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Q. Find the area of the triangle formed by the points (0, 0), (4, 0), and (0, 3). (2022) 2022

A. 6
B. 12
C. 8
D. 10

Solution

Area = 1/2 * base * height = 1/2 * 4 * 3 = 6.

Correct Answer: A — 6

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Q. Find the area of the triangle formed by the points A(0, 0, 0), B(1, 0, 0), and C(0, 1, 0). (2023)

A. 0.5
B. 1
C. 2
D. 0

Solution

Area = 0.5 * base * height = 0.5 * 1 * 1 = 0.5 square units.

Correct Answer: A — 0.5

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Q. Find the area of the triangle formed by the points A(1, 2, 3), B(4, 5, 6), and C(7, 8, 9). (2022)

A. 0
B. 6
C. 12
D. 3

Solution

The points are collinear, hence the area = 0.

Correct Answer: A — 0

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Q. Find the coefficient of x^0 in the expansion of (2x - 3)^3.

A. -27
B. -24
C. -18
D. -12

Solution

The coefficient of x^0 is (-3)^3 = -27.

Correct Answer: A — -27

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Q. Find the coefficient of x^1 in the expansion of (x + 4)^3.

A. 12
B. 48
C. 36
D. 24

Solution

The coefficient of x^1 is C(3,1) * (4)^2 = 3 * 16 = 48.

Correct Answer: A — 12

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Q. Find the coefficient of x^2 in the expansion of (2x + 3)^6.

A. 540
B. 720
C. 810
D. 960

Solution

The coefficient of x^2 is given by 6C2 * (2)^2 * (3)^4 = 15 * 4 * 81 = 4860.

Correct Answer: A — 540

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Q. Find the coefficient of x^2 in the expansion of (x + 4)^5. (2023)

A. 80
B. 100
C. 120
D. 160

Solution

The coefficient of x^2 is C(5,2)(4)^3 = 10 * 64 = 640.

Correct Answer: A — 80

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Q. Find the coefficient of x^2 in the expansion of (x - 5)^5.

A. 100
B. 150
C. 200
D. 250

Solution

The coefficient of x^2 is C(5,2) * (-5)^3 = 10 * (-125) = -1250.

Correct Answer: A — 100

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Q. Find the coefficient of x^3 in the expansion of (2x - 3)^4. (2022)

A. -54
B. -108
C. 108
D. 54

Solution

The coefficient of x^3 is C(4,3) * (2)^3 * (-3)^1 = 4 * 8 * (-3) = -96.

Correct Answer: B — -108

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Q. Find the coefficient of x^3 in the expansion of (x - 1)^7.

A. -35
B. -21
C. -7
D. -49

Solution

The coefficient of x^3 is given by 7C3 * (-1)^4 = 35.

Correct Answer: A — -35

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Q. Find the coefficient of x^3 in the expansion of (x - 2)^5.

A. -40
B. -80
C. -60
D. -100

Solution

The coefficient of x^3 is C(5,3) * (-2)^2 = 10 * 4 = 40.

Correct Answer: A — -40

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Q. Find the coefficient of x^4 in the expansion of (2x - 3)^5.

A. -240
B. -360
C. -480
D. -600

Solution

The coefficient of x^4 is C(5,4) * (2)^4 * (-3)^1 = 5 * 16 * (-3) = -240.

Correct Answer: A — -240

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Q. Find the coefficient of x^4 in the expansion of (2x - 3)^6.

A. 540
B. 720
C. 810
D. 900

Solution

The coefficient of x^4 is C(6,4) * (2)^4 * (-3)^2 = 15 * 16 * 9 = 2160.

Correct Answer: A — 540

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Q. Find the coefficient of x^4 in the expansion of (3x + 2)^5. (2022)

A. 240
B. 360
C. 480
D. 600

Solution

The coefficient of x^4 is C(5,4)(3)^4(2)^1 = 5 * 81 * 2 = 810.

Correct Answer: B — 360

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Q. Find the coefficient of x^4 in the expansion of (x + 1)^6.

A. 15
B. 20
C. 30
D. 35

Solution

The coefficient of x^4 is C(6,4) * 1^2 = 15.

Correct Answer: B — 20

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Q. Find the coefficient of x^4 in the expansion of (x + 2)^6.

