Q. The quadratic equation 5x^2 + 3x - 2 = 0 has roots that can be expressed in which form? (2023)
-
A.
Rational
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B.
Irrational
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C.
Complex
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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
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B.
(x - 3)^2
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C.
(x + 6)^2
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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
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B.
k > 9
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C.
k = 9
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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
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B.
k < 9
-
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. The quadratic equation x^2 - 4x + 4 = 0 can be expressed in which of the following forms? (2022)
-
A.
(x - 2)^2
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B.
(x + 2)^2
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C.
(x - 4)^2
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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
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B.
3 and 3
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C.
0 and 3
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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)
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A.
3 and 7
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B.
4 and 6
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C.
5 and 5
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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
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B.
≥ 1
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C.
≤ 1
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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 sum of the roots of the equation x^2 - 7x + k = 0 is 7. What is the value of k if the product of the roots is 10? (2023)
Solution
Using Vieta's formulas, the sum of the roots is 7 and the product is k = 10.
Correct Answer: A — 10
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Q. The sum of the roots of the quadratic equation 3x^2 + 12x + 12 = 0 is equal to what? (2022)
Solution
Using Vieta's formulas, the sum of the roots is -b/a = -12/3 = -4.
Correct Answer: A — -4
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Q. What is the 3rd term in the expansion of (a + b)^6?
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A.
15ab^5
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B.
20ab^5
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C.
30ab^5
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D.
6ab^5
Solution
The 3rd term is given by 6C2 * a^4 * b^2 = 15 * a^4 * b^2 = 15ab^5.
Correct Answer: B — 20ab^5
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Q. What is the 3rd term in the expansion of (x + 2)^6?
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A.
60x^4
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B.
90x^4
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C.
120x^4
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D.
180x^4
Solution
The 3rd term is given by C(6, 2) * (x)^2 * (2)^4 = 15 * x^2 * 16 = 240x^2.
Correct Answer: B — 90x^4
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Q. What is the 4th term in the expansion of (3x + 2)^6?
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A.
540x^4
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B.
540x^3
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C.
720x^4
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D.
720x^3
Solution
The 4th term is given by C(6,3) * (3x)^3 * (2)^3 = 20 * 27x^3 * 8 = 4320x^3.
Correct Answer: A — 540x^4
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Q. What is the 5th term in the expansion of (3x - 2)^6?
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A.
-540x^5
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B.
540x^5
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C.
-486x^5
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D.
486x^5
Solution
The 5th term is given by C(6, 4) * (3x)^4 * (-2)^2 = 15 * 81x^4 * 4 = 4860x^4.
Correct Answer: A — -540x^5
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Q. What is the absolute value of -12? (2023)
Solution
The absolute value of -12 is 12.
Correct Answer: C — 12
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Q. What is the absolute value of -7? (2023)
Solution
The absolute value of -7 is 7.
Correct Answer: C — 7
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Q. What is the coefficient of x^0 in the expansion of (x - 1)^5?
Solution
The coefficient of x^0 in (x - 1)^5 is given by 5C5 * (-1)^5 = -1.
Correct Answer: C — -5
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Q. What is the coefficient of x^2 in the expansion of (2x + 5)^4?
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A.
60
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B.
80
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C.
100
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D.
120
Solution
Using the binomial theorem, the coefficient of x^2 in (2x + 5)^4 is given by 4C2 * (2)^2 * (5)^2 = 6 * 4 * 25 = 600.
Correct Answer: A — 60
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Q. What is the coefficient of x^2 in the expansion of (2x - 5)^5? (2019)
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A.
-300
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B.
-600
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C.
600
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D.
300
Solution
The coefficient of x^2 in (2x - 5)^5 is given by 5C2 * (2)^2 * (-5)^3 = 10 * 4 * (-125) = -5000.
Correct Answer: B — -600
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Q. What is the coefficient of x^3 in the expansion of (3x + 2)^5? (2023)
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A.
90
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B.
180
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C.
270
-
D.
360
Solution
The coefficient of x^3 in (3x + 2)^5 is given by 5C3 * (3)^3 * (2)^2 = 10 * 27 * 4 = 1080.
Correct Answer: B — 180
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Q. What is the conjugate of the complex number z = 7 - 4i? (2021)
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A.
7 + 4i
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B.
7 - 4i
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C.
-7 + 4i
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D.
-7 - 4i
Solution
The conjugate of z is given by z* = 7 + 4i.
Correct Answer: A — 7 + 4i
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Q. What is the product of the roots of the equation 2x^2 - 8x + 6 = 0?
Solution
The product of the roots of the equation ax^2 + bx + c = 0 is given by c/a. Here, c = 6 and a = 2, so the product is 6/2 = 3.
Correct Answer: A — 3
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Q. What is the product of the roots of the equation x² - 4x + 3 = 0? (2021)
Solution
The product of the roots is given by c/a = 3/1 = 3.
Correct Answer: A — 3
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Q. What is the product of the roots of the quadratic equation x^2 - 7x + 10 = 0? (2023)
Solution
Using Vieta's formulas, the product of the roots is c/a = 10/1 = 10.
Correct Answer: A — 10
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Q. What is the real part of the complex number z = -2 + 3i? (2020)
Solution
The real part of z is -2.
Correct Answer: A — -2
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Q. What is the real part of the complex number z = 4 + 5i? (2022)
Solution
The real part of z = 4 + 5i is 4.
Correct Answer: A — 4
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