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Three - dimensional Geometry

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Class 12 Maths Chapter 11
Three - Dimensional Geometry
Important Questions

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Three-Dimensional Geometry typically covers a range of topics related to geometry in three dimensions. Here are CBSE class 12 maths Three dimensional Geometry important questions you should be familiar with.Questions are designed to test understanding of the fundamental concepts in this chapter. Engaging with various question formats enhances students’ readiness for the exams.questions cover a broad range of key concepts in three-dimensional geometry and can serve as a good starting point for revision or practice.

Introduction

Three-dimensional (3D) geometry is a branch of mathematics that deals with shapes and objects existing in three-dimensional space. Unlike two-dimensional geometry, which focuses on flat shapes like squares and circles, 3D geometry involves depth, making it a more complex and realistic representation of the physical world around us. Students delve into the basics of point,lines and planes,3D Shapes and Solids.This segment often extends to cover 3D operations,, including their properties and applications within the realm of 3D applications.

What are Three - Dimentional Geometry?

Three-dimensional (3D) geometry is an area of mathematics that deals with shapes and structures in a three-dimensional space. Unlike two-dimensional geometry, which focuses on flat shapes like squares and circles, 3D geometry involves depth in addition to length and width, making it a more complex and realistic representation of the physical world. 3D geometry is not just a theoretical construct but has practical applications in various fields:
  • Engineering and Architecture: For designing structures and machines.
  • Computer Graphics: In animation, video games, and virtual reality.
  • Physics and Astronomy: To model and understand the movement and interaction of objects in space.
  • Medical Imaging: In techniques like MRI and CT scans for creating 3D images of the human body.

Class 12 Three - Dimentional Geometry Important Questions and Answers

Q1. The direction cosines of the y-axis are:
Options
(a) (9, 0, 0)
(b) (1, 0, 0)
(c) (0, 1, 0)
(d) (0, 0, 1)

Ans. (c) (0, 1, 0)

Q2. The equation x² – x – 2 = 0 in three-dimensional space is represented by:
Options
(a) A pair of parallel planes
(b) A pair of straight lines
(c) A pair of the perpendicular plane
(d) None of these

Ans. (a) A pair of parallel planes

Q3. Show that the points A (2, 3, – 4), B (1, – 2, 3) and C (3, 8, – 11) are collinear.

Explanation:
We know that the direction ratios of the line passing through two points P(x_1, y_1, z1)\text{ and }Q(x_2, y_2, z_2) are given by:
x_2 – x_1, y_2 – y_1, z_2 – z_1\text{ or }x_1 – x_2, y_1 – y_2, z_1 – z_2 \\[2.5 bp] Given points are A (2, 3, – 4), B (1, – 2, 3) and C (3, 8, – 11).
Direction ratios of the line joining A and B are:
1 – 2, – 2 – 3, 3 + 4
i.e. – 1, – 5, 7.
The direction ratios of the line joining B and C are:
3 –1, 8 + 2, – 11 – 3
i.e., 2, 10, – 14.
From the above, it is clear that direction ratios of AB and BC are proportional.
That means AB is parallel to BC. But point B is common to both AB and BC.
Hence, A, B, C are collinear points.

Q4. Show that the lines
\dfrac{(x – 5)}{7} = \dfrac{(y + 2)}{-5} = \dfrac{z}{1}\text{ and }\dfrac{x}{1} = \dfrac{y}{2} = \dfrac{z}{3} are perpendicular to each other.

Explanation:
Given lines are:
\dfrac{(x – 5)}{7} = \dfrac{(y + 2)}{-5} =\dfrac{z}{1}\text{ and }\dfrac{x}{1} = \dfrac{y}{2} = \dfrac{z}{3} \\ The direction ratios of the given lines are 7, -5, 1 and 1, 2, 3, respectively.
We know that,
Two lines with direction ratios a_1, b_1, c_1 \text{ and }a_2, b_2, c_2 are perpendicular to each other if a_1a_2 + b_1b_2 + c_1c_2 = 0 \\[2.5 bp] Therefore, 7(1) + (-5) (2) + 1 (3)
= 7 – 10 + 3
= 0
Hence, the given lines are perpendicular to each other.

Q5. Find the intercepts cut off by the plane 2x + y – z = 5.

Explanation:
Given plane is 2x + y – z = 5 ……(i)
Dividing both sides of the equation (i) by 5,
\left(\dfrac{1}{2}\right)x + \left(\dfrac{y}{5}\right) – \left(\dfrac{z}{5}\right) = 1 \\[4.5 bp] \dfrac{x}{\dfrac{5}{2}} + \dfrac{y}{5} + \dfrac{z}{-5} = 1 .....\text{(ii)} \\ We know that,
The equation of a plane in intercept form is \left(\dfrac{x}{a}\right) + \left(\dfrac{y}{b}\right) + \left(\dfrac{z}{c}\right) = 1, where a, b, c are intercepts cut off by the plane at x, y, z-axes respectively.
For the given equation,
a = \dfrac{5}{2}, b = 5, c = -5
Hence, the intercepts cut off by the plane are \dfrac{5}{2}, 5 and -5.

CBSE Class 12 Maths Chapter wise Important Questions

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Conclusion

Three-dimensional (3D) geometry stands as a fundamental and immensely influential branch of mathematics, playing a pivotal role in our understanding and interaction with the world around us. Exploring CBSE class 12 maths Three dimensional geometry important questions, becomes essential for a comprehensive grasp of this fundamental aspect of mathematics.Platforms like Oswal.io provide a rich collection of questions and materials, extending beyond theoretical knowledge to offer practical insights and a deeper understanding of 3D geometry.

Frequently Asked Questions

Ans: 3D geometry is a branch of mathematics that deals with shapes and objects in three-dimensional space. It involves studying points, lines, planes, and figures that have length, width, and depth.
Ans: While 2D geometry deals with flat shapes like squares and circles, 3D geometry involves depth, making it a more complex representation of the real world. In 3D geometry, objects have volume and occupy space.
Ans: Common 3D shapes include cubes, spheres, cylinders, cones, pyramids, and prisms. Each of these has unique properties like volume and surface area.
Ans: In 3D geometry, points are represented by a set of three coordinates (x, y, z), which indicate their position in three-dimensional space along the width, height, and depth axes.
Ans: Vectors in 3D geometry have three components and are used to represent direction and magnitude in space. They are essential in defining movement, force, and positions relative to other objects.
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