3D Geometry
Q. Calculate the distance from the point P(1, 2, 3) to the origin O(0, 0, 0). (2023)
Solution
Distance = √[(1-0)² + (2-0)² + (3-0)²] = √[1 + 4 + 9] = √14.
Correct Answer: B — √14
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Q. Determine the coordinates of the centroid of the triangle with vertices A(0, 0, 0), B(4, 0, 0), C(0, 3, 0). (2023)
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A.
(1, 1, 0)
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B.
(2, 1, 0)
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C.
(4/3, 1, 0)
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D.
(0, 1, 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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Q. Determine the coordinates of the centroid of the triangle with vertices A(0, 0, 0), B(6, 0, 0), and C(0, 8, 0). (2023)
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A.
(2, 2, 0)
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B.
(2, 3, 0)
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C.
(3, 2, 0)
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D.
(0, 0, 0)
Solution
Centroid = ((0+6+0)/3, (0+0+8)/3, (0+0+0)/3) = (2, 2.67, 0).
Correct Answer: A — (2, 2, 0)
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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)
Solution
The points are collinear, hence the area = 0.
Correct Answer: A — 0
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Q. Find the coordinates of the midpoint of the line segment joining A(2, 3, 4) and B(4, 5, 6). (2023)
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A.
(3, 4, 5)
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B.
(2, 3, 4)
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C.
(4, 5, 6)
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D.
(5, 6, 7)
Solution
Midpoint M = ((2+4)/2, (3+5)/2, (4+6)/2) = (3, 4, 5).
Correct Answer: A — (3, 4, 5)
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Q. Find the distance between the parallel planes x + 2y + 3z = 4 and x + 2y + 3z = 10. (2023)
Solution
Distance = |d1 - d2| / √(a² + b² + c²) = |4 - 10| / √(1² + 2² + 3²) = 6 / √14.
Correct Answer: A — 2
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Q. Find the length of the diagonal of a rectangular box with dimensions 2, 3, and 6 units. (2022)
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A.
√49
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B.
√45
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C.
√36
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D.
√50
Solution
Diagonal = √(2² + 3² + 6²) = √(4 + 9 + 36) = √49 = 7 units.
Correct Answer: A — √49
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Q. Find the length of the diagonal of a rectangular box with dimensions 2, 3, and 6. (2023)
-
A.
√49
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B.
√36
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C.
√45
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D.
√50
Solution
Diagonal = √(2² + 3² + 6²) = √(4 + 9 + 36) = √49 = 7.
Correct Answer: A — √49
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Q. What is the area of the base of a cone with radius 3? (2023)
Solution
Area of base = πr² = π(3²) = 9π square units.
Correct Answer: A — 9π
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Q. What is the area of the base of a cylinder with radius 4 units? (2020)
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A.
16π
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B.
8π
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C.
12π
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D.
20π
Solution
Area of base = πr² = π(4)² = 16π square units.
Correct Answer: A — 16π
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Q. What is the distance between the parallel planes 2x + 3y - z = 5 and 2x + 3y - z = 10? (2021)
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A.
5/√14
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B.
10/√14
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C.
15/√14
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D.
20/√14
Solution
Distance = |d1 - d2| / √(A² + B² + C²) = |5 - 10| / √(2² + 3² + (-1)²) = 5/√14.
Correct Answer: B — 10/√14
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Q. What is the distance between the points A(1, 2, 3) and B(4, 5, 6)? (2023)
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A.
3√2
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B.
3√3
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C.
3
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D.
√27
Solution
Distance = √[(4-1)² + (5-2)² + (6-3)²] = √[3² + 3² + 3²] = √27 = 3√3.
Correct Answer: A — 3√2
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Q. What is the equation of the plane passing through the point (1, 2, 3) with normal vector (1, -1, 1)? (2023)
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A.
x - y + z = 0
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B.
x + y + z = 6
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C.
x - y + z = 1
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D.
x + y - z = 0
Solution
Equation of the plane: 1(x-1) - 1(y-2) + 1(z-3) = 0 => x - y + z = 1.
Correct Answer: C — x - y + z = 1
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Q. What is the equation of the plane passing through the points (1, 2, 3), (4, 5, 6), and (7, 8, 9)? (2021)
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A.
0 = 0
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B.
x + y + z = 12
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C.
x + y + z = 10
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D.
x + y + z = 9
Solution
The points are collinear, hence the equation of the plane is 0 = 0.
Correct Answer: A — 0 = 0
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Q. What is the volume of a cube with side length 3 units? (2023)
Solution
Volume = side³ = 3³ = 27 cubic units.
Correct Answer: B — 27
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Q. What is the volume of a cylinder with radius 2 and height 5? (2023)
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A.
20π
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B.
10π
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C.
15π
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D.
25π
Solution
Volume = πr²h = π(2²)(5) = 20π cubic units.
Correct Answer: A — 20π
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Q. What is the volume of a sphere with radius 5 units? (2021)
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A.
500/3π
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B.
125/3π
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C.
100/3π
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D.
200/3π
Solution
Volume = (4/3)πr³ = (4/3)π(5)³ = (4/3)π(125) = 500/3π cubic units.
Correct Answer: A — 500/3π
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