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CBSE · Class 11 · Mathematics

Introduction to Three-dimensional Geometry

Standard difficulty · 20 questions · 40 marks

Name: Date: Score: ____ / 40

Show your working for Sections B and C. Answers are on the last page — try every question first!

Section A — Choose the correct answer8 × 1 = 8 marks
1.
The point (−2, 9, 2) lies in octant:
  1. II
  2. VI
  3. VIII
  4. VII
2.
The point (−8, 0, −2) lies on:
  1. the xz-plane
  2. the z-axis
  3. the yz-plane
  4. the x-axis
3.
The distance between (3, 4, 8) and (2, −8, 20) is:
  1. 17
  2. 18
  3. 289
  4. 25
4.
The distance of the point (3, −5, −6) from the y-axis is:
  1. √46
  2. 5
  3. √70
  4. 3√5
5.
The three coordinate planes divide space into:
  1. three parts
  2. six regions
  3. four quadrants
  4. eight octants
6.
The perpendicular distance of (7, 5, 8) from the yz-plane is:
  1. √138
  2. 8
  3. 7
  4. 5
7.
The distance between (k, 2, 4) and (−4, 10, −4) is 12. The possible values of k are:
  1. 0 only
  2. −8 and 0
  3. −9 and 1
  4. 0 and 8
8.
Assertion (A): The distance between (−5, 6, 7) and (−1, 14, 15) is 20.
Reason (R): The distance between two points is √((x2 − x1)2 + (y2 − y1)2 + (z2 − z1)2).
  1. Both A and R are true, and R is the correct explanation of A.
  2. Both A and R are true, but R is not the correct explanation of A.
  3. A is true, but R is false.
  4. A is false, but R is true.
Section B — Short answer8 × 2 = 16 marks
9.
Find the distance between (1, 8, 7) and (4, 5, −2).
10.
Show that the points A(1, 0, −8), B(0, 2, −6) and C(−3, 8, 0) are collinear.
11.
Find the lengths of the sides of the triangle with vertices A(−2, 0, 7), B(2, 2, 7) and C(2, −2, 7). Is it right-angled, isosceles, both or neither?
12.
Find the equation satisfied by the points P(x, y, z) that are equidistant from A(8, 7, 5) and B(−7, 1, 2).
13.
Name the octant in which each point lies: (8, −2, −9), (3, 8, 5), (−6, −7, −1).
14.
Find the coordinates of the feet of the perpendiculars from (8, 2, −4) to the three coordinate planes.
15.
Find k if the distance between (k, −9, 15) and (−7, 0, 3) is 17.
16.
Find the distance of the point (8, 3, 8) from (a) the origin, (b) the x-axis, (c) the y-axis, (d) the z-axis.
Section C — Word problems4 × 4 = 16 marks
17.
A triangle has vertices A(8, 1, −6), B(5, −3, −4) and C(1, 8, 5). Find AB, BC and CA. Is the triangle right-angled? Is it isosceles?
18.
Find the point on the x-axis that is equidistant from (4, −4, 8) and (−2, 4, −8).
19.
Show that A(−5, 4, −6), B(−4, 6, −4) and C(−3, 8, −2) are collinear, and find the ratio AB : BC.
20.
A cuboid has one vertex at the origin and edges of length 2, 5 and 14 along the positive x-, y- and z-axes. Write the coordinates of all eight vertices and find the length of a diagonal of the cuboid.
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CBSE · Class 11 · Mathematics

Answer key

Introduction to Three-dimensional Geometry · AW-11M-TDG-2026-V1 · For parents — fold back or tear off before handing the sheet to your child.

QAnswerWorking
1(a) IIsigns (−, +, +)
2(a) the xz-planelook at which coordinates are zero
3(a) 17√(12 + 122 + 122) = √289
4(d) 3√5√(32 + (−6)2)
5(d) eight octants—
6(c) 7the x-coordinate, ignoring the sign
7(b) −8 and 0(k + 4)2 = 144 − 64 − 64 = 16
8(d) A is false, but R is true.A is false: √(42 + 82 + 82) = 12. R is true.
93√11√(32 + (−3)2 + (−9)2) = √99
10AB = 3, BC = 9, AC = 12; AB + BC = AC, so they are collinearfind the three distances
11AB = 2√5, BC = 4, CA = 2√5; isoscelesAB2 = 20, BC2 = 16, CA2 = 20
12−30x − 12y − 6z = −84PA2 = PB2; the x2, y2, z2 terms cancel
13(8, −2, −9): VIII; (3, 8, 5): I; (−6, −7, −1): VII—
14xy-plane (8, 2, 0); yz-plane (0, 2, −4); xz-plane (8, 0, −4)set the missing coordinate to 0
15k = −15 or 1(k − (−7))2 = 289 − 81 − 144 = 64
16(a) √137; (b) √73; (c) 8√2; (d) √73distance from the x-axis is √(y2 + z2), and so on
17AB = √29, BC = √218, CA = √219. Not right-angled. Not isosceles.AB2 = 29, BC2 = 218, CA2 = 219
18(1, 0, 0)(x − 4)2 + 80 = (x − (−2))2 + 80
19AB = 3, BC = 3, AC = 6; AB + BC = AC, so they are collinear; AB : BC = 1 : 1compute the three distances
20(0, 0, 0), (2, 0, 0), (0, 5, 0), (0, 0, 14), (2, 5, 0), (2, 0, 14), (0, 5, 14), (2, 5, 14); diagonal 15√(22 + 52 + 142)