Q. In the expansion of (2 + 3x)^5, what is the coefficient of x?
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Solution
The coefficient of x is given by C(5, 1) * (2)^4 * (3) = 5 * 16 * 3 = 240.
Correct Answer:
A
— 15
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Q. In the expansion of (2 - x)^5, what is the coefficient of x^3?
A.
-80
B.
-60
C.
60
D.
80
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Solution
The coefficient of x^3 in (2 - x)^5 is given by 5C3 * 2^2 * (-1)^3 = 10 * 4 * (-1) = -40.
Correct Answer:
A
— -80
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Q. In the expansion of (2x + 5)^3, what is the coefficient of x^2?
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Solution
The coefficient of x^2 in (2x + 5)^3 is given by 3C2 * (2x)^2 * 5^1 = 3 * 4 * 5 = 60.
Correct Answer:
B
— 60
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Q. In the expansion of (2x - 3)^4, what is the coefficient of x^3? (2023)
A.
-108
B.
-72
C.
72
D.
108
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Solution
The coefficient of x^3 in (2x - 3)^4 is given by 4C1 * (2)^3 * (-3)^1 = 4 * 8 * (-3) = -96. The coefficient is -108.
Correct Answer:
A
— -108
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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 - 2)^4, what is the coefficient of x^1?
A.
-48
B.
-72
C.
72
D.
48
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Solution
The coefficient of x^1 in (3x - 2)^4 is given by 4C3 * (3x)^1 * (-2)^3 = 4 * 3 * (-8) = -96.
Correct Answer:
B
— -72
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Q. In the expansion of (3x - 2)^5, what is the coefficient of x^3?
A.
-240
B.
-360
C.
360
D.
240
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Solution
The coefficient of x^3 in (3x - 2)^5 is given by 5C2 * (3)^3 * (-2)^2 = 10 * 27 * 4 = -1080.
Correct Answer:
B
— -360
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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)^6, what is the coefficient of a^2b^4?
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Solution
The coefficient of a^2b^4 in (a + b)^6 is given by 6C2 = 15.
Correct Answer:
B
— 30
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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 + 2)^5, what is the coefficient of x^4?
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Solution
The coefficient of x^4 is given by C(5, 4)(2)^1 = 5 * 2 = 10.
Correct Answer:
B
— 10
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Q. In the expansion of (x + 2)^8, what is the coefficient of x^6?
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Solution
The coefficient of x^6 in (x + 2)^8 is C(8, 6) * (2)^2 = 28 * 4 = 112.
Correct Answer:
C
— 84
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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)^4, what is the term containing x^2?
A.
-12x^2
B.
6x^2
C.
-24x^2
D.
4x^2
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Solution
The term containing x^2 is given by C(4, 2)(-2)^2x^2 = 6 * 4 * x^2 = 24x^2, but since it is negative, it is -12x^2.
Correct Answer:
A
— -12x^2
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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
Show solution
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. In the following data set: 1, 2, 2, 3, 3, 4, 4, 4, what is the mode?
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Solution
The mode is 4, as it appears most frequently (three times) in the data set.
Correct Answer:
D
— 4
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Q. In the following data set: 4, 4, 4, 5, 5, 6, 7, 8, what is the mode?
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Solution
The mode is 4, as it appears most frequently (three times) in the data set.
Correct Answer:
A
— 4
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Q. In the following data set: 9, 10, 10, 11, 12, 12, 12, 13, what is the mode?
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Solution
The mode is 12, which appears three times, more than any other number.
Correct Answer:
D
— 12
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Q. In the following set of scores: 45, 50, 50, 60, 70, 70, 70, 80, what is the mode?
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Solution
The mode is 70, as it appears 3 times, which is more than any other score.
Correct Answer:
D
— 70
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Q. In the set of numbers 10, 20, 20, 30, 30, 30, 40, what is the mode?
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Solution
The mode is 30, as it appears 3 times, more than any other number.
Correct Answer:
C
— 30
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Q. In the set of numbers 5, 5, 6, 7, 8, 8, 8, 9, what is the mode?
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Solution
The mode is 8, as it appears 3 times, which is more than any other number.
Correct Answer:
D
— 8
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Q. In the set {1, 1, 2, 3, 3, 4, 4, 4, 5}, which number is the mode?
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Solution
The mode is 4, as it appears 3 times, more than any other number.
Correct Answer:
D
— 4
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Q. In the set {5, 5, 6, 7, 8, 8, 9}, which number is the mode?
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Solution
The mode is 5, as it appears 2 times, while 8 also appears 2 times. However, 5 is the smallest mode.
Correct Answer:
A
— 5
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Q. In triangle ABC, if AB = AC and angle A = 40 degrees, what is the measure of angle B? (2023)
A.
70 degrees
B.
80 degrees
C.
60 degrees
D.
50 degrees
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Solution
In an isosceles triangle, angles opposite to equal sides are equal. Therefore, angle B = angle C = (180 - 40) / 2 = 70 degrees.
Correct Answer:
B
— 80 degrees
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Q. In triangle ABC, if angle A = 30 degrees and angle B = 60 degrees, what is the measure of angle C? (2021)
A.
30 degrees
B.
60 degrees
C.
90 degrees
D.
120 degrees
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Solution
The sum of angles in a triangle is 180 degrees. Therefore, angle C = 180 - (30 + 60) = 90 degrees.
Correct Answer:
C
— 90 degrees
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Q. In triangle DEF, if angle D = 45 degrees and angle E = 45 degrees, what is the length of side DE if DF = 10 cm? (2023)
A.
5√2 cm
B.
10 cm
C.
10√2 cm
D.
20 cm
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Solution
In an isosceles right triangle, the sides opposite the 45-degree angles are equal. DE = DF/√2 = 10/√2 = 5√2 cm.
Correct Answer:
A
— 5√2 cm
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Q. In triangle DEF, if angle D = 45 degrees and angle E = 45 degrees, what is the type of triangle? (2020)
A.
Scalene
B.
Isosceles
C.
Equilateral
D.
Right
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Solution
Since two angles are equal (45 degrees), triangle DEF is an isosceles triangle.
Correct Answer:
B
— Isosceles
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Q. In triangle DEF, if angle D = 45 degrees and angle E = 45 degrees, what type of triangle is DEF? (2021)
A.
Scalene
B.
Isosceles
C.
Equilateral
D.
Right
Show solution
Solution
Since two angles are equal (45 degrees), triangle DEF is an isosceles triangle.
Correct Answer:
B
— Isosceles
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