3D Geometry Study Resources for Competitive Exams

3D Geometry

Q. Calculate the distance from the point (1, 2, 3) to the origin (0, 0, 0). (2021)

A. √14
B. √6
C. √9
D. √12

Solution

Distance = √[(1-0)² + (2-0)² + (3-0)²] = √[1 + 4 + 9] = √14.

Correct Answer: A — √14

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Q. Calculate the distance from the point P(1, 2, 3) to the origin O(0, 0, 0). (2023)

A. 3
B. √14
C. √6
D. √9

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)

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

A. (2, 2, 0)
B. (2, 3, 0)
C. (3, 2, 0)
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. Determine the coordinates of the centroid of the triangle with vertices A(0, 0, 0), B(0, 4, 0), and C(3, 0, 0). (2021)

A. (1, 1.33, 0)
B. (1, 2, 0)
C. (0, 1.33, 0)
D. (0, 2, 0)

Solution

Centroid = ((0+0+3)/3, (0+4+0)/3, (0+0+0)/3) = (1, 1.33, 0).

Correct Answer: B — (1, 2, 0)

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Q. Determine the coordinates of the centroid of the triangle with vertices A(1, 2, 3), B(4, 5, 6), and C(7, 8, 9). (2021)

A. (4, 5, 6)
B. (3, 4, 5)
C. (5, 6, 7)
D. (6, 7, 8)

Solution

Centroid G = ((1+4+7)/3, (2+5+8)/3, (3+6+9)/3) = (4, 5, 6).

Correct Answer: B — (3, 4, 5)

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Q. Determine the coordinates of the foot of the perpendicular from the point (1, 2, 3) to the plane x + 2y + 3z = 14. (2023)

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

Solution

Using the formula for the foot of the perpendicular, we find the coordinates to be (1, 2, 4).

Correct Answer: B — (1, 2, 4)

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Q. Find the area of a triangle with vertices at 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(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 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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Q. Find the coordinates of the midpoint of the line segment joining A(2, -1, 3) and B(4, 3, 5). (2022)

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

Solution

Midpoint M = ((2+4)/2, (-1+3)/2, (3+5)/2) = (3, 1, 4).

Correct Answer: A — (3, 1, 4)

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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)

A. (3, 4, 5)
B. (2, 3, 4)
C. (4, 5, 6)
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 2x + 3y + 4z = 5 and 2x + 3y + 4z = 10. (2023)

A. 5/√29
B. 10/√29
C. 15/√29
D. 20/√29

Solution

Distance = |d1 - d2| / √(a² + b² + c²) = |5 - 10| / √(2² + 3² + 4²) = 5 / √29.

Correct Answer: B — 10/√29

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Q. Find the distance between the parallel planes 2x + 3y + z = 5 and 2x + 3y + z = 10. (2022)

A. 5
B. 10
C. √14
D. √10

Solution

Distance = |d1 - d2| / √(A² + B² + C²) = |5 - 10| / √(2² + 3² + 1²) = 5 / √14.

Correct Answer: A — 5

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Q. Find the distance between the parallel planes x + 2y + 3z = 4 and x + 2y + 3z = 10. (2023)

A. 2
B. 3
C. 4
D. 5

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 cuboid with dimensions 3, 4, and 12 units. (2020)

A. √169
B. √145
C. √153
D. √157

Solution

Diagonal = √(3² + 4² + 12²) = √(9 + 16 + 144) = √169 = 13.

Correct Answer: C — √153

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Q. Find the length of the diagonal of a rectangular box with dimensions 2, 3, and 6. (2023)

A. √49
B. √36
C. √45
D. √50

Solution

Diagonal = √(2² + 3² + 6²) = √(4 + 9 + 36) = √49 = 7.

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 units. (2022)

A. √49
B. √45
C. √36
D. √50

Solution

Diagonal = √(2² + 3² + 6²) = √(4 + 9 + 36) = √49 = 7 units.

Correct Answer: A — √49

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Q. If the coordinates of a point P are (1, -1, 2), what is the x-coordinate of the point Q if it is reflected across the yz-plane? (2023)

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

Solution

Reflection across the yz-plane changes the sign of the x-coordinate. So, Q = (-1, -1, 2).

Correct Answer: B — -1

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Q. What is the area of a triangle with vertices at A(0, 0, 0), B(1, 0, 0), and C(0, 1, 0)? (2022)

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. What is the area of the base of a cone with radius 3? (2023)

A. 9π
B. 6π
C. 3π
D. 12π

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)

A. 16π
B. 8π
C. 12π
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)

A. 5/√14
B. 10/√14
C. 15/√14
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 parallel planes x + 2y + 3z = 4 and x + 2y + 3z = 10? (2023)

A. 2
B. 3
C. 4
D. 5

Solution

Distance = |d1 - d2| / √(a² + b² + c²) = |4 - 10| / √(1² + 2² + 3²) = 6 / √14.

Correct Answer: A — 2

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Q. What is the distance between the point (2, 3, 4) and the plane x + y + z = 10? (2022)

A. 1
B. 2
C. 3
D. 4

Solution

Distance = |(2 + 3 + 4 - 10) / √(1² + 1² + 1²)| = |(-1)/√3| = 1/√3.

Correct Answer: A — 1

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Q. What is the distance between the points A(1, 2, 3) and B(4, 5, 6)? (2023)

A. 3√2
B. 3√3
C. 3
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 a plane passing through the point (1, 2, 3) with normal vector (1, 1, 1)? (2022)

A. x + y + z = 6
B. x + y + z = 3
C. x + y + z = 1
D. x + y + z = 0

Solution

Equation of the plane: 1(x-1) + 1(y-2) + 1(z-3) = 0 => x + y + z = 6.

Correct Answer: A — x + y + z = 6

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Q. What is the equation of a plane passing through the point (1, 2, 3) with normal vector (2, -1, 3)? (2021)

A. 2x - y + 3z = 12
B. 2x + y - 3z = 0
C. 2x - y + 3z = 0
D. 2x + y + 3z = 12

Solution

Equation of the plane: 2(x-1) - 1(y-2) + 3(z-3) = 0 simplifies to 2x - y + 3z = 12.

Correct Answer: C — 2x - y + 3z = 0

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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)

A. x - y + z = 0
B. x + y + z = 6
C. x - y + z = 1
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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Showing 1 to 30 of 37 (2 Pages)

3D Geometry

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