Introduction to Three Dimensional Geometry — Class 11 MCQs with Answers

Introduction to Three Dimensional Geometry — important questions & MCQs with answers (Class 11 Mathematics)

14 solved questions from this chapter's difficulty levels, each with its answer and explanation. The other 76 are timed and scored when you sign in.

1. Q1Easy

   Number of octants in 3D:

   A.8✓ Correct

   B.4

   C.6

   D.2

   Answer: A. 8

   Explanation: Three mutually perpendicular coordinate planes cut space into 2³ = 8 regions called octants — confusing this with the four planar quadrants (2² = 4) is the usual slip.

2. Q2Easy

   In which octant is the point (2, 3, 4)?

   A.First✓ Correct

   B.Second

   C.Third

   D.Eighth

   Answer: A. First

   Explanation: All three coordinates of (2, 3, 4) are positive, and the all-positive octant is by convention labelled the first — a sign change in even one coordinate moves it elsewhere.

3. Q3Easy

   Distance between (x₁, y₁, z₁) and (x₂, y₂, z₂):

   A.x₁ + y₁ + z₁

   B.|x₂−x₁| + |y₂−y₁| + |z₂−z₁|

   C.(x₂−x₁)(y₂−y₁)(z₂−z₁)

   D.√[(x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²]✓ Correct

   Answer: D. √[(x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²]

   Explanation: Distance between two points in 3D extends Pythagoras with a third squared term: √[(x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²] — dropping the z-term collapses it into the 2D formula by mistake.

4. Q4Easy

   Point dividing segment from (x₁, y₁, z₁) to (x₂, y₂, z₂) in ratio m:n (internally):

   A.Midpoint formula only

   B.((mx₂ + nx₁)/(m+n), (my₂ + ny₁)/(m+n), (mz₂ + nz₁)/(m+n))✓ Correct

   C.Cannot generalise

   D.Same as 2D

   Answer: B. ((mx₂ + nx₁)/(m+n), (my₂ + ny₁)/(m+n), (mz₂ + nz₁)/(m+n))

   Explanation: The 3D section formula divides each coordinate the way 2D does, with a third coordinate added: ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n), (mz₂+nz₁)/(m+n)) for internal division — not a new rule, just an extension.

5. Q5Easy

   The distance between the points (2, 5, 7) and (2, 5, 1) is:

   A.6✓ Correct

   B.8

   C.√6

   D.4

   Answer: A. 6

   Explanation: Only the z-coordinate differs, by 6.

6. Q6Easy

   Point on xz-plane has y =

   A.−1

   B.1

   C.0✓ Correct

   D.x+z

   Answer: C. 0

   Explanation: Any point on the xz-plane has zero displacement along the y-axis, so y = 0 while x and z stay free — picking x+z instead describes a line, not the defining condition.

7. Q7Medium

   Projection of point (3, 4, 5) on xy-plane:

   A.(3, 0, 5)

   B.(3, 4, 0)✓ Correct

   C.(0, 4, 5)

   D.(3, 4, 5)

   Answer: B. (3, 4, 0)

   Explanation: Projecting onto the xy-plane means dropping the height, so only z is set to zero: (3, 4, 0) — zeroing x or y instead moves the point onto the wrong coordinate plane.

8. Q8Medium

   Point on z-axis equidistant from (1, −2, 3) and (4, 0, 1):

   A.(0, 0, 1)

   B.(0, 0, 0)

   C.(0, 0, −3/4)✓ Correct

   D.(0, 0, 2)

   Answer: C. (0, 0, −3/4)

   Explanation: Setting distances equal for P(0,0,z) gives z²−6z+14 = z²−2z+17, so −4z = 3 and z = −3/4 — mishandling the constant terms while cancelling z² is where sign errors creep in.

9. Q9Medium

   Point dividing segment (1, 2, 3) and (5, 6, 9) externally in ratio 2:3:

   A.(−7, −6, −9)✓ Correct

   B.(7, 6, 9)

   C.(−2, 0, −3)

   D.(2, 3, 4)

   Answer: A. (−7, −6, −9)

   Explanation: External division flips a sign: (mx₂−nx₁)/(m−n) coordinate-wise. With m=2, n=3: x=(2·5−3·1)/(−1)=−7, y=(2·6−3·2)/(−1)=−6, z=(2·9−3·3)/(−1)=−9, giving (−7,−6,−9) — using the internal formula here is the usual trap.

10. Q10Medium

    Distance between (1,2,3) and (1,2,5):

    A.4

    B.3

    C.2✓ Correct

    D.5

    Answer: C. 2

    Explanation: x and y match at both points, so only z changes: distance = √(0² + 0² + (5−3)²) = √4 = 2 — don't add every coordinate difference blindly when two axes coincide.

11. Q11Medium

    Projection of point (3, 4, 5) on xz-plane:

    A.(3, 0, 5)✓ Correct

    B.(3, 4, 0)

    C.(0, 4, 5)

    D.(3, 0, 0)

    Answer: A. (3, 0, 5)

    Explanation: Projecting onto the xz-plane keeps x and z and zeroes the y-coordinate: (3, 0, 5) — zeroing z by mistake lands the point on the xy-plane instead of the xz-plane.

12. Q12Hard

    Origin coordinates in 3D:

    A.(1,1,1)

    B.(0,0,0)✓ Correct

    C.(0,0)

    D.(x,y,z)

    Answer: B. (0,0,0)

    Explanation: The origin, where all three coordinate axes meet, is always (0, 0, 0) in space — writing it as (0, 0) confuses it with the 2D origin, which has only two coordinates.

13. Q13Hard

    Triangle ABC with A(0,0,0), B(2,0,0), C(0,2,0). Find ∠BAC:

    A.90°✓ Correct

    B.60°

    C.45°

    D.30°

    Answer: A. 90°

    Explanation: Vectors AB = (2,0,0) and AC = (0,2,0) have zero dot product, so the angle between them is 90° — any nonzero dot product would mean the angle isn't a right angle.

14. Q14Hard

    YZ-plane divides line segment from (3, 1, 5) to (−6, 4, 11) in ratio:

    A.1:2 externally

    B.2:1 internally

    C.1:2 internally✓ Correct

    D.2:1 externally

    Answer: C. 1:2 internally

    Explanation: Setting x = 0 gives (−6k+3)/(k+1) = 0, so k = 1/2 — a positive k means internal division, in the ratio 1:2, unlike the external case where k comes out negative.

Introduction to Three Dimensional Geometry — FAQs

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