Q. The equation x^2 - 2x + k = 0 has roots that are both positive. What is the range of k?
-
A.
k < 0
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B.
k > 0
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C.
k > 1
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D.
k < 1
Solution
For both roots to be positive, k must be greater than 1 (from Vieta's formulas).
Correct Answer: C — k > 1
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Q. The equation x^2 - 4x + k = 0 has no real roots if k is:
-
A.
< 4
-
B.
≥ 4
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C.
≤ 4
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D.
> 4
Solution
The discriminant must be less than zero: (-4)^2 - 4*1*k < 0 leads to k > 4.
Correct Answer: A — < 4
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Q. The equation x^2 - 7x + 10 = 0 has roots that are:
-
A.
1 and 10
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B.
2 and 5
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C.
3 and 4
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D.
5 and 2
Solution
Factoring the equation gives (x - 2)(x - 5) = 0, so the roots are 2 and 5.
Correct Answer: C — 3 and 4
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Q. The following data represents the number of books read by students in a month: 1, 2, 2, 3, 3, 3, 4, 4, 5. What is the mode?
Solution
The mode is 3, as it appears most frequently (3 times) in the data set.
Correct Answer: C — 3
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Q. The following data represents the number of books read by students in a month: 4, 5, 6, 5, 4, 5, 7, 8, 5. What is the mode?
Solution
The mode is 5, as it appears 4 times, more than any other number.
Correct Answer: B — 5
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Q. The following data set has the numbers: 2, 3, 4, 4, 5, 5, 5, 6, 7. What is the mode of this data set? (2020)
Solution
The number 5 appears most frequently (3 times), so the mode is 5.
Correct Answer: B — 5
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Q. The following data set represents the ages of participants in a survey: 22, 25, 22, 30, 25, 22, 28. What is the mode?
Solution
The mode is 22, as it appears most frequently (three times) in the data set.
Correct Answer: A — 22
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Q. The following data shows the number of goals scored by a team in 7 matches: 1, 2, 2, 3, 2, 4, 5. What is the mode?
Solution
The mode is 2, as it appears 3 times, more than any other number.
Correct Answer: B — 2
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Q. The following data shows the number of pets owned by families: 1, 1, 2, 2, 3, 3, 3, 4. What is the mode?
Solution
The mode is 3, as it appears most frequently (3 times) in the data set.
Correct Answer: C — 3
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Q. The following numbers represent the number of books read by students: 1, 2, 2, 3, 4, 4, 4, 5. What is the mode?
Solution
The mode is 4, as it appears three times, which is more than any other number.
Correct Answer: C — 4
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Q. The following numbers represent the number of hours studied by students: 1, 2, 2, 3, 3, 4, 4, 4, 5. What is the mode?
Solution
The mode is 4, as it appears most frequently (3 times) in the data set.
Correct Answer: C — 4
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Q. The following numbers represent the number of hours studied by students: 5, 6, 5, 7, 8, 5, 6. What is the mode?
Solution
The mode is 5, as it appears 3 times, more than any other number.
Correct Answer: A — 5
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Q. The following numbers represent the number of languages spoken by individuals: 1, 2, 2, 3, 3, 3, 4, 5. What is the mode?
Solution
The mode is 3, as it appears most frequently (3 times) in the data set.
Correct Answer: C — 3
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Q. The function f(x) = 1/(x-1) is continuous on which of the following intervals?
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A.
(-∞, 1)
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B.
(1, ∞)
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C.
(-∞, ∞)
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D.
(-∞, 0)
Solution
The function f(x) = 1/(x-1) is discontinuous at x = 1, hence it is continuous on (1, ∞).
Correct Answer: B — (1, ∞)
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Q. The function f(x) = x^2 + 3 is continuous for which of the following intervals? (2023)
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A.
(-∞, ∞)
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B.
(0, 1)
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C.
(1, 2)
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D.
(2, 3)
Solution
f(x) = x^2 + 3 is a polynomial function and is continuous for all x in (-∞, ∞).
Correct Answer: A — (-∞, ∞)
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Q. The function f(x) = x^2 is continuous at which of the following points? (2023)
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A.
x = -1
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B.
x = 0
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C.
x = 1
-
D.
All of the above
Solution
The function f(x) = x^2 is a polynomial function and is continuous at all points, including -1, 0, and 1.
