Q. The mean of 8, 12, 16, and 20 is what? (2019)
Solution
Mean = (8 + 12 + 16 + 20) / 4 = 56 / 4 = 14.
Correct Answer: B — 15
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Q. The mean of a data set is 10 and the variance is 4. What is the standard deviation? (2019)
Solution
Standard deviation is the square root of variance. Therefore, SD = √4 = 2.
Correct Answer: A — 2
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Q. The mean of five consecutive integers is 12. What is the smallest of these integers?
Solution
Let the integers be x, x+1, x+2, x+3, x+4. Mean = (5x + 10) / 5 = 12. Solving gives x = 10.
Correct Answer: A — 10
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Q. The mean of the first four prime numbers is what? (2019)
Solution
The first four prime numbers are 2, 3, 5, and 7. Mean = (2 + 3 + 5 + 7) / 4 = 17 / 4 = 4.25.
Correct Answer: C — 5
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Q. The mean of the numbers 4, 8, 10, 12, and x is 10. What is the value of x? (2021)
Solution
Mean = (4 + 8 + 10 + 12 + x) / 5 = 10. Solving gives x = 14.
Correct Answer: C — 14
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Q. The mean of the numbers 4, 8, 12, and x is 10. What is the value of x?
Solution
Mean = (4 + 8 + 12 + x) / 4 = 10. Therefore, 4 + 8 + 12 + x = 40. So, x = 40 - 24 = 16.
Correct Answer: C — 16
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Q. The mean of three numbers is 15. If one of the numbers is 10, what is the mean of the other two numbers? (2020)
Solution
Let the other two numbers be a and b. Then, (10 + a + b) / 3 = 15. Solving gives (a + b) = 35, so Mean of a and b = 35 / 2 = 17.5.
Correct Answer: B — 20
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Q. The mean of three numbers is 30. If one of the numbers is 20, what is the mean of the other two numbers? (2023)
Solution
Let the three numbers be a, b, c. Then, (a + b + 20) / 3 = 30. Thus, a + b = 90, and the mean of a and b = 90 / 2 = 45.
Correct Answer: A — 25
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Q. The median of the data set 3, 7, 8, 12, x is 8. What is the value of x? (2023)
Solution
For the median to be 8, x must be less than or equal to 8. The only value that satisfies this is x = 6.
Correct Answer: A — 6
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Q. The median of the data set 3, 7, 9, 12, 15 is: (2022)
Solution
Median = 9 (middle value of the ordered data set)
Correct Answer: B — 9
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Q. The minimum value of the function f(x) = x^2 - 4x + 6 occurs at x = ? (2020)
Solution
The vertex of the parabola occurs at x = -b/(2a) = 4/2 = 2. The minimum value is f(2) = 2^2 - 4*2 + 6 = 2.
Correct Answer: B — 2
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Q. The product of the roots of the quadratic equation x^2 + 5x + 6 = 0 is: (2021)
Solution
The product of the roots is given by c/a = 6/1 = 6.
Correct Answer: A — 6
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Q. The quadratic equation 2x^2 - 4x + k = 0 has no real roots. What is the condition on k? (2022)
-
A.
k < 0
-
B.
k > 0
-
C.
k > 8
-
D.
k < 8
Solution
The discriminant must be less than zero: (-4)^2 - 4*2*k < 0 leads to k > 8.
Correct Answer: C — k > 8
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Q. The quadratic equation 2x^2 - 4x + k = 0 has no real roots. What is the condition for k? (2022)
-
A.
k < 0
-
B.
k > 0
-
C.
k > 8
-
D.
k < 8
Solution
The discriminant must be less than zero: (-4)^2 - 4*2*k < 0 leads to k > 8.
Correct Answer: C — k > 8
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Q. The quadratic equation 5x^2 + 3x - 2 = 0 has roots that can be expressed in which form? (2023)
-
A.
Rational
-
B.
Irrational
-
C.
Complex
-
D.
Imaginary
Solution
The discriminant is 3^2 - 4*5*(-2) = 9 + 40 = 49, which is a perfect square, hence the roots are rational.
Correct Answer: A — Rational
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Q. The quadratic equation x^2 + 6x + 9 = 0 can be expressed in which of the following forms? (2020)
-
A.
(x + 3)^2
-
B.
(x - 3)^2
-
C.
(x + 6)^2
-
D.
(x - 6)^2
Solution
This is a perfect square trinomial: (x + 3)(x + 3) = 0.
