Q. Determine the scalar product of the vectors A = (2, 2, 2) and B = (3, 3, 3).
Solution
A · B = 2*3 + 2*3 + 2*3 = 6 + 6 + 6 = 18.
Correct Answer: A — 12
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Q. Determine the slope of the tangent line to f(x) = x^2 at x = 3. (2023)
Solution
f'(x) = 2x; thus, f'(3) = 2(3) = 6.
Correct Answer: B — 6
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Q. Determine the solution for the inequality -2x + 6 > 0.
-
A.
x < 3
-
B.
x > 3
-
C.
x < -3
-
D.
x > -3
Solution
-2x + 6 > 0 => -2x > -6 => x < 3.
Correct Answer: B — x > 3
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Q. Determine the solution for the inequality -2x + 6 ≥ 0.
-
A.
x ≤ 3
-
B.
x ≥ 3
-
C.
x ≤ -3
-
D.
x ≥ -3
Solution
-2x + 6 ≥ 0 => -2x ≥ -6 => x ≤ 3.
Correct Answer: B — x ≥ 3
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Q. Determine the solution for the inequality -3x + 1 ≤ 4.
-
A.
x ≥ -1
-
B.
x ≤ -1
-
C.
x ≥ 1
-
D.
x ≤ 1
Solution
-3x + 1 ≤ 4 => -3x ≤ 3 => x ≥ -1.
Correct Answer: B — x ≤ -1
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Q. Determine the solution for the inequality -3x + 4 ≤ 1.
-
A.
x ≥ 1
-
B.
x ≤ 1
-
C.
x ≥ -1
-
D.
x ≤ -1
Solution
-3x + 4 ≤ 1 => -3x ≤ -3 => x ≥ 1.
Correct Answer: B — x ≤ 1
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Q. Determine the solution for the inequality 2x + 3 ≤ 7.
-
A.
x ≤ 2
-
B.
x ≥ 2
-
C.
x ≤ 3
-
D.
x ≥ 3
Solution
2x + 3 ≤ 7 => 2x ≤ 4 => x ≤ 2.
Correct Answer: A — x ≤ 2
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Q. Determine the solution for the inequality 6 - x > 2.
-
A.
x < 4
-
B.
x > 4
-
C.
x < 6
-
D.
x > 6
Solution
6 - x > 2 => -x > -4 => x < 4.
Correct Answer: A — x < 4
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Q. Determine the solution for the inequality 6 - x ≤ 3.
-
A.
x ≥ 3
-
B.
x ≤ 3
-
C.
x ≥ 6
-
D.
x ≤ 6
Solution
6 - x ≤ 3 => -x ≤ -3 => x ≥ 3.
Correct Answer: B — x ≤ 3
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Q. Determine the solution for the inequality 7 - 3x < 1.
-
A.
x < 2
-
B.
x > 2
-
C.
x ≤ 2
-
D.
x ≥ 2
Solution
7 - 3x < 1 => -3x < -6 => x > 2.
Correct Answer: A — x < 2
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Q. Determine the solution for the inequality 7x - 2 ≤ 5x + 6.
-
A.
x ≤ 4
-
B.
x ≥ 4
-
C.
x ≤ 3
-
D.
x ≥ 3
Solution
7x - 2 ≤ 5x + 6 => 2x ≤ 8 => x ≤ 4.
Correct Answer: A — x ≤ 4
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Q. Determine the solution of the differential equation dy/dx = y^2 - 1.
-
A.
y = tan(x + C)
-
B.
y = 1/(C - x)
-
C.
y = 1/(C + x)
-
D.
y = e^(x + C)
Solution
This is separable. Separating and integrating gives y = 1/(C - x).
Correct Answer: B — y = 1/(C - x)
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Q. Determine the solution set for the inequality 2(x - 1) ≥ 3.
-
A.
x ≤ 2
-
B.
x ≥ 2
-
C.
x ≤ 3
-
D.
x ≥ 3
Solution
2(x - 1) ≥ 3 => x - 1 ≥ 1.5 => x ≥ 2.
Correct Answer: B — x ≥ 2
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Q. Determine the solution set for the inequality 2x + 3 > 5.
-
A.
x < 1
-
B.
x > 1
-
C.
x < 2
-
D.
x > 2
Solution
2x + 3 > 5 => 2x > 2 => x > 1.
Correct Answer: B — x > 1
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Q. Determine the solution set for the inequality 2x + 3 > 7.
-
A.
x < 2
-
B.
x > 2
-
C.
x < 3
-
D.
x > 3
Solution
2x + 3 > 7 => 2x > 4 => x > 2.
