Q. What is the value of k if the roots of the equation x^2 + kx + 4 = 0 are -2 and -2?
Solution
The sum of the roots is -2 + -2 = -4, so k = 4.
Correct Answer: C — 6
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Q. What is the value of k if the roots of the equation x^2 + kx + 9 = 0 are imaginary?
-
A.
k < 0
-
B.
k > 0
-
C.
k = 0
-
D.
k ≤ 0
Solution
The discriminant must be less than zero: k^2 - 4*1*9 < 0 leads to k^2 < 36, hence k < 0 or k > 0.
Correct Answer: A — k < 0
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Q. What is the value of k if the roots of the equation x^2 - 5x + k = 0 are equal? (2020)
Solution
For the roots to be equal, the discriminant must be zero. Thus, (-5)^2 - 4(1)(k) = 0. Solving gives k = 6.25.
Correct Answer: A — 6.25
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Q. What is the value of k if the roots of the equation x^2 - kx + 16 = 0 are real and distinct?
Solution
For the roots to be real and distinct, the discriminant must be positive: k^2 - 64 > 0, leading to k > 8 or k < -8.
Correct Answer: C — 12
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Q. What is the value of k if the roots of the equation x^2 - kx + 8 = 0 are 2 and 4? (2023)
Solution
The sum of the roots is 2 + 4 = 6, so k = 6.
Correct Answer: A — 6
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Q. What is the value of k if the roots of the equation x^2 - kx + 9 = 0 are 3 and 3?
Solution
The sum of the roots is 3 + 3 = 6, so k = 6.
Correct Answer: A — 6
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Q. What is the value of log_10(0.01)?
Solution
log_10(0.01) = log_10(10^-2) = -2.
Correct Answer: B — -2
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Q. What is the value of log_2(1/8)? (2023)
Solution
log_2(1/8) = log_2(2^-3) = -3.
Correct Answer: A — -3
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Q. What is the value of log_4(16)?
Solution
log_4(16) = log_4(4^2) = 2.
Correct Answer: B — 2
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Q. What is the value of log_5(1)?
-
A.
0
-
B.
1
-
C.
5
-
D.
undefined
Solution
log_5(1) = 0 because 5^0 = 1.
Correct Answer: A — 0
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Q. What is the value of p for the parabola defined by the equation y^2 = 20x?
Solution
In the equation y^2 = 4px, we have 4p = 20, thus p = 5.
Correct Answer: A — 5
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Q. What is the value of sin(180° - θ)?
-
A.
sin θ
-
B.
cos θ
-
C.
tan θ
-
D.
sec θ
Solution
Using the identity, sin(180° - θ) = sin θ.
Correct Answer: A — sin θ
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Q. What is the value of sin² θ + cos² θ? (2021)
Solution
According to the Pythagorean identity, sin² θ + cos² θ = 1 for any angle θ.
Correct Answer: B — 1
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Q. What is the value of the 5th term in the expansion of (x + 2)^7?
-
A.
672
-
B.
672x^4
-
C.
672x^3
-
D.
672x^2
Solution
The 5th term is C(7,4) * (2)^4 * x^3 = 35 * 16 * x^3 = 560x^3.
Correct Answer: C — 672x^3
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Q. What is the value of the coefficient of x^4 in the expansion of (x + 5)^6?
-
A.
150
-
B.
300
-
C.
600
-
D.
750
Solution
The coefficient of x^4 in (x + 5)^6 is given by 6C4 * 5^2 = 15 * 25 = 375.
Correct Answer: B — 300
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Q. What is the value of the discriminant for the equation 3x^2 + 12x + 9 = 0?
Solution
The discriminant is b^2 - 4ac = 12^2 - 4*3*9 = 144 - 108 = 36.
Correct Answer: A — 0
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Q. What is the value of the discriminant for the quadratic equation 3x^2 + 12x + 9 = 0? (2019)
Solution
The discriminant D = b^2 - 4ac = 12^2 - 4*3*9 = 0, indicating equal roots.
Correct Answer: A — 0
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Q. What is the value of the discriminant for the quadratic equation 3x^2 + 6x + 2 = 0? (2023)
Solution
The discriminant is b^2 - 4ac = 6^2 - 4(3)(2) = 36 - 24 = 12.
Correct Answer: B — 4
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Q. What is the value of x if the matrix A = [[x, 2], [3, 4]] is singular?
Solution
A matrix is singular if its determinant is zero. Det(A) = (x*4) - (2*3) = 4x - 6 = 0. Solving gives x = 1.5.
Correct Answer: C — 3
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Q. What is the value of x in the equation 3x - 7 = 2x + 5? (2023)
Solution
Rearranging gives 3x - 2x = 5 + 7, thus x = 12.
Correct Answer: B — 6
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Q. What is the value of x in the equation 4x + 3 = 19? (2023)
Solution
Solving for x gives 4x = 16, thus x = 4.
Correct Answer: A — 4
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Q. What is the value of x in the equation 4x - 5 = 11? (2022)
Solution
Add 5 to both sides: 4x = 16. Then divide by 4: x = 4.
Correct Answer: B — 3
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Q. What is the value of x in the equation 4x - 7 = 9? (2022)
Solution
Adding 7 to both sides gives 4x = 16. Dividing by 4 gives x = 4.
Correct Answer: A — 4
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Q. What is the value of √(144) + √(64)?
Solution
√(144) = 12 and √(64) = 8, so 12 + 8 = 20.
Correct Answer: C — 18
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Q. What is the value of √(16) + √(25)? (2021)
Solution
√(16) = 4 and √(25) = 5, so 4 + 5 = 9.
Correct Answer: C — 11
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Q. What is the variance of the following numbers: 1, 1, 1, 1, 1?
Solution
All values are the same, so variance = 0.
Correct Answer: A — 0
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Q. What is the variance of the following numbers: 1, 2, 3, 4, 5, 6? (2022)
-
A.
2.5
-
B.
3.5
-
C.
4.5
-
D.
5.5
Solution
Mean = (1 + 2 + 3 + 4 + 5 + 6) / 6 = 3.5. Variance = [(1-3.5)² + (2-3.5)² + (3-3.5)² + (4-3.5)² + (5-3.5)² + (6-3.5)²] / 6 = 2.92 (approx 2.5).
Correct Answer: A — 2.5
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Q. What is the variance of the following numbers: 5, 5, 5, 5? (2019)
Solution
All values are the same, so variance = 0.
Correct Answer: A — 0
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Q. What is the variance of the following numbers: 5, 7, 9, 11? (2020)
Solution
Mean = (5 + 7 + 9 + 11) / 4 = 8. Variance = [(5-8)² + (7-8)² + (9-8)² + (11-8)²] / 4 = 4.
Correct Answer: A — 4
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Q. What is the variance of the numbers: 1, 1, 1, 1, 1? (2023)
Solution
All values are the same, so variance = 0.
Correct Answer: A — 0
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