Q. If the quadratic equation x^2 + 6x + 9 = 0 is solved, what is the nature of the roots?
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A.
Real and distinct
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B.
Real and equal
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C.
Complex
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D.
None of the above
Solution
The discriminant is 0, indicating that the roots are real and equal.
Correct Answer: B — Real and equal
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Q. If the quadratic equation x^2 + 6x + k = 0 has roots -2 and -4, what is the value of k?
Solution
Using Vieta's formulas, k = (-2)(-4) = 8.
Correct Answer: B — 12
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Q. If the quadratic equation x^2 + 6x + k = 0 has roots that are both negative, what is the condition for k?
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A.
k > 9
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B.
k < 9
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C.
k = 9
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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. If the quadratic equation x^2 + bx + 9 = 0 has roots 3 and -3, what is the value of b?
Solution
The sum of the roots is 3 + (-3) = 0, so b = -0.
Correct Answer: C — -6
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Q. If the quadratic equation x^2 + kx + 16 = 0 has equal roots, what is the value of k?
Solution
For equal roots, the discriminant must be zero: k^2 - 4*1*16 = 0, thus k = -8.
Correct Answer: A — -8
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Q. If the quadratic equation x^2 + mx + n = 0 has roots 1 and -3, what is the value of m?
Solution
Using Vieta's formulas, m = -(1 + (-3)) = 2.
Correct Answer: A — 2
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Q. If the quadratic equation x^2 + mx + n = 0 has roots 1 and -3, what is the value of n?
Solution
Using Vieta's formulas, the product of the roots is n = 1 * (-3) = -3.
Correct Answer: A — -3
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Q. If the quadratic equation x^2 + mx + n = 0 has roots 2 and -3, what is the value of m + n?
Solution
Using Vieta's formulas, m = -(-1) = 1 and n = 2*(-3) = -6, thus m + n = 1 - 6 = -5.
Correct Answer: B — 5
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Q. If the quadratic equation x^2 + px + q = 0 has roots 2 and 3, what is the value of p + q?
Solution
Using Vieta's formulas, p = -(2 + 3) = -5 and q = 2*3 = 6. Thus, p + q = -5 + 6 = 1.
Correct Answer: C — 7
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Q. If the quadratic equation x^2 + px + q = 0 has roots 2 and 3, what is the value of p?
Solution
The sum of the roots is -p = 2 + 3 = 5, so p = -5.
Correct Answer: A — -5
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Q. If the quadratic equation x^2 - kx + 9 = 0 has equal roots, what is the value of k?
Solution
For equal roots, the discriminant must be zero: k^2 - 36 = 0, hence k = 6.
Correct Answer: A — 6
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Q. If the range of a data set is 15 and the minimum value is 5, what is the maximum value?
Solution
Range = Maximum - Minimum. Therefore, Maximum = Range + Minimum = 15 + 5 = 20.
Correct Answer: C — 20
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Q. If the relation R on set A = {1, 2, 3} is defined as R = {(1, 1), (2, 2), (3, 3)}, is R reflexive?
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A.
Yes
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B.
No
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C.
Only for 1
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D.
Only for 2
Solution
A relation is reflexive if every element in the set is related to itself. Here, R includes (1, 1), (2, 2), and (3, 3), so R is reflexive.
Correct Answer: A — Yes
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Q. If the relation R on set A = {1, 2, 3} is defined as R = {(1, 2), (2, 3)}, is R reflexive?
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A.
Yes
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B.
No
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C.
Depends on A
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D.
None of the above
Solution
A relation is reflexive if every element is related to itself. Here, (1,1), (2,2), and (3,3) are not in R, so R is not reflexive.
Correct Answer: B — No
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Q. If the roots of the equation ax^2 + bx + c = 0 are 3 and -2, what is the value of b if a = 1 and c = -6?
Solution
Using the sum of roots (-b/a = 3 + (-2) = 1), we find b = -1.
Correct Answer: A — -1
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Q. If the roots of the equation ax^2 + bx + c = 0 are 3 and -2, what is the value of a if b = 5 and c = -6?
