NCERT Solutions for Class 12 Maths Chapter 11

NCERT Solutions for Class 12 Maths Chapter 11 Three dimensional geometry (3D) Exercise 11.1, 11.2, 11.3 & Miscellaneous exercises in PDF form to free download updated for new academic session 2020-21.

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NCERT Solutions for Class 12 Maths Chapter 11

Class:12
Subject:Maths
Chapter 11:Three dimensional geometry

12th Maths Chapter 11 Solutions

NCERT Solutions for Class 12 Maths Chapter 11 are given below to download in PDF form free. Join the discussion forum to ask your doubts related to NIOS board and CBSE board. Download latest NCERT Books 2020-2021 following the current CBSE Syllabus.

Previous Years Important Questions

1. Find the Cartesian and Vector equations of the line which passes through the point (-2, 4, -5) and parallel to the line given by (x+3)/3 = (y-4)/5 = (8-z)/-6. [CBSE Sample Paper 2017]
2. If the vectors p = ai + j + k, q = i + bj + k and r = i + j + ck are co-planar, then for a, b, c ≠ 1, show that 1/(1-a) + 1/(1-b) + 1/(1-c) = 1. [CBSE Sample Paper 2017]
3. A plane meets the coordinate axes in A, B and C such that the centroid of triangle ABC is the point (α, β, γ). Show that the equation of the plane is x/α + y/β + z/γ = 3. [CBSE Sample Paper 2017]
4. Define skew lines. Using only vector approach, find the shortest distance between the following two skew lines: r = (8 + 3m)i – (9 + 16m)j + (10 + 7m)k and r = 15i + 29j + 5k + n(3i + 8j – 5k). [CBSE Sample Paper 2017]
5. Find the vector equation of the line passing through the point A(1, 2, –1) and parallel to the line 5x – 25 = 14 – 7y = 35z. [Delhi 2017]

Questions from Board Papers

1. Find the vector and Cartesian equations of a line passing through (1, 2, –4) and perpendicular to the two lines (x – 8)/3 = (y + 19)/-16 = (z – 10)/7 and (x – 15)/3 = (y – 29)/8 = (z – 5)/-5. [Delhi 2017]
2. Find the vector equation of a plane which is at a distance of 5 units from the origin and its normal vector is 2i – 3j + 6k. [Delhi 2016]
3. Show that the vectors a, b, and c are co-planar if a + b + c and c + a are co-planar. [Delhi 2016]
4. Find the coordinate of the point P where the line through A(3, – 4, –5) and B (2, –3, 1) crosses the plane passing through three points L(2, 2, 1), M(3, 0, 1) and N(4, –1, 0). Also, find the ratio in which P divides the line segment AB. [Delhi 2016]
5. Find the equation of the plane through the line of intersection of the planes x + y + z = 1 and 2x + 3y + 4z = 5 which is perpendicular to the plane x – y + z = 0. Then find the distance of plane thus obtained from the point A(1, 3, 6). [Delhi 2015C]

Important Questions on 12th Maths Chapter 11

What is the relation between the direction cosines of a line?
If l, m, n are the direction cosines of a line, then l² + m² + n² = 1
What is meant by Direction ratios of a line?
Direction ratios of a line are the numbers which are proportional to the direction cosines of the line.
What are Skew lines in 3-D geometry?
Skew lines are lines in the space which are neither parallel nor intersecting. They lie in the different planes.
What is the shortest distance between two skew lines?
The shortest distance between two skew lines is the length of the line segment perpendicular to both the lines.
Find the Equation of a plane which is at a distance p from the origin with direction cosines of the normal to the plane as l, m, n.
Equation of a plane which is at a distance p from the origin with direction cosines of the normal to the plane as l, m, n is lx + my + nz = p.