Algebra
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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Q. Determine the solution set for the inequality 7x + 3 < 4x + 12.
-
A.
x < 3
-
B.
x > 3
-
C.
x ≤ 3
-
D.
x ≥ 3
Solution
7x + 3 < 4x + 12 => 3x < 9 => x < 3.
Correct Answer: B — x > 3
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Q. Determine the solution set for the inequality 7x - 4 ≥ 10.
-
A.
x ≥ 2
-
B.
x < 2
-
C.
x > 2
-
D.
x ≤ 2
Solution
Add 4 to both sides: 7x ≥ 14. Then divide by 7: x ≥ 2.
Correct Answer: A — x ≥ 2
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Q. Determine the value of \( k \) such that \( \begin{vmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & k \end{vmatrix} = 0 \).
Solution
Setting the determinant to zero and solving gives \( k = 10 \).
Correct Answer: B — 10
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Q. Evaluate cos(tan^(-1)(1)).
-
A.
√2/2
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B.
1/√2
-
C.
1
-
D.
0
Solution
Let θ = tan^(-1)(1). Then, cos(θ) = 1/√(1 + tan^2(θ)) = 1/√(1 + 1) = 1/√2.
Correct Answer: A — √2/2
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Q. Evaluate cos(tan^(-1)(3/4)).
-
A.
4/5
-
B.
3/5
-
C.
5/4
-
D.
3/4
Solution
Using the triangle with opposite = 3 and adjacent = 4, hypotenuse = 5. Thus, cos(tan^(-1)(3/4)) = 4/5.
Correct Answer: A — 4/5
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Q. Evaluate cos(tan^(-1)(5/12)).
-
A.
12/13
-
B.
5/13
-
C.
13/12
-
D.
5/12
Solution
Using the right triangle definition, cos(tan^(-1)(5/12)) = adjacent/hypotenuse = 12/13.
Correct Answer: A — 12/13
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Q. Evaluate sin(cos^(-1)(1/2)).
Solution
sin(cos^(-1)(1/2)) = √(1 - (1/2)^2) = √(3/4) = √3/2.
Correct Answer: A — √3/2
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Q. Evaluate sin(tan^(-1)(3/4)).
-
A.
3/5
-
B.
4/5
-
C.
1/5
-
D.
5/5
Solution
Using the right triangle definition, sin(tan^(-1)(3/4)) = opposite/hypotenuse = 3/5.
Correct Answer: A — 3/5
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Q. Evaluate sin(tan^(-1)(x)).
-
A.
x/√(1+x^2)
-
B.
√(1-x^2)
-
C.
1/x
-
D.
x
Solution
Using the identity, sin(tan^(-1)(x)) = x/√(1+x^2).
Correct Answer: A — x/√(1+x^2)
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Q. Evaluate sin^(-1)(-1/2) + cos^(-1)(1/2).
Solution
sin^(-1)(-1/2) = -π/6 and cos^(-1)(1/2) = π/3. Therefore, -π/6 + π/3 = π/6.
Correct Answer: B — π/2
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Q. Evaluate sin^(-1)(sin(5π/6)).
-
A.
5π/6
-
B.
π/6
-
C.
7π/6
-
D.
0
Solution
Since 5π/6 is in the range of sin^(-1), sin^(-1)(sin(5π/6)) = 5π/6.
Correct Answer: A — 5π/6
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Q. Evaluate sin^(-1)(sin(π/3)).
-
A.
π/3
-
B.
2π/3
-
C.
π/6
-
D.
0
Solution
sin^(-1)(sin(π/3)) = π/3, since π/3 is in the range of sin^(-1).
Correct Answer: A — π/3
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Q. Evaluate sin^(-1)(√3/2) + cos^(-1)(1/2).
-
A.
π/3
-
B.
π/2
-
C.
π/4
-
D.
2π/3
Solution
sin^(-1)(√3/2) + cos^(-1)(1/2) = π/2
Correct Answer: B — π/2
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Q. Evaluate tan(sin^(-1)(3/5)).
-
A.
3/4
-
B.
4/3
-
C.
5/3
-
D.
3/5
Solution
Let θ = sin^(-1)(3/5). Then sin(θ) = 3/5 and using the Pythagorean theorem, cos(θ) = 4/5. Therefore, tan(θ) = sin(θ)/cos(θ) = (3/5)/(4/5) = 3/4.
Correct Answer: B — 4/3
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Q. Evaluate tan^(-1)(1) + tan^(-1)(1).
-
A.
π/2
-
B.
π/4
-
C.
π/3
-
D.
0
Solution
tan^(-1)(1) = π/4, thus tan^(-1)(1) + tan^(-1)(1) = π/4 + π/4 = π/2.
Correct Answer: A — π/2
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Q. Evaluate tan^(-1)(1) + tan^(-1)(√3).
-
A.
π/3
-
B.
π/4
-
C.
π/2
-
D.
π/6
Solution
tan^(-1)(1) = π/4 and tan^(-1)(√3) = π/3. Therefore, π/4 + π/3 = π/2.
Correct Answer: C — π/2
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Q. Evaluate the determinant \( \begin{pmatrix} 2 & 1 & 3 \\ 1 & 0 & 2 \\ 3 & 2 & 1 \end{pmatrix} \).
Solution
The determinant is calculated as \( 2(0*1 - 2*2) - 1(1*1 - 2*3) + 3(1*2 - 0*3) = 2(0 - 4) - 1(1 - 6) + 3(2) = -8 + 5 + 6 = 3 \).
Correct Answer: A — -12
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Q. Evaluate the determinant \( \begin{vmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{vmatrix} \)
Solution
The determinant of the identity matrix is 1.
Correct Answer: B — 1
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