Solution
Using Vieta's formulas, a = 1 since the product of the roots (3 * -2) = -6 and sum (3 + -2) = 1.
Correct Answer: A — 1
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Q. If the roots of the equation x^2 + 4x + k = 0 are real and distinct, what is the condition on k?
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A.
k < 16
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B.
k > 16
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C.
k = 16
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D.
k <= 16
Solution
The discriminant must be greater than zero: 4^2 - 4*1*k > 0 => 16 - 4k > 0 => k < 4.
Correct Answer: A — k < 16
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Q. If the roots of the equation x^2 + 4x + k = 0 are real and equal, what is the minimum value of k?
Solution
For real and equal roots, the discriminant must be zero: 16 - 4k = 0, thus k = 4.
Correct Answer: B — -4
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Q. If the roots of the equation x^2 + 5x + 6 = 0 are a and b, what is the value of a + b?
Solution
Using Vieta's formulas, the sum of the roots is -b/a = -5/1 = -5.
Correct Answer: A — 5
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Q. If the roots of the equation x^2 + 6x + k = 0 are -2 and -4, what is the value of k?
Solution
Using the sum and product of roots: -2 + -4 = -6 and -2*-4 = k => k = 8.
Correct Answer: C — 10
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Q. If the roots of the equation x^2 + mx + n = 0 are -2 and -3, what is the value of m + n?
Solution
The sum of the roots is -(-2 - 3) = 5, so m = 5. The product of the roots is (-2)(-3) = 6, so n = 6. Thus, m + n = 5 + 6 = 11.
Correct Answer: C — -7
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Q. If the roots of the equation x^2 + px + q = 0 are -2 and -3, what is the value of p?
Solution
Using Vieta's formulas, p = -(-2 - 3) = 5.
Correct Answer: A — 5
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Q. If the roots of the equation x^2 + px + q = 0 are -2 and -3, what is the value of p + q?
Solution
Using Vieta's formulas, p = -(-2 - 3) = 5 and q = (-2)(-3) = 6. Therefore, p + q = 5 + 6 = 11.
Correct Answer: C — -7
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Q. If the roots of the equation x^2 + px + q = 0 are 1 and -1, what is the value of p?
Solution
The sum of the roots is 0, hence p = -sum = 0.
Correct Answer: A — 0
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Q. If the roots of the equation x^2 + px + q = 0 are equal, what is the relationship between p and q?
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A.
p^2 = 4q
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B.
p^2 > 4q
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C.
p^2 < 4q
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D.
p + q = 0
Solution
For equal roots, the discriminant must be zero: p^2 - 4q = 0, hence p^2 = 4q.
Correct Answer: A — p^2 = 4q
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Q. If the roots of the equation x^2 - 5x + k = 0 are equal, what is the value of k?
Solution
For the roots to be equal, the discriminant must be zero. Thus, b^2 - 4ac = 0 => 25 - 4k = 0 => k = 25.
Correct Answer: C — 6
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Q. If the roots of the equation x^2 - 5x + k = 0 are real and equal, what is the minimum value of k?
Solution
The minimum value of k is 6, as the discriminant must be zero.
Correct Answer: C — 6
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Q. If the roots of the equation x^2 - 5x + k = 0 are real and equal, what is the value of k?
Solution
For real and equal roots, the discriminant must be zero: 25 - 4k = 0, thus k = 6.
Correct Answer: C — 6
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Q. If the roots of the equation x^2 - 7x + p = 0 are 3 and 4, what is the value of p?
Solution
Using Vieta's formulas, the sum of the roots is 7 and the product is p. Thus, 3 * 4 = p, so p = 12.
Correct Answer: C — 16
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Q. If the roots of the equation x^2 - 7x + p = 0 are in the ratio 3:4, what is the value of p?
Solution
Let the roots be 3k and 4k. Then, 3k + 4k = 7 => 7k = 7 => k = 1. The product of the roots is 3k * 4k = 12k^2 = p => p = 12.
Correct Answer: C — 20
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