Correct Answer: D — All of the above
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Q. The function f(x) = x^3 - 3x is continuous at which of the following points? (2023)
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A.
x = -2
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B.
x = 0
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C.
x = 2
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D.
All of the above
Solution
The function f(x) = x^3 - 3x is a polynomial function and is continuous at all points, including -2, 0, and 2.
Correct Answer: D — All of the above
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Q. The function f(x) = { x^2, x < 0; 0, x = 0; x + 1, x > 0 } is:
-
A.
Continuous
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B.
Not continuous
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C.
Continuous from the left
-
D.
Continuous from the right
Solution
The left limit as x approaches 0 is 0, but the right limit is 1. Hence, it is not continuous at x = 0.
Correct Answer: B — Not continuous
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Q. The function f(x) = { x^2, x < 0; 2, x = 0; x + 1, x > 0 } is continuous at x = 0?
-
A.
Yes
-
B.
No
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C.
Only left continuous
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D.
Only right continuous
Solution
At x = 0, lim x→0- f(x) = 0 and lim x→0+ f(x) = 1, hence it is discontinuous at x = 0.
Correct Answer: B — No
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Q. The lengths of the sides of triangle ABC are 5 cm, 12 cm, and 13 cm. What is the area of the triangle? (2019)
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A.
30 cm²
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B.
60 cm²
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C.
24 cm²
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D.
40 cm²
Solution
Using Heron's formula, s = (5 + 12 + 13)/2 = 15. Area = √[s(s-a)(s-b)(s-c)] = √[15(15-5)(15-12)(15-13)] = √[15*10*3*2] = √[900] = 30 cm².
Correct Answer: A — 30 cm²
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Q. The lengths of the sides of triangle ABC are 7 cm, 24 cm, and 25 cm. Is this triangle a right triangle? (2020)
-
A.
Yes
-
B.
No
-
C.
Cannot be determined
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D.
Only if angle A is 90 degrees
Solution
Using the Pythagorean theorem, 7^2 + 24^2 = 49 + 576 = 625 = 25^2. Therefore, triangle ABC is a right triangle.
Correct Answer: A — Yes
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Q. The lengths of the sides of triangle GHI are 5 cm, 12 cm, and 13 cm. What is the area of the triangle? (2019)
-
A.
30 cm²
-
B.
60 cm²
-
C.
65 cm²
-
D.
78 cm²
Solution
This is a right triangle. Area = 1/2 * base * height = 1/2 * 5 * 12 = 30 cm².
Correct Answer: A — 30 cm²
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Q. The magnitude of the vector A = 4i - 3j + 12k is:
Solution
Magnitude |A| = √(4^2 + (-3)^2 + 12^2) = √(16 + 9 + 144) = √169 = 13.
Correct Answer: B — 14
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Q. The mean of 5, 10, 15, and 20 is what?
-
A.
10
-
B.
12.5
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C.
15
-
D.
17.5
Solution
Mean = (5 + 10 + 15 + 20) / 4 = 50 / 4 = 12.5.
Correct Answer: C — 15
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Q. The mean of 5, 10, 15, and x is 12. What is the value of x?
Solution
Mean = (5 + 10 + 15 + x) / 4 = 12. Therefore, 30 + x = 48, so x = 18.
Correct Answer: A — 8
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Q. The mean of 5, 15, 25, and x is 20. Find the value of x. (2023)
Solution
Mean = (5 + 15 + 25 + x) / 4 = 20. Solving gives x = 30.
Correct Answer: C — 30
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Q. The mean of 7, 9, 11, 13, and x is 10. Find the value of x.
Solution
Mean = (7 + 9 + 11 + 13 + x) / 5 = 10. Solving gives x = 5.
Correct Answer: A — 5
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Q. The mean of 7, 9, 11, 13, and x is 10. What is the value of x?
Solution
Mean = (7 + 9 + 11 + 13 + x) / 5 = 10.\n\n40 + x = 50\n\nx = 50 - 40 = 10.
Correct Answer: A — 5
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Q. The mean of 7, 9, 11, and 13 is what? (2019)
Solution
Mean = (7 + 9 + 11 + 13) / 4 = 40 / 4 = 10.
Correct Answer: B — 10
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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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