Algebra
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Q. If the equation 2x^2 + 3x + k = 0 has roots 1 and -2, what is the value of k?
Solution
Using Vieta's formulas, k = 2*1*(-2) = -4.
Correct Answer: D — 4
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Q. If the equation 2x^2 + 3x - 5 = 0 has roots r1 and r2, what is the value of r1 + r2?
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A.
-3/2
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B.
3/2
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C.
5/2
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D.
-5/2
Solution
Using Vieta's formulas, r1 + r2 = -b/a = -3/2.
Correct Answer: A — -3/2
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Q. If the equation x^2 + px + q = 0 has roots 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 expansion of (x + a)^n has a term 15x^3a^2, what is the value of n?
Solution
The term is given by C(n, 3) * a^2 * x^3. Setting C(n, 3) * a^2 = 15 gives n = 6.
Correct Answer: B — 6
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Q. If the first term of an arithmetic series is 5 and the common difference is 3, what is the 15th term?
Solution
a_n = a + (n-1)d = 5 + (15-1) * 3 = 5 + 42 = 47.
Correct Answer: A — 44
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Q. If the nth term of a sequence is given by a_n = n^2 + n, what is a_4?
Solution
a_4 = 4^2 + 4 = 16 + 4 = 20.
Correct Answer: A — 20
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Q. If the polynomial P(x) = x^3 - 6x^2 + 11x - 6 has a root at x = 1, what is P(2)?
Solution
P(2) = 2^3 - 6(2^2) + 11(2) - 6 = 8 - 24 + 22 - 6 = 0.
Correct Answer: D — 3
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Q. If the quadratic equation ax^2 + bx + c = 0 has roots 3 and -2, what is the value of a?
Solution
Using the fact that the product of the roots is c/a and the sum is -b/a, we can set a = 1, b = -1, c = -6.
Correct Answer: A — 1
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Q. If the quadratic equation x^2 + 2px + p^2 - 4 = 0 has real roots, what is the condition on p?
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A.
p > 2
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B.
p < 2
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C.
p = 2
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D.
p >= 2
Solution
The discriminant must be non-negative: (2p)^2 - 4(1)(p^2 - 4) >= 0 => 4p^2 - 4p^2 + 16 >= 0, which is always true. Thus, p can be any real number.
Correct Answer: D — p >= 2
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Q. If the quadratic equation x^2 + 2px + p^2 - 4 = 0 has roots that are equal, what is the value of p?
Solution
Setting the discriminant to zero: (2p)^2 - 4(1)(p^2 - 4) = 0 leads to p = ±2.
Correct Answer: C — -2
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Q. If the quadratic equation x^2 + 2x + k = 0 has equal roots, what is the value of k?
Solution
For equal roots, the discriminant must be zero: 2^2 - 4*1*k = 0, leading to k = 1.
Correct Answer: C — -1
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Q. If the quadratic equation x^2 + 2x + k = 0 has no real roots, what is the condition on k?
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A.
k < 0
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B.
k > 0
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C.
k >= 0
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D.
k <= 0
Solution
For no real roots, the discriminant must be less than zero: 2^2 - 4*1*k < 0, hence k > 1.
Correct Answer: A — k < 0
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Q. If the quadratic equation x^2 + 2x + k = 0 has no real roots, what is the condition for k?
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A.
k < 0
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B.
k > 0
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C.
k >= 0
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D.
k <= 0
Solution
For no real roots, the discriminant must be less than zero: 2^2 - 4*1*k < 0 => 4 - 4k < 0 => k > 1.
Correct Answer: A — k < 0
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Q. If the quadratic equation x^2 + 2x + k = 0 has roots that are equal, what is the value of k?
Solution
For equal roots, the discriminant must be zero: 2^2 - 4*1*k = 0 leads to k = -1.
Correct Answer: D — -2
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Q. If the quadratic equation x^2 + 4x + c = 0 has one root equal to -2, what is the value of c?
Solution
If one root is -2, then substituting x = -2 gives: (-2)^2 + 4(-2) + c = 0 => 4 - 8 + c = 0 => c = 4.
Correct Answer: A — 0
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Q. If the quadratic equation x^2 + 4x + k = 0 has roots -2 and -2, what is the value of k?
Solution
Using the formula for roots, k = (-2)^2 - 4*(-2) = 4 + 8 = 12.
Correct Answer: B — 4
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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 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 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 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?
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 + 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 - 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 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 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 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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