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Edited By Lovekush kumar saini | Updated on Jan 25, 2022 11:53 AM IST

The RD Sharma solution books are every class 12 students’ best companion. It helps them in clarifying their doubts, helping with the homework, and sharpening their knowledge for the exams. And the syllabus of the subject mathematics is a significant threat to the class 12 students. Moreover, a concept like The Plane tends to confuse the students a lot. RD Sharma solutions In such circumstances, the RD Sharma Class 12th Exercise 28.11 reference material plays a major role.

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The Plane Exercise 28.11 Question 1

As we know that the angle between the line and plane

Here,

and

The angle between them is

The Plane Exercise 28.11 Question 2

We know that, the angle between the line and plane is

The Plane Exercise 28.11 Question 3

So,

The given line is parallel to the vector and given line is normal to the vector

We know that, the angle between the line and plane is given by

The Plane exercise 28.11 question 4

Plane

If the line is parallel to the plane, the normal to the plane is perpendicular to the line

The Plane exercise 28.11 question 5

The given plane is

So, the normal vector

Now,

So, is perpendicular to

So, the given line is parallel to the given plane.

The distance between the line and the parallel plane.

Then, d = length of perpendicular from the point to the plane

The Plane exercise 28.11 question 6

Therefore, it is parallel to the normal

Thus, the required line passes through the position vector and parallel to the vector

So, its vector equation is

The Plane Excercise 28.11 Question 7

……………….. (1)

So, equation of plane passing through (2,3,-4) is

…………….…… (2)

It also passes through (1,-1,3)

……………….... (3)

We know that line

is parallel to plane

if ………………….. (4)

Here equation (2) is parallel to x- axis

…………………… (5)

Using (2) and (5) in equation (4) we get

Putting the value of ‘a’ in equation (3) we get

Now, putting the value of a and b in (2) we get

Dividing by c we have

The Plane Excercise 28.11 Question 8

……………….. (1)

So, equation of plane passing through (0,0,0) is

…………………. (2)

It also passes through

So, equation (2) must satisfy the point

………………… (3)

We know that line

is parallel to plane if

…………........ (4)

Here, the plane is parallel to line

So,

…………….… (5)

Solving equation (3) and (5) by cross multiplication we have,

Putting the value in equation (2) we get

Dividing by k we have

The required equation is

The Plane Excercise 28.11 Question 9

or

……………….. (1)

We know that line

is parallel to plane

If ………………… (2)

Here, line (1) is parallel to plane

So,

…………………. (3)

Also, line (1) is parallel to plane,

So,

………………….. (4)

Solving equation (3) and (4) by cross multiplication

We have,

Putting the value in equation (1) we have

Multiplying by k we have

The required equation is

or

The Plane Exercise 28.11 Question 10

As we know, that if two planes are perpendicular with direction ratios as and then

Since line lies in both the planes, so it is perpendicular to both planes

…………….. (1)

…………….. (2)

Solving equation (1) and (2) by cross multiplication

We have,

We know that line

is parallel to plane

if ………….. (3)

Here, line with direction ratios is parallel to plane

Therefore, the line of section is parallel to the plane.

The Plane Exercise 28.11 Question 11

Given that the line passes through …………….. (2)

Since, line (1) is perpendicular to the plane

So, normal to plane is parallel to the line.

In vector form,

as, is any scalar.

