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RS Aggarwal Class 12 Chapter 28 Solutions (The Plane)

RS Aggarwal Solutions for Class 12 Chapter 28 ‘The Plane ‘will help you to solve all the problems given in this chapter. You will learn concepts related to planes such as coplanarity of two lines, the angle between two planes, the distance of a point from the plane etc. The chapter contains 181 questions and 11 exercises along with more than 100 solved examples for your practice. 

RS Aggarwal Solutions for Chapter 28 ‘The Plane’ is intended to provide precise and accurate answers from the CBSE examination point of view. These solutions are available in simple language to each question of all the exercises. The answers by Instasolv are reliable and hence, will help you score substantially better in your exams. The exercise solutions constitute thorough discussion on all the concepts covered in the syllabus.

We, at Instasolv, are committed to providing the best reference resource for you to constitute a one-stop solution for all the doubts that might arise while solving the chapter. The subject matter experts at Instasolv continuously upgrade themselves to pace up with the changing trends of the exam pattern. With RS Aggarwal Class 12 Solutions, you will find thorough coverage of the topics in your syllabus for the boards and entrance examinations like JEE and NEET.  

Topics Covered in RS Aggarwal Solutions for Class 12 Chapter 28 The Plane

Introduction to the Plane:

A plane can be defined distinctively if any one of the following conditions seems to be fulfilled:

  1. If the distance of the plane and the normal to the plane with respect to the Origin is given. This is also called the equation of a plane in normal form.
  2. If the plane passes through a point and is perpendicular to a given vector.
  3. If it passes through 3 given non-collinear points.

Equation of a Plane in Normal Form:

  • Vector Form of the Equation:

When n is a unit normal vector along the given normal from the origin to the plane. Consider the perpendicular distance of this given plane as d. Consider an arbitrary point P on the plane such that its position vector r is perpendicular to the normal of the plane from the origin as discussed above.

Then, the vector form of the equation of the plane is given by 

r .n=d

  • Cartesian Form of the Equation:

If l, m, and n are the direction cosine of n, then the cartesian equation of the plane when the distance of the plane and the normal to the plane with respect to the Origin is given as,

lx + my + nz =d

Equation of a Plane Plane passes through a Point and is Perpendicular to a given Vector:

  • Vector Form of the Equation of the Plane

If the point through which the plane passes is A with a position vector a and if the plane is normal to the vector N. Consider an arbitrary point P on the plane such that its position vector r, then the equation of the given plane can be represented as:

  • Cartesian Form from the Vector Equation

If the coordinates of the point P and A are given as (x, y, z) and (x1, y1, z1) and the direction cosines of Nare A, B and C, then the Cartesian equation of such a plane is given as:

Equation of Plane Passing Thorugh 3 Non-Collinear Points

If R, S, and T are three non-collinear points with position vectors a , b , and c respectively, then the vector equation is given as,

Important Highlights of the Chapter:

  • Intercept Form of the Equation of the Plane:

If the plane intercepts the axes x, y and z at points a, b and c respectively, then intercept form of the equation of the plane is given as:

  • Plane Passing through the intersection of two planes:

Exercise Wise Discussion of RS Aggarwal Solutions for Class 12 Chapter 28 The Plane

  1. The first exercise of the chapter focuses on finding the equation of a plane based on the coordinates of points given in the questions. There are 9 questions in this exercise. 
  2. Exercise 2 has 30 questions on finding the vector and cartesian equations of a plane along with the unit vector. 
  3. Exercise 3 has 13 questions in which you have to find the distance of a point from the given plane. 
  4. In the next exercise, you have to find the answers consist of a diverse set of problems related to the intercept form of the equation of the plane.
  5. Exercise 5 discusses the concept of the plane passing through the intersection of two planes and the application of its equations.
  6. Exercise 6 is based on finding the angle between two planes. There are 18 questions in this exercise. 
  7. Exercise 7 has 15 questions where you need to find the equation of a plane that is perpendicular to a point and is parallel to a line. 
  8. In exercise 8, there are 5 questions on vector and cartesian equations of a plane passing through a point and parallel to a line. 
  9. In exercise 9, there are just 9 questions where you need to prove that given lines are coplanar. 
  10. Exercise 10 has 26 questions of very small answer type based on the topics of the chapter for your quick revision. 
  11. Then there are 30 objective-type questions as a separate exercise for you. 

Benefits of RS Aggarwal Solutions for Class 12 Maths Chapter 28 by Instasolv

  1. At Instasolv, you will be able to solve all the issues related to the syllabus and will get an opportunity of robust practice before you sit for your CBSE exam.
  2. This is a highly recommended reference source for you if you are appearing in the CBSE board exam or are preparing for competitive entrance exams like NEET, JEE Main or JEE Advanced. 
  3. At Instasolv, despite the provision of detailed solutions, you will get ample chance to brush your analytical abilities.