Introduction to 3D Geometry — Chapter Test | Class 11 IMO

Q1

How many octants are formed by the three coordinate planes?

Three mutually perpendicular coordinate planes divide space into 8 octants.

Q2

The coordinates of the foot of the perpendicular drawn from the point (2, 5, 7) onto the x-axis are:

The foot of the perpendicular on the x-axis retains the x-coordinate of the point, but its y and z coordinates become 0. Thus, it is (2, 0, 0).

Q3

The distance between points (0,0,5) and (0,12,5) is:

Distance = √(0²+12²+0²)=12.

Q4

Distance between (4,4,4) and (1,0,0) equals:

Distance = √(3²+4²+4²)=√41.

Q5

Which among the strictly following defined spatial points sits nearest to the absolute spatial origin (0, 0, 0)?

Compare the squared absolute distances: A is 4+4+1 = 9. B is 9. C is 1+1+1 = 3. D is 0+4+4 = 8. The smallest squared distance is 3, proving (1, 1, 1) is absolutely closest.

Q6

Calculate purely mathematically the exact spatial coordinate mapping defining the absolute reflection of central origin directly across and through the fixed spatial point (1, 2, 3).

Under strict spatial symmetry, the mirroring node functionally acts practically as an absolute midpoint between original and reflection. Resolving standard (0+x)/2 = 1 yields strictly x=2. Symmetrical parallel processing defines y=4 and rigidly z=6. Concluding broadly as (2, 4, 6).

Q7

A commercial cold-storage containment structurally maps internal dimensions strictly matching 10m × 8m × 6m natively. A solitary flying drone structurally powering directly diagonally linking opposite bounding internal corners sequentially calculates a straight absolute flight measuring:

Calculating total maximum diagonal internal length structurally requires evaluating absolute spatial diagonal mathematically: d = √(l² + b² + h²). Translating strictly implies √(10² + 8² + 6²) uniformly equating identically to √(100 + 64 + 36) = √200 logically filtering to functionally 10√2 meters precisely.

Q8

Which of the following points definitively lies within the xy-plane?

Any point lying in the xy-plane has no displacement along the z-axis, meaning its z-coordinate must be exactly 0. Thus, (2, 3, 0) is the correct point.

Q9

Two strictly sequential vertices defining a parallelogram scale as A(1, 2, −4) and B(3, 1, −2). Given the rigid diagonals bisect mathematically at (2, −1, 3), determine the exact vertex C.

A standard parallelogram strictly bisects its diagonals. The provided midpoint is intrinsically the precise midpoint of AC. Assigning C as (x,y,z): (1+x)/2 = 2 → x = 3; (2+y)/2 = −1 → y = −4; (−4+z)/2 = 3 → z = 10. Validating C(3, −4, 10).

Q10

Which of the following equations accurately represents the x-axis in 3D geometry?

On the x-axis, both the y and z coordinates of any point are exactly zero. Therefore, the equations y = 0 and z = 0 together represent the x-axis.

Q11

The vertices of a triangle are A(1, 1, 1), B(1, 2, 3) and C(2, 3, 1). This triangle is fundamentally:

Calculating lengths: AB = √(0² + 1² + 2²) = √5. BC = √(1² + 1² + (−2)²) = √6. AC = √(1² + 2² + 0²) = √5. Since AB = AC, it is an isosceles triangle.

Q12

What is the total number of mutually perpendicular coordinate planes defined within a standard 3D Cartesian space?

There are exactly 3 fundamental mutually perpendicular coordinate planes in standard 3D geometry: the xy-plane, the yz-plane, and the zx-plane.

Q13

The coordinates of the midpoint of A(−2, 3, 5) and B(4, 1, 7) are:

Midpoint = ((−2+4)/2, (3+1)/2, (5+7)/2) = (1, 2, 6).

Q14

Distance between (0,0,0) and (6,8,0) is:

Distance = √(6²+8²+0²)=√100=10.

Q15

Find the z-coordinate of the point on the z-axis which is equidistant from the points (1, 5, 7) and (5, 1, −4).

Let point be P(0, 0, z). P to (1, 5, 7): 1² + 5² + (z−7)² = z² − 14z + 75. P to (5, 1, −4): 5² + 1² + (z+4)² = z² + 8z + 42. Equating: z² − 14z + 75 = z² + 8z + 42 → 22z = 33 → z = 33/22 = 3/2.