Correct Answer: B — x > 2
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Q. Determine the solution set for the inequality 2x + 3 ≤ 7.
-
A.
x ≤ 2
-
B.
x ≥ 2
-
C.
x < 2
-
D.
x > 2
Solution
2x + 3 ≤ 7 => 2x ≤ 4 => x ≤ 2.
Correct Answer: A — x ≤ 2
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Q. Determine the solution set for the inequality 3x - 4 < 2x + 5.
-
A.
x < 9
-
B.
x > 9
-
C.
x ≤ 9
-
D.
x ≥ 9
Solution
3x - 4 < 2x + 5 => x < 9.
Correct Answer: A — x < 9
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Q. Determine the solution set for the inequality 4x - 1 > 3.
-
A.
x < 1
-
B.
x > 1
-
C.
x ≤ 1
-
D.
x ≥ 1
Solution
4x - 1 > 3 => 4x > 4 => x > 1.
Correct Answer: B — x > 1
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Q. Determine the solution set for the inequality 4x - 1 > 3x + 2.
-
A.
x < 3
-
B.
x > 3
-
C.
x < 1
-
D.
x > 1
Solution
4x - 1 > 3x + 2 => x > 3.
Correct Answer: B — x > 3
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Q. Determine the solution set for the inequality 4x - 1 < 3.
-
A.
x < 1
-
B.
x > 1
-
C.
x ≤ 1
-
D.
x ≥ 1
Solution
4x - 1 < 3 => 4x < 4 => x < 1.
Correct Answer: A — x < 1
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Q. Determine the solution set for the inequality 4x - 8 < 0.
-
A.
x < 2
-
B.
x > 2
-
C.
x ≤ 2
-
D.
x ≥ 2
Solution
4x - 8 < 0 => 4x < 8 => x < 2.
Correct Answer: A — x < 2
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Q. Determine the solution set for the inequality 5 - 2x ≤ 3.
-
A.
x < 1
-
B.
x > 1
-
C.
x ≤ 1
-
D.
x ≥ 1
Solution
5 - 2x ≤ 3 => -2x ≤ -2 => x ≥ 1.
Correct Answer: C — x ≤ 1
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Q. Determine the solution set for the inequality 5 - x ≥ 2.
-
A.
x ≤ 3
-
B.
x < 3
-
C.
x ≥ 3
-
D.
x > 3
Solution
5 - x ≥ 2 => -x ≥ -3 => x ≤ 3.
Correct Answer: C — x ≥ 3
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Q. Determine the solution set for the inequality 5x - 1 > 4.
-
A.
x < 1
-
B.
x > 1
-
C.
x ≤ 1
-
D.
x ≥ 1
Solution
5x - 1 > 4 => 5x > 5 => x > 1.
Correct Answer: B — x > 1
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Q. Determine the solution set for the inequality 5x - 7 < 3.
-
A.
x < 2
-
B.
x > 2
-
C.
x ≤ 2
-
D.
x ≥ 2
Solution
5x - 7 < 3 => 5x < 10 => x < 2.
Correct Answer: A — x < 2
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Q. Determine the solution set for the inequality 5x - 7 ≥ 3.
-
A.
x ≥ 2
-
B.
x < 2
-
C.
x > 2
-
D.
x ≤ 2
Solution
5x - 7 ≥ 3 => 5x ≥ 10 => x ≥ 2.
Correct Answer: A — x ≥ 2
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Q. Determine the solution set for the inequality 6x + 4 < 10.
-
A.
x < 1
-
B.
x > 1
-
C.
x < 2
-
D.
x > 2
Solution
6x + 4 < 10 => 6x < 6 => x < 1.
Correct Answer: A — x < 1
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Q. Determine the solution set for the inequality 6x - 4 < 2x + 8.
-
A.
x < 3
-
B.
x > 3
-
C.
x < 2
-
D.
x > 2
Solution
6x - 4 < 2x + 8 => 4x < 12 => x < 3.
Correct Answer: B — x > 3
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Q. Determine the solution set for the inequality 7 - 3x < 1.
-
A.
x > 2
-
B.
x < 2
-
C.
x > 3
-
D.
x < 3
Solution
7 - 3x < 1 => -3x < -6 => x > 2.
Correct Answer: B — x < 2
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Q. Determine the solution set for the inequality 7x + 2 ≥ 4.
-
A.
x ≥ 0
-
B.
x ≤ 0
-
C.
x ≥ 1/7
-
D.
x ≤ 1/7
Solution
Subtract 2 from both sides: 7x ≥ 2. Then divide by 7: x ≥ 2/7.
Correct Answer: C — x ≥ 1/7
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