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

The CBSE board institutions recommend that their students use the RD Sharma solution books to refer to the concepts when they encounter doubts at home. This helps the majority of the students who do not find time to visit tuitions after school hours and who cannot afford to go to tuitions. When topics like the plane in mathematics are a threat to most students, those who possess the RD Sharma Class 12th Exercise 28.13 solution book do not worry about any complexities.

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The Plane Exercise 28.13 Question 1

Answer:

Hint: use vector cross product to prove

Given:

Solution:

It is given that

Consider:

It can be written as

on further calculation

So we get

.....(i)

Similary :

....(ii)

Using both the equation we get

So the lines & are coplanar

Hence the equation of plane containing

Hence, the given lines are coplanar and the equation of the plane determined by these lines is

The Plane Exercise 28.13 Question 2

Answer: Proud L.H.S =R.H.S

Hint: use vector cross product

Given:

Solution :two lines

are coplanar if

Here,

The given lines are coplarer . equation of the plane containing the given line is

The Plane Exercise 28.13 Question 3

Answer:Hint: use simultaneous equation to solve

Given: and the point

Solution: let the equation of the plane passing through be

-----------(1)

The line passes through and its direction ratios are proportional to

Since plane (1) contains this line, it must pass through the point

......(2)

Since plane (1) contains this line, it must be parallel to the line

............(3)

Solving (1)(2) and (3) we get

The Plane Exercise 28.13 Question 4

Answer:

Hint: use simultaneous equation to solve

Given:

Solution: we know that equation of plane passing through is given by

....(i)

Since required plane contains lines , so we required plane passes through (4, 3, 2) and (3, -8, 0) so equation of required plane is

....(2)

Plane (2) also passes through (3, -3, 0) , so

....(3)

Now plane (2) is also parallel to the line with direction ratio (1, -4, 5) so

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

Multiplying by 3

Put a, b, c in equation (2)

Dividing by

So equation of required plane is

The Plane Exercise 28.13 Question 5

**Solution**: we have equation of the line is

Point on the line is given by -------(1)

Another equation of line is

Let a, b, c be the direction ratio of the lines it will be perpendicular to normal of and so using

............(2)

Agian

Solving (2) and (3) by cross-multiplication

Directional ratios are proportional to

Solving (i) and (ii) by elimination method

Put y in equation (i)

So, the equation of the line (2) is symmetrical form

Put the general point of a line from equation (i)

The equation of the plane is

There point of intersection is (2, 4, -3)

The Plane Exercise 28.13 Question 6

Given equation of plane;

and equation of line

So,

Since and so the line is the given plane hence proved

The Plane Exercise 28.13 Question 7

**Answer: **

**Hint**: use simultaneous equation;**Given**: **Solution:**

Let and

Equation of two lines .let the plane be -------(i)

Given that the required plane through the intersection of the lines L1 and L2 hence the normal to the plane is perpendicular to the line L1 and L2

Using cross multiplication, we get

The Plane Exercise 28.13 Question 8

Answer:Hint: first find the coordinates

Given: (3,4,2) and (7,0,6)

Solution: let the equation of the plane be --------(1)

Plane is passing through (3,4,2) and (7,0,6)

Required plane is perpendicular to

Solving the above equation

Substituting the values (1)

Vector equation of the plane is

The line passes through B(1,3,-2)

The point B lies on the plane the line; lies on the plane

The Plane Exercise 28.13 Question 9(i)

Answer:Hint: use vector cross product.

Given:

Solution: the direction ratio of the line

The direction ratio of the line

Since the line and are perpendicular so

The equation of the line are and

The equation of the plane containing the line perpendicular lines and

The Plane Exercise 28.13 Question 9(ii)

The line pass through the point (1,2,3) so putting is the equation we get

Therefore the equation of the plane containing the lines is

The Plane Exercise 28.13 Question 10(i)

**Hint: **use vector dot product

Since

**Solution **:any point on the line is of the form

if the point lies in the plane

Thus, the coordinates of the point of intersection of the line and the planes are

Let 0 be the angle between the line and the plane thus

Where l, m, and n are the direction ratio of the line and x, y and z are the direction ratios of the normal to the plane

The Plane Exercise 28.13 Question 11(i)

and

Thus

As we know that cross product of the line vectors gives a perpendicular vector so

