Q. What is the value of \( \begin{vmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & 6 \end{vmatrix} \)?
Solution
The determinant is 0 because the first column is repeated.
Correct Answer: A — 0
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Q. What is the value of |z| if z = -3 + 4i?
Solution
|z| = √((-3)^2 + (4)^2) = √(9 + 16) = √25 = 5.
Correct Answer: A — 5
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Q. What is the value of |z| if z = 4 - 3i?
Solution
|z| = √(4^2 + (-3)^2) = √(16 + 9) = √25 = 5.
Correct Answer: A — 5
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Q. What is the value of |z| if z = 4e^(iπ/3)?
Solution
The modulus |z| = 4, as it is the coefficient in the polar form.
Correct Answer: A — 4
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Q. What is the value of |z|^2 if z = 4 - 3i?
Solution
|z|^2 = 4^2 + (-3)^2 = 16 + 9 = 25.
Correct Answer: A — 25
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Q. What is the vertex of the parabola represented by the equation y = 2x^2 - 8x + 5?
-
A.
(2, -3)
-
B.
(2, -7)
-
C.
(4, -3)
-
D.
(4, -7)
Solution
The vertex can be found using x = -b/(2a) = 4. Substituting x = 4 into the equation gives y = -3.
Correct Answer: A — (2, -3)
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Q. Which of the following equations has no real roots?
-
A.
x^2 + 2x + 1 = 0
-
B.
x^2 - 4 = 0
-
C.
x^2 + 4x + 5 = 0
-
D.
x^2 - 1 = 0
Solution
The discriminant for x^2 + 4x + 5 is negative (16 - 20 < 0), indicating no real roots.
Correct Answer: C — x^2 + 4x + 5 = 0
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Q. Which of the following is a purely imaginary number?
-
A.
2 + 3i
-
B.
0
-
C.
4
-
D.
0 + 5i
Solution
A purely imaginary number has no real part, so 0 + 5i is purely imaginary.
Correct Answer: D — 0 + 5i
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Q. Which of the following is a root of the equation z^2 + 1 = 0?
Solution
The roots are z = ±i.
Correct Answer: A — i
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Q. Which of the following is equivalent to log_a(b^c)?
-
A.
c * log_a(b)
-
B.
log_a(b) / c
-
C.
log_a(c) + log_a(b)
-
D.
log_a(b) - c
Solution
log_a(b^c) = c * log_a(b) by the power rule of logarithms.
Correct Answer: A — c * log_a(b)
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Q. Which of the following is the solution set for the inequality x + 2 > 1?
-
A.
x > -1
-
B.
x < -1
-
C.
x > 1
-
D.
x < 1
Solution
x + 2 > 1 => x > -1.
Correct Answer: A — x > -1
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Q. Which of the following is the solution set for the inequality x + 2 > 7?
-
A.
x > 5
-
B.
x < 5
-
C.
x > 9
-
D.
x < 9
Solution
x + 2 > 7 => x > 5.
Correct Answer: A — x > 5
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Q. Which of the following is the solution set for the inequality x + 5 > 2?
-
A.
x > -3
-
B.
x < -3
-
C.
x > 3
-
D.
x < 3
Solution
x + 5 > 2 => x > -3.
Correct Answer: A — x > -3
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Q. Which of the following is the solution set for the inequality: 3x + 2 > 11?
-
A.
x > 3
-
B.
x < 3
-
C.
x > 2
-
D.
x < 2
Solution
3x + 2 > 11 => 3x > 9 => x > 3.
Correct Answer: A — x > 3
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Q. Which of the following is the solution to the inequality 3x + 2 ≥ 11?
-
A.
x < 3
-
B.
x > 3
-
C.
x ≤ 3
-
D.
x ≥ 3
Solution
3x + 2 ≥ 11 => 3x ≥ 9 => x ≥ 3.
Correct Answer: D — x ≥ 3
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Q. Which of the following is the solution to the inequality x + 2 > 7?
-
A.
x < 5
-
B.
x > 5
-
C.
x < 9
-
D.
x > 9
Solution
x + 2 > 7 => x > 5.
Correct Answer: B — x > 5
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Q. Which of the following represents the solution to the inequality 4 - x < 1?
-
A.
x > 3
-
B.
x < 3
-
C.
x > 4
-
D.
x < 4
Solution
4 - x < 1 => -x < -3 => x > 3
Correct Answer: A — x > 3
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Q. Which of the following represents the solution to the inequality 4 - x < 2?
-
A.
x < 2
-
B.
x > 2
-
C.
x < 4
-
D.
x > 4
Solution
4 - x < 2 => -x < -2 => x > 2.
Correct Answer: B — x > 2
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Q. Which of the following represents the solution to the inequality 6x - 4 ≤ 2x + 8?
-
A.
x ≤ 3
-
B.
x ≥ 3
-
C.
x < 3
-
D.
x > 3
Solution
6x - 4 ≤ 2x + 8 => 4x ≤ 12 => x ≤ 3.
Correct Answer: B — x ≥ 3
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Q. Which of the following represents the solution to the inequality 7x - 4 < 10?
-
A.
x < 2
-
B.
x > 2
-
C.
x < 3
-
D.
x > 3
Solution
7x - 4 < 10 => 7x < 14 => x < 2.
Correct Answer: C — x < 3
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Q. Which of the following represents the solution to the inequality x/3 + 2 < 5?
-
A.
x < 9
-
B.
x > 9
-
C.
x < 6
-
D.
x > 6
Solution
x/3 + 2 < 5 => x/3 < 3 => x < 9.
Correct Answer: A — x < 9
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Q. Which of the following represents the solution to the inequality x/3 - 2 < 1?
-
A.
x < 9
-
B.
x > 9
-
C.
x ≤ 9
-
D.
x ≥ 9
Solution
x/3 - 2 < 1 => x/3 < 3 => x < 9.
Correct Answer: A — x < 9
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