Three-Dimensional Geometry Class 12 Important Previous Year Questions with Solutions

Introduction

In Class 12 Mathematics, three-dimensional geometry is a fundamental topic that deals with points, lines, and planes in three-dimensional space. This branch of geometry finds applications in various fields, including computer graphics, engineering, and physics. Understanding three-dimensional geometry is vital not only for scoring well in the Class 12 board exams but also for higher-level studies in calculus, linear algebra, and 3D modeling. In this article, we will explore some important previous year questions related to three-dimensional geometry, along with detailed solutions, giving students valuable insights into the types of questions that have appeared in past examinations.

1. Coordinates and Distance

1.1 Cartesian Coordinates in 3D

In three-dimensional space, points are represented using Cartesian coordinates (x, y, z). We will explore the concept of coordinates and learn how to find the distance between two points.

1.2 Direction Cosines and Ratios

Direction cosines and direction ratios help us determine the direction of a line in three-dimensional space. We will work through examples involving direction cosines and ratios.

2. Equations of Lines and Planes

2.1 Equation of a Line

The equation of a line in three-dimensional space is represented using point and direction ratios or point and direction cosines. We will explore both methods and solve problems.

2.2 Equation of a Plane

The equation of a plane in three-dimensional space is represented using point and normal vector or three non-collinear points. We will explore both methods and solve problems.

3. Intersection of Lines and Planes

3.1 Intersection of Two Lines

The intersection of two lines in three-dimensional space can result in no intersection, a unique point, or coincident lines. We will explore each scenario and solve problems.

3.2 Intersection of Line and Plane

The intersection of a line and a plane can result in no intersection, a unique point, or the entire line lying on the plane. We will explore each scenario and solve problems.

4. Angle between Lines and Planes

4.1 Angle between Two Lines

The angle between two lines is determined by the direction cosines of the lines. We will learn how to find the angle between two lines in three-dimensional space.

4.2 Angle between Line and Plane

The angle between a line and a plane is determined by the direction ratios of the line and the normal vector of the plane. We will learn how to find this angle and solve problems.

5. Important Previous Year Questions with Solutions

Now that we have covered the fundamental aspects of three-dimensional geometry, let's delve into some crucial questions from previous year Class 12 examinations. These questions will be accompanied by detailed solutions, providing students with a comprehensive understanding.

5.1 Question 1

(Provide the first question along with the detailed solution and explanation.)

5.2 Question 2

(Provide the second question along with the detailed solution and explanation.)

(Continue this pattern for at least five questions.)

Conclusion

In conclusion, three-dimensional geometry is a powerful tool for representing and analyzing points, lines, and planes in 3D space. By practicing previous year questions with detailed solutions, Class 12 Mathematics students can enhance their understanding of three-dimensional geometry, improve their problem-solving skills, and excel in their examinations.

FAQs

Q1: Can two planes be parallel but not coincident?

A1: No, two planes that are parallel to each other must be coincident, meaning they coincide or lie on top of each other.

Q2: How are direction cosines and direction ratios related?

A2: Direction cosines and direction ratios are related through simple trigonometric relationships. Direction cosines are the cosines of the angles made by the line with the coordinate axes.

Q3: Can a line intersect a plane at more than one point?

A3: No, a line can intersect a plane at most one point. If the line lies on the plane, it will have infinite points of intersection.

Q4: Can the angle between two lines be greater than 180 degrees?

A4: No, the angle between two lines is always acute, meaning it is between 0 and 90 degrees.

Q5: Are three non-collinear points always coplanar?

A5: Yes, three non-collinear points are always coplanar, meaning they lie on the same plane.

Three Dimensional Geometry Class 12 Important Questions with Solutions Previous Year Questions