Thus, the equation of required line,

The Plane Exercise 28.11 Question 12

……………….. (1)

So, equation of plane passing through is

………………… (2)

It also passes through

So, equation (2) passes through

…………………. (3)

We know that line

is parallel to plane

if …………… (4)

Here, the plane (2) is parallel to line having direction ratios 7, 0 ,6

So,

………………… (5)

Putting the value of a in equation (3) we get

Putting the value of a and b in equation (2) we get

Multiplying by we have

Equation of required plane is

The Plane exercise 28.11 question 13

is parallel to plane

If

and the angle between them is given by

…………….. (1)

Now, given equation by line is

So,

Equation of plane is

So,

The angle between the plane and the line is

The Plane exercise 28.11 question 14

So, equation of plane passing through the intersection of planes is

………………. (1)

We know that line

Is parallel to plane

Given the plane is parallel to line with direction ratios 1, 2, 1

Putting the value of k in equation (1) we get

We know that the distance (D) of point from plane is given by

So, distance of Point from plane is

Taking the mod value we have

The Plane exercise 28.11 question 15**Answer:** Therefore, the line and plane are parallel. And line

if ……………… (1)

Given, equation of line and equation of plane is

So,

Now,

So, the line and plane are parallel.

We know that, the distance ‘D’ of plane is from a point is given by

We take the mod value

The Plane Exercise 28.11 Question 16

Given, equation of line and equation of plane is

So,

Now,

So, the line and plane are parallel

we know that the Distance ‘D’ of a plane from a point is given by

We take the mod value,

So,

The Plane Exercise 28.11 Question 17

So, equation of plane passing through the intersection of planes

………………. (1)

We know that line

is parallel to plane

Given the plane is parallel to line

Or, (dividing the equation by 6)

Putting the value of k in equation (1) we get

The required equation is

The Plane Exercise 28.11 Question 18

A vector in a direction perpendicular to and

Equation of the plane is

Substitution , we get the Cartesian from as

The distance of the point from the plane

The Plane Excercise 28.11 Question 19

……………… (1)

So, equation of plane passing through is

……………… (2)

It also passes through

So, equation (2) must satisfy the point

……………… (3)

We know that line is parallel to plane if

……………… (4)

So,

………………. (5)

Solving equation (3) and (5) by cross a multiplication, we have

Putting the value in equation (2) we get

Dividing by k we have

Equation of required plane is

The Plane Excercise 28.11 Question 20

Substituting in the equation of the plane , We get,

Hence, the coordinates of intersection is

Direction ratios of the line are

Direction ratios of the line perpendicular to the plane are

The angle between the plane and line is

The Plane Excercise 28.11 Question 21

Given that, the line is passing through

So,

It is given that line is perpendicular to plane

So, normal to plane is parallel to

So, Let

Putting in (1), equation of line is

The Plane Exercise 28.11 Question 22

Direction ratio of a line perpendicular to the plane

As we know that the angle between the line and plane

is given by

The Plane Exercise 28.11 Question 23

………….. (1)

We know that line

is parallel to plane if

…………… (2)

Here line (1) is parallel to plane

So,

…………... (3)

Also, line (1) parallel to plane

So,

…………… (4)

Solving equation (3) and (4) by cross multiplication we have,

Putting the value in equation (1) we get

Multiplying by k we have

The required equation is

The Plane Exercise 28.11 Question 24

is perpendicular to plane

We know that line

is perpendicular to plane if

So, normal vector of plane is parallel to line.

So, direction ratios of normal to plane are proportional to the direction of line.

Here,

By cross multiplying the last two we have

The Plane Excercise 28.11 Question 25

…………. (1)

So, equation of plane passing through is

…………. (2)

It also passes through

So, equation (2) must satisfy the point

……….. (3)

We know that line

is parallel to plane

If ………… (4)

Here the plane is parallel to line

So,

………… (5)

Solving equation (3) and (5) by cross multiplication we have,

Putting the value in equation (2) we get

Dividing by k we have,

The required equation is

The Plane Excercise 28.11 Question 26

Line of intersection

Scalar triple product,

Position Vector of – Position Vector of

is required equation of plane

Now, the angle between the plane and the y-axis is

The chapter 28 portion for class 12 mathematics contains fifteen exercises. Whereas, in the eleventh exercise, 28.11, most of the concepts revolve around finding the angle between the line and plane, angle between the point and the plane, finding the vector and cartesian equations of the line passing through the points and parallel to the plane. All the solutions for the 26 questions present in this exercise can be found in the RD Sharma Class 12 Chapter 28 Exercise 28.11 solution book. Moreover, the questions are given only under the Level 1 section, reducing the complexity.

