22
B.
6.022 x 10^23
C.
1.2044 x 10^23
D.
3.011 x 10^23
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Solution
Number of molecules = moles x Avogadro's number = 0.1 moles x 6.022 x 10^23 molecules/mole = 6.022 x 10^22 molecules.
Correct Answer: A — 6.022 x 10^22
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