Q. In the expansion of (2x - 3y)^5, what is the coefficient of x^3y^2?
A.
-720
B.
-540
C.
540
D.
720
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Solution
The coefficient is C(5,3) * (2)^3 * (-3)^2 = 10 * 8 * 9 = 720.
Correct Answer: C — 540
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Q. In the expansion of (3x - 4)^7, what is the coefficient of x^5? (1920)
A.
1260
B.
1440
C.
1680
D.
1920
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Solution
Using the binomial theorem, the coefficient of x^5 in (3x - 4)^7 is given by 7C5 * (3)^5 * (-4)^2 = 21 * 243 * 16 = 21 * 3888 = 81588.
Correct Answer: A — 1260
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Q. In the expansion of (a + b)^n, if the coefficient of a^3b^2 is 60, what is the value of n?
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Solution
C(n,3) * b^2 = 60. For n = 5, C(5,3) = 10, which does not satisfy. For n = 6, C(6,3) = 20, which does not satisfy. For n = 7, C(7,3) = 35, which does not satisfy. For n = 8, C(8,3) = 56, which satisfies.
Correct Answer: B — 6
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Q. In the expansion of (x + 3)^5, what is the coefficient of x^3?
A.
60
B.
90
C.
100
D.
120
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Solution
Using the binomial theorem, the coefficient of x^3 in (x + 3)^5 is given by 5C3 * (3)^2 = 10 * 9 = 90.
Correct Answer: B — 90
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Q. In the expansion of (x + 3)^6, what is the coefficient of x^4?
A.
540
B.
720
C.
810
D.
900
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Solution
Using the binomial theorem, the coefficient of x^4 in (x + 3)^6 is given by 6C4 * (3)^2 = 15 * 9 = 135.
Correct Answer: B — 720
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Q. In the expansion of (x - 1)^8, what is the coefficient of x^5?
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Solution
The coefficient of x^5 in (x - 1)^8 is C(8,5) * (-1)^3 = 56 * (-1) = -56.
Correct Answer: A — -56
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Q. In the expansion of (x - 2)^6, what is the term containing x^2?
A.
-60x^2
B.
90x^2
C.
-80x^2
D.
80x^2
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Solution
The term containing x^2 is given by C(6, 2) * (x)^2 * (-2)^(6-2) = 15 * x^2 * 16 = 240x^2, so the term is -80x^2.
Correct Answer: C — -80x^2
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Q. The equation x^2 - 2x + k = 0 has roots that are both positive. What is the range of k?
A.
k < 0
B.
k > 0
C.
k > 1
D.
k < 1
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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
C.
≤ 4
D.
> 4
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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
B.
2 and 5
C.
3 and 4
D.
5 and 2
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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 product of the roots of the quadratic equation x^2 + 5x + 6 = 0 is: (2021)
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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 for k? (2022)
A.
k < 0
B.
k > 0
C.
k > 8
D.
k < 8
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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 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
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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 roots that are both negative. What is the condition for k? (2020)
A.
k > 9
B.
k < 9
C.
k = 9
D.
k = 0
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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
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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
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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 + 3x + k = 0 are -1 and -2. What is the value of k? (2021)
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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?
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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
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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
Show solution
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)
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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)
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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)
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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?
A.
15ab^5
B.
20ab^5
C.
30ab^5
D.
6ab^5
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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?
A.
60x^4
B.
90x^4
C.
120x^4
D.
180x^4
Show solution
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?
A.
540x^4
B.
540x^3
C.
720x^4
D.
720x^3
Show solution
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?
A.
-540x^5
B.
540x^5
C.
-486x^5
D.
486x^5
Show solution
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 -7? (2023)
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Solution
The absolute value of -7 is 7.
Correct Answer: C — 7
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Q. What is the coefficient of x^2 in the expansion of (2x + 5)^4?
A.
60
B.
80
C.
100
D.
120
Show solution
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)
A.
-300
B.
-600
C.
600
D.
300
Show solution
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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