Correct Answer: A — (x + 3)^2
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Q. The quadratic equation x^2 + 6x + k = 0 has no real roots. What is the condition on k? (2020)
-
A.
k < 9
-
B.
k > 9
-
C.
k = 9
-
D.
k ≤ 9
Solution
For no real roots, the discriminant must be less than zero: 6^2 - 4*1*k < 0, which gives k > 9.
Correct Answer: B — k > 9
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Q. The quadratic equation x^2 + 6x + k = 0 has roots that are both negative. What is the condition for k? (2020)
-
A.
k > 9
-
B.
k < 9
-
C.
k = 9
-
D.
k = 0
Solution
For both roots to be negative, k must be greater than the square of half the coefficient of x, hence k > 9.
Correct Answer: A — k > 9
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Q. The quadratic equation x^2 - 4x + 4 = 0 can be expressed in which of the following forms? (2022)
-
A.
(x - 2)^2
-
B.
(x + 2)^2
-
C.
(x - 4)^2
-
D.
(x + 4)^2
Solution
The equation can be factored as (x - 2)(x - 2) = 0, which is (x - 2)^2.
Correct Answer: A — (x - 2)^2
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Q. The roots of the equation 4x^2 - 12x + 9 = 0 are: (2019)
-
A.
1 and 2
-
B.
3 and 3
-
C.
0 and 3
-
D.
2 and 1
Solution
The equation can be factored as (2x - 3)(2x - 3) = 0, hence the roots are 3 and 3.
Correct Answer: B — 3 and 3
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Q. The roots of the equation x^2 + 2x + k = 0 are -1 and -3. What is the value of k?
Solution
The sum of the roots is -1 + (-3) = -4, so -2 = -4, which is correct. The product of the roots is (-1)(-3) = 3, so k = 3.
Correct Answer: B — 3
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Q. The roots of the equation x^2 + 3x + k = 0 are -1 and -2. What is the value of k? (2021)
Solution
The sum of the roots is -1 + (-2) = -3, and the product is (-1)(-2) = 2. Thus, k = 2.
Correct Answer: A — 2
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Q. The roots of the equation x^2 + 4x + k = 0 are equal. What is the value of k?
Solution
For the roots to be equal, the discriminant must be zero. Thus, 4^2 - 4*1*k = 0 => 16 - 4k = 0 => k = 4.
Correct Answer: B — 8
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Q. The roots of the equation x^2 - 10x + 21 = 0 are: (2020)
-
A.
3 and 7
-
B.
4 and 6
-
C.
5 and 5
-
D.
2 and 8
Solution
Factoring gives (x - 3)(x - 7) = 0, so the roots are 3 and 7.
Correct Answer: A — 3 and 7
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Q. The roots of the equation x^2 - 2x + k = 0 are real and distinct if k is:
-
A.
< 1
-
B.
≥ 1
-
C.
≤ 1
-
D.
> 1
Solution
The discriminant must be greater than zero: (-2)^2 - 4*1*k > 0, which simplifies to k < 1.
Correct Answer: A — < 1
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Q. The roots of the quadratic equation x^2 - 4x + k = 0 are 2 and 2. What is the value of k? (2022)
Solution
If the roots are both 2, then k = 2^2 - 4*2 = 4 - 8 = -4. Thus, k = 4.
Correct Answer: C — 4
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Q. The scalar product of two unit vectors is 0. What can be inferred about these vectors?
-
A.
They are parallel
-
B.
They are orthogonal
-
C.
They are collinear
-
D.
They are equal
Solution
If the scalar product is 0, the vectors are orthogonal.
Correct Answer: B — They are orthogonal
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Q. The scores of a test are: 20, 30, 40, 50. What is the standard deviation? (2020)
Solution
Mean = (20 + 30 + 40 + 50) / 4 = 35. Variance = [(20-35)² + (30-35)² + (40-35)² + (50-35)²] / 4 = 125. SD = √125 = 11.18 (approx 10).
Correct Answer: B — 15
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Q. The scores of five students are: 20, 22, 24, 26, 28. What is the standard deviation? (2020)
Solution
Mean = (20 + 22 + 24 + 26 + 28) / 5 = 24. Variance = [(20-24)² + (22-24)² + (24-24)² + (26-24)² + (28-24)²] / 5 = 8. SD = √8 = 2.83 (approximately 3).
Correct Answer: A — 2
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Q. The scores of five students are: 20, 22, 24, 26, 28. What is the variance? (2020)
Solution
Mean = (20 + 22 + 24 + 26 + 28) / 5 = 24. Variance = [(20-24)² + (22-24)² + (24-24)² + (26-24)² + (28-24)²] / 5 = 8.
Correct Answer: A — 4
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