Introduction to Three Dimensional Geometry — Class 11 MCQs with Answers

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

6 solved questions from this chapter's difficulty levels, each with its answer and explanation. Tap an option to check yourself.

0 of 6 answered

1. Q1 of 6Easy

   Number of octants in 3D:

   A8B4C6D2
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   Answer: 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. Q2 of 6Easy

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

   AFirstBSecondCThirdDEighth
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   Answer: 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. Q3 of 6Medium

   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)D(0, 0, 2)
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   Answer: (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.

4. Q4 of 6Medium

   Assertion (A): The perpendicular distance of the point (3, −5, 4) from the ZX-plane is 3 units. Reason (R): If P(x, y, z) is any point, then |x|, |y| and |z| are the perpendicular distances of P from the YZ, ZX and XY-planes respectively.

   ABoth A and R are true and R is the correct explanation of ABBoth A and R are true but R is not the correct explanation of ACA is true but R is falseDA is false but R is true
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   Answer: A is false but R is true

   Explanation: A is false: the distance from the ZX-plane is given by the y-coordinate, so it is |−5| = 5 units, not 3 (3 is the distance from the YZ-plane). R is true: the size of each coordinate is the distance from the corresponding coordinate plane.

5. Q5 of 6Hard

   Point dividing (1,2,3),(7,8,9) in ratio 2:1 internally:

   A(5,6,7)B(3,4,5)C(4,5,6)D(7,8,9)
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   Answer: (5,6,7)

   Explanation: Internal section formula gives (mx₂+nx₁)/(m+n) coordinate-wise: ((2·7+1·1)/3, (2·8+1·2)/3, (2·9+1·3)/3) = (5, 6, 7) — swapping m and n flips which endpoint gets more weight.

6. Q6 of 6Hard

   Assertion (A): For the points A(−1, 2, 1), B(1, −2, 5), C(4, −7, 8) and D(2, −3, 4), AB = CD and BC = DA. Reason (R): The distance between P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) is √[(x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²].

   ABoth A and R are true and R is the correct explanation of ABBoth A and R are true but R is not the correct explanation of ACA is true but R is falseDA is false but R is true
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   Answer: Both A and R are true and R is the correct explanation of A

   Explanation: R is the distance formula, which is true. Using it, AB² = 2² + (−4)² + 4² = 36 and CD² = (−2)² + 4² + (−4)² = 36, so AB = CD = 6; BC² = 3² + (−5)² + 3² = 43 and DA² = (−3)² + 5² + (−3)² = 43, so BC = DA = √43. A is true, and it follows directly from R, so R explains A.

Introduction to Three Dimensional Geometry notes: key concepts (Class 11 Mathematics)

Coordinate geometry steps into space: three mutually perpendicular axes fix any point by an ordered triple (x, y, z). This chapter builds the coordinate planes and octants, the distance formula between two points, and the section formula locating a point that divides a segment in a given ratio.

Coordinate axes

Three mutually perpendicular lines — the x, y and z axes — meet at the origin O, and together they let us pin any point in space to a unique triple.

Coordinate planes

Taken in pairs the axes span three planes: XY (z = 0), YZ (x = 0) and ZX (y = 0); a point's distance from each plane is one of its coordinates.

Octants

The three coordinate planes cut space into eight regions called octants; the sign pattern of (x, y, z) tells you which octant a point lies in.

Coordinates of a point

A point P is the ordered triple (x, y, z): its signed perpendicular distances from the YZ, ZX and XY planes respectively.

Sign convention

In the first octant all three coordinates are positive; an axis point has its other two coordinates zero, and the origin is (0, 0, 0).

Distance formula

The distance between P(x₁,y₁,z₁) and Q(x₂,y₂,z₂) is the square root of the sum of the squared differences of their matching coordinates.

Distance from the origin

The distance of P(x, y, z) from O is √(x² + y² + z²), a special case of the distance formula with the second point at the origin.

Collinearity test

Three points are collinear when the largest of the three pairwise distances equals the sum of the other two — no triangle can be formed.

Section formula (internal)

A point dividing PQ internally in ratio m:n takes each coordinate as (m·second + n·first)/(m+n), a weighted average of the endpoints.

Section formula (external)

For external division in ratio m:n replace n by −n: each coordinate becomes (m·second − n·first)/(m−n), placing the point beyond the segment.

Midpoint

The midpoint of PQ divides it in ratio 1:1, so each coordinate is simply the average of the two endpoints' matching coordinates.

Centroid of a triangle

The centroid of a triangle with vertices A, B, C is the average of their coordinates and is the balance point of the three vertices.