A. 15
B. 30
C. 45
D. 60

Solution

The coefficient of x^4 is C(6,4) * (2)^2 = 15 * 4 = 60.

Correct Answer: B — 30

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Q. Find the coefficient of x^4 in the expansion of (x + 3)^6.

A. 81
B. 162
C. 243
D. 729

Solution

The coefficient of x^4 is C(6,4) * (3)^2 = 15 * 9 = 135.

Correct Answer: C — 243

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Q. Find the coefficient of x^4 in the expansion of (x + 5)^7.

A. 210
B. 1260
C. 1750
D. 2450

Solution

The coefficient of x^4 is given by C(7,4) * 5^3 = 35 * 125 = 4375.

Correct Answer: B — 1260

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Q. Find the coefficient of x^4 in the expansion of (x - 1)^5.

A. -5
B. 10
C. -10
D. 5

Solution

The coefficient of x^4 is C(5,4) * (-1)^1 = 5 * (-1) = -5.

Correct Answer: C — -10

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Q. Find the coefficient of x^4 in the expansion of (x - 5)^6.

A. 150
B. 200
C. 250
D. 300

Solution

The coefficient of x^4 is given by C(6,4) * (-5)^2 = 15 * 25 = 375.

Correct Answer: B — 200

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Q. Find the coefficient of x^5 in the expansion of (2x - 3)^6. (2022)

A. -540
B. 540
C. -720
D. 720

Solution

The coefficient of x^5 is C(6,5) * (2)^5 * (-3)^1 = 6 * 32 * (-3) = -576.

Correct Answer: A — -540

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Q. Find the coefficient of x^5 in the expansion of (2x - 3)^7. (2023)

A. 168
B. 252
C. 336
D. 504

Solution

The coefficient of x^5 is C(7,5) * (2)^5 * (-3)^2 = 21 * 32 * 9 = 6048.

Correct Answer: B — 252

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Q. Find the coefficient of x^5 in the expansion of (2x - 3)^8.

A. -6720
B. 6720
C. -3360
D. 3360

Solution

The coefficient of x^5 is C(8,5) * (2)^5 * (-3)^3 = 56 * 32 * (-27) = -6720.

Correct Answer: A — -6720

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Q. Find the coefficient of x^5 in the expansion of (x + 1)^7.

A. 7
B. 21
C. 35
D. 42

Solution

The coefficient of x^5 is C(7,5) = 21.

Correct Answer: C — 35

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Q. Find the coefficient of x^5 in the expansion of (x + 2)^7.

A. 21
B. 42
C. 56
D. 70

Solution

The coefficient of x^5 is C(7,5) * 2^2 = 21 * 4 = 84.

Correct Answer: C — 56

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Q. Find the coefficient of x^6 in the expansion of (x + 1)^10. (2023)

A. 10
B. 45
C. 120
D. 210

Solution

The coefficient of x^6 is given by 10C6 = 210.

Correct Answer: D — 210

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Q. Find the constant term in the expansion of (3x - 4/x)^5.

A. -64
B. 0
C. 32
D. 48

Solution

The constant term occurs when the power of x is zero. The term is given by 5C2 * (3x)^2 * (-4/x)^3 = 10 * 9 * (-64) = -5760.

Correct Answer: A — -64

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Q. Find the constant term in the expansion of (x - 2/x)^6. (2022)

A. -64
B. 32
C. 0
D. -32

Solution

The constant term occurs when the power of x is zero. Setting 6 - 2k = 0 gives k = 3. The term is C(6,3)(-2)^3 = -64.

Correct Answer: A — -64

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Q. Find the coordinates of the centroid of the triangle with vertices (0, 0), (6, 0), and (0, 8). (2022)

A. (2, 2)
B. (2, 3)
C. (3, 2)
D. (4, 4)

Solution

Centroid = ((0+6+0)/3, (0+0+8)/3) = (2, 2).

Correct Answer: A — (2, 2)

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Q. Find the coordinates of the centroid of the triangle with vertices A(0, 0, 0), B(4, 0, 0), C(0, 3, 0). (2021)

A. (4/3, 1, 0)
B. (2, 1, 0)
C. (1, 1, 0)
D. (0, 0, 0)

Solution

Centroid G = ((0+4+0)/3, (0+0+3)/3, (0+0+0)/3) = (4/3, 1, 0).

Correct Answer: B — (2, 1, 0)

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