So the equation of the required plane is

Also we have to find the coordinates of the point of intersection of the plane and the line

any point on the line

The Plane Exercise 28.13 Question 11(ii)

is of the form lies in the plane

Thus the required point of intersection is put value in the equation

The Plane Exercise 28.13 Question 13

-----(i)

--------------(ii)

Plane also contains the line

So it passes through the point (3, 2, 10)

------------------(2)

Also plane will be parallel to

Solving (2) and (3) by cross multiplication,

Put in equation (ii) we get

The Plane Exercise 28.13 Question 14(i)

Are coplanar if

So the given lines are coplanar , the equation of plane contains line is

Divided by -5

The Plane Exercise 28.13 Question 15

Answer:Hint: use simultaneous equation to solve

Given:

Solution: we know that the lines

and

Multiply equation (1) by 2 and equation (2) by 3 and then subtract we get

The Plane Exercise 28.13 Question 16(i)

Answer:Hint: use vector cross product

Given: and

Solution: we know that the lines

Coplanar if

Here:

The Plane Exercise 28.13 Question 16(ii)

Let then

Put value of t

is neglected because direction cosine cannot be imaginary

The Plane Exercise 28.13 Question 17(i)

Answer:Hint: use vector cross product

Given: and

Solution: we know that the lines

are coplar of

The Plane Exercise 28.13 Question 18(i)

Answer:Hint: use vector cross product

Given: and

Solution: we know that the lines and are coplanar if

here

The Plane Exercise 28.13 Question 18(ii)

The equation of plane contain lines is

When k=2

The equation of plane contain line

When k=-2

The Plane Exercise 28.13 Question 19(i)

**Answer:** units

**Hint:** use vector cross products

**Given: **

**Solution:** equation of given line is

and ...................(i)

where

Again:

Where

The vector equation of the plane containing the line (i) and (ii) is given by

The portions in the class 12 mathematics chapter 28 include fifteen exercises. The thirteenth exercise or the ex 28.13 consists of nineteen questions given in the maths textbook which deals with the concepts like to show the lines are coplanar, find the vector equation of the plane, and find the coordinates of the given points. All these concepts have various questions under level 1; there are no level 2 questions in this exercise. To clarify the doubts in this concept, the RD Sharma Class 12 Chapter 28 Exercise 28.13 will lend a helping hand.

The intensity of the questions gets more profound in exercise 13, where all the previously learned concepts are included. This makes the students struggle to solve the sums if they are unclear about the previous exercises. In such cases, they can refer to the RD Sharma Class 12th Exercise 28.13 solution book and the previous exercises to understand the concepts in-depth. All these books are based on the NCERT pattern, which explains why CBSE students must prefer these books.

Whenever students encounter a doubt, they can immediately refer to the Class 12 RD Sharma Chapter 28 Exercise 28.13 Solution material to clear those doubts. Many staff members and experts have diligently checked the accuracy of every solution given by them in the RD Sharma solution books. Any student who finds the concept of the plane challenging can use these reference materials to develop their knowledge in this topic.

Many previous batch students suggest the RD Sharma Class 12 Solutions the Plane Ex 28.13 for their juniors to practice the sums in the 28th chapter. The career 360 website provides free access to the RD Sharma Class 12th Exercise 28.13 solution material. The students can also download it in PDF format for later reference.

The RD Sharma Class 12 Solutions Chapter 28 Ex 28.13 material is used by staff to prepare questions for the tests and exams. Therefore, using this book makes the students exam-ready without any extra effort.

- 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 is the prescribed book to find the solved sums of class 12, mathematics chapter 28, ex 28.13?

The RD Sharma Class 12th Exercise 28.13 material is the prescribed book to refer to the solved sums of this exercise.

2. Which website provides the RD Sharma solution books for free?

The Career360 website gives access for everyone to view and download the RD Sharma solution books for free of cost.

3. How many questions from the 13th exercise of chapter 28 are solved in the RD Sharma solution book?

The ex 28.13 consists of nineteen questions, and the solutions for all the sum are given in the RD Sharma Class 12th Exercise 28.13 reference material.

4. What is the specialty of the RD Sharma solution books?

The solutions for every sum are given in all possible methods in the RD Sharma books. This lets the student adapt to the method they feel is easy.

5. What do the RD Sharma solution books provide the class 12 students?

The solutions for every question asked in the textbook.

Various practice questions.

An in-depth explanation of every concept is given.

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