Exercise 11.1 — Introduction to Three Dimensional Geometry

1. 11.1.11 markNCERT Class 11 Mathematics, Introduction to Three Dimensional Geometry, Reprint 2026-27

   A point is on the x-axis. What are its y-coordinate and z-coordinate?

   Hint. Any point on the x-axis has zero distance from both the ZX-plane's partner axes — think about what its perpendicular distance from the YZ-plane's neighbouring axes must be.

   A point on the x-axis has the form (x,0,0), since it lies on both the XY-plane (z=0) and the XZ-plane (y=0) simultaneously.

   ✦ Working through each part gives: Both the y-coordinate and the z-coordinate are 0.

2. 11.1.21 markNCERT Class 11 Mathematics, Introduction to Three Dimensional Geometry, Reprint 2026-27

   A point is in the XZ-plane. What can you say about its y-coordinate?

   Hint. The XZ-plane is defined as the set of points at zero distance from it in the y-direction.

   Every point in the XZ-plane has the form (x,0,z), since its perpendicular distance from the XZ-plane itself is zero, and that distance is measured by the y-coordinate.

   ✦ Working through each part gives: The y-coordinate is 0.

3. 11.1.33 marksNCERT Class 11 Mathematics, Introduction to Three Dimensional Geometry, Reprint 2026-27

   Name the octants in which the following points lie: (1,2,3), (4,-2,3), (4,-2,-5), (4,2,-5), (-4,2,-5), (-4,2,5), (-3,-1,6), (-2,-4,-7).

   Hint. Read off the sign of each coordinate and match the (x,y,z) sign pattern against the table of eight octants.

   Matching each point's sign pattern against the octant table: (1,2,3) is (+,+,+) -> I. (4,-2,3) is (+,-,+) -> IV. (4,-2,-5) is (+,-,-) -> VIII. (4,2,-5) is (+,+,-) -> V. (-4,2,-5) is (-,+,-) -> VI. (-4,2,5) is (-,+,+) -> II. (-3,-1,6) is (-,-,+) -> III. (-2,-4,-7) is (-,-,-) -> VII.

   ✦ Working through each part gives: (1,2,3): I; (4,-2,3): IV; (4,-2,-5): VIII; (4,2,-5): V; (-4,2,-5): VI; (-4,2,5): II; (-3,-1,6): III; (-2,-4,-7): VII.

4. 11.1.42 marksNCERT Class 11 Mathematics, Introduction to Three Dimensional Geometry, Reprint 2026-27

   Fill in the blanks: (i) The x-axis and y-axis taken together determine a plane known as _______. (ii) The coordinates of points in the XY-plane are of the form _______. (iii) Coordinate planes divide the space into ______ octants.

   Hint. Each blank follows directly from how the coordinate axes and planes were defined in this chapter.

   (i) The x-axis and y-axis together determine the XY-plane. (ii) Any point in the XY-plane has zero distance from that plane, i.e. z=0, so its coordinates have the form (x,y,0). (iii) The three coordinate planes divide space into eight regions.

   ✦ Working through each part gives: (i) XY-plane; (ii) (x, y, 0); (iii) eight (8) octants.

Solutions written by the tuition.in editorial team and checked against the NCERT Class 11 Mathematics textbook, Reprint 2026-27 (kemh111.pdf) — Exercise 11.1 (4 questions), Exercise 11.2 (5 questions), plus the chapter's Miscellaneous Exercise (4 questions), 13 questions total, the shortest solutions file in this project to date, matching the book's own shortest chapter (9 pages, only two numbered exercises). Confirmed against the CBSE curriculum PDF that the section formula is formative-only for this chapter and confirmed via full-text search that it is completely absent from the book (zero hits for 'section formula', 'internal division', or 'ratio m'). The old stub had fabricated a third numbered exercise ('Exercise 11.3', 5 questions on the section formula) that does not exist in the book at all — removed. Midpoint and centroid, while never given a named boxed formula in the book, are confirmed as genuinely required techniques via the book's own Example 7 (parallelogram diagonals bisecting each other), Example 9 (centroid used to find a missing vertex), and the Miscellaneous Exercise's own Q1 (parallelogram's fourth vertex) and Q2 (median lengths).. Questions are referenced from the NCERT textbook for identification.