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#### Chapters

Chapter 2: Relations and Functions

Chapter 3: Trigonometric Functions

Chapter 4: Principle of Mathematical Induction

Chapter 5: Complex Numbers and Quadratic Equations

Chapter 6: Linear Inequalities

Chapter 7: Permutations and Combinations

Chapter 8: Binomial Theorem

Chapter 9: Sequences and Series

Chapter 10: Straight Lines

Chapter 11: Conic Sections

Chapter 12: Introduction to Three Dimensional Geometry

Chapter 13: Limits and Derivatives

Chapter 14: Mathematical Reasoning

Chapter 15: Statistics

Chapter 16: Probability

## Solutions for Chapter 12: Introduction to Three Dimensional Geometry

Below listed, you can find solutions for Chapter 12 of CBSE, Karnataka Board PUC NCERT for Class 11 Mathematics.

### NCERT solutions for Class 11 Mathematics Chapter 12 Introduction to Three Dimensional Geometry Exercise 12.1 [Page 271]

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

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

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)

Fill in the blanks:

The x-axis and y-axis taken together determine a plane known as_______.

Fill in the blanks:

The coordinates of points in the XY-plane are of the form _______.

Fill in the blanks:

Coordinate planes divide the space into ______ octants.

### NCERT solutions for Class 11 Mathematics Chapter 12 Introduction to Three Dimensional Geometry Exercise 12.2 [Page 273]

Find the distance between the pairs of points

(2, 3, 5) and (4, 3, 1)

Find the distance between the pairs of points:

(–3, 7, 2) and (2, 4, –1)

Find the distance between the pairs of points:

(–1, 3, –4) and (1, –3, 4)

Find the distance between the following pairs of points:

(2, –1, 3) and (–2, 1, 3)

Show that the points (–2, 3, 5), (1, 2, 3) and (7, 0, –1) are collinear.

Verify the following: (0, 7, –10), (1, 6, –6) and (4, 9, –6) are the vertices of an isosceles triangle.

Verify the following: (0, 7, 10), (–1, 6, 6) and (–4, 9, 6) are the vertices of a right angled triangle.

Verify the following: (–1, 2, 1), (1, –2, 5), (4, –7, 8) and (2, –3, 4) are the vertices of a parallelogram.

Find the equation of the set of points which are equidistant from the points (1, 2, 3) and (3, 2, –1).

Find the equation of the set of points P, the sum of whose distances from A (4, 0, 0) and B (–4, 0, 0) is equal to 10.

### NCERT solutions for Class 11 Mathematics Chapter 12 Introduction to Three Dimensional Geometry Exercise 12.3 [Page 277]

Find the coordinates of the point which divides the line segment joining the points (–2, 3, 5) and (1, –4, 6) in the ratio (i) 2:3 internally, (ii) 2:3 externally.

Given that P (3, 2, –4), Q (5, 4, –6) and R (9, 8, –10) are collinear. Find the ratio in which Q divides PR.

Find the ratio in which the YZ-plane divides the line segment formed by joining the points (–2, 4, 7) and (3, –5, 8).

Using section formula, show that the points A (2, –3, 4), B (–1, 2, 1) and `C(0, 1/3 , 2)` are collinear.

Find the coordinates of the points which trisect the line segment joining the points P (4, 2, –6) and Q (10, –16, 6).

### NCERT solutions for Class 11 Mathematics Chapter 12 Introduction to Three Dimensional Geometry Miscellaneous Exercise [Pages 278 - 279]

Three vertices of a parallelogram ABCD are A (3, –1, 2), B (1, 2, –4) andC (–1, 1, 2). Find the coordinates of the fourth vertex.

Find the lengths of the medians of the triangle with vertices A (0, 0, 6), B (0, 4, 0) and (6, 0, 0).

If the origin is the centroid of the triangle PQR with vertices P (2*a*, 2, 6), Q (–4, 3*b*, –10) and R (8, 14, 2*c*), then find the values of *a*, *b* and *c*

Find the coordinates of a point on *y*-axis which are at a distance of `5sqrt2` from the point P (3, –2, 5).

A point R with *x*-coordinate 4 lies on the line segment joining the pointsP (2, –3, 4) and Q (8, 0, 10). Find the coordinates of the point R.

[**Hint** suppose R divides PQ in the ratio *k*: 1. The coordinates of the point R are given by `((8k + 2)/(k+1), (-3)/(k+1), (10k + 4)/(k+1))`

If A and B be the points (3, 4, 5) and (–1, 3, –7), respectively, find the equation of the set of points P such that PA^{2} + PB^{2} = *k*^{2}, where *k* is a constant.

## Solutions for Chapter 12: Introduction to Three Dimensional Geometry

## NCERT solutions for Class 11 Mathematics chapter 12 - Introduction to Three Dimensional Geometry

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Concepts covered in Class 11 Mathematics chapter 12 Introduction to Three Dimensional Geometry are Coordinate Axes and Coordinate planes, Coordinates of a Point in Space, Distance Between Two Points, Section Formula, Three Dimessional Space.

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