The RD Sharma Class 12th Exercise 28.11 consists of multiple questions for practice apart from the ones given in the textbook. With a bit of practice in these concepts, the students will be able to score higher than their previous performance. This eventually elevates their confidence to face the public exam. As the RD Sharma books follow the NCERT pattern, it becomes easier for the CBSE school students to adapt to its methods.

The benefits of the RD Sharma books are ineffable as most of the students have benefitted and grown to great heights based on their 12th public exams. Any confusion regarding The Plane concept will be cleared once the students refer to the Class 12 RD Sharma Chapter 28 Exercise 28.11 Solution book. The answers are given by the teachers and experts who have a piece of in-depth knowledge in this particular domain. Therefore, the students tend to learn the concepts easier without any extra effort.

The RD Sharma Class 12 Solutions the Plane Ex 28.11 reference book consists of various methods to solve a sum. The students can try and figure out the ones that they find easy to adapt. In such a way, everyone would have their own easy way to arrive at a solution, even for a challenging sum. In this way, the RD Sharma Class 12th Exercise 28.11 book helps them a lot.

And the most exciting benefit is that the RD Sharma Class 12 Solutions Chapter 28 Ex 28.11 can be found free of cost at the Career 360 website. So, visit the site and download the set of RD Sharma books that you require.

- Chapter 1 - Relations
- Chapter 2 - Functions
- Chapter 3 - Inverse Trigonometric Functions
- Chapter 4 - Algebra of Matrices
- Chapter 5 - Determinants
- Chapter 6 - Adjoint and Inverse of a Matrix
- Chapter 7 - Solution of Simultaneous Linear Equations
- Chapter 8 - Continuity
- Chapter 9 - Differentiability
- Chapter 10 - Differentiation
- Chapter 11 - Higher Order Derivatives
- Chapter 12 - Derivative as a Rate Measurer
- Chapter 13 - Differentials, Errors and Approximations
- Chapter 14 - Mean Value Theorems
- Chapter 15 - Tangents and Normals
- Chapter 16 - Increasing and Decreasing Functions
- Chapter 17 - Maxima and Minima
- Chapter 18 - Indefinite Integrals
- Chapter 19 - Definite Integrals
- Chapter 20 - Areas of Bounded Regions
- Chapter 21 - Differential Equations
- Chapter 22 - Algebra of Vectors
- Chapter 23 - Scalar Or Dot Product
- Chapter 24 - Vector or Cross Product
- Chapter 25 - Scalar Triple Product
- Chapter 26 - Direction Cosines and Direction Ratios
- Chapter 27 - Straight Line in Space
- Chapter 28 - The Plane
- Chapter 29 - Linear programming
- Chapter 30- Probability
- Chapter 31 - Mean and Variance of a Random Variable

1. Which solution book is the must for the students to be clear of the concepts in class 12, mathematics, chapter 28?

The RD Sharma Class 12th Exercise 28.11 reference book provides the optimal solution for Chapter 28 in mathematics.

2. How many questions are present in chapter 28, ex 28.11 in mathematics?

There are around nineteen questions given in the textbook for exercise 28.11. The RD Sharma Class 12th Exercise 28.11 given the accurate solutions for these questions.

3. Is the concept of the plane in mathematics easy to solve?

Any mathematical concept can be solved easily with a good amount of practice referring to the proper solutions book, and the concept of the plane is no exception to it.

4. Where can the RD Sharma books be found online?

The Career 360 website contains the best set of RD Sharma Solutions books for every grade and subject.

5. Is it easy for the CBSE students to use the RD Sharma solution books?

The RD Sharma solution books follow the NCERT pattern, which makes it convenient for the CBSE students to follow it.

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