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ApplyAs per latest 2024 syllabus. Physics formulas, equations, & laws of class 11 & 12th chapters
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
Answer: equation of the plane point of intersection isSolution: 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
Answer: L.H.S = R.H.SSince 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: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: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)
Answer: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)
Answer p(1,1,-2)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
Answer:The Plane Exercise 28.13 Question 14(i)
Answer:So the given lines are coplanar , the equation of plane contains line is
Divided by -5
The Plane Exercise 28.13 Question 15
Answer:Multiply equation (1) by 2 and equation (2) by 3 and then subtract we get
The Plane Exercise 28.13 Question 16(i)
Answer:The Plane Exercise 28.13 Question 16(ii)
The Plane Exercise 28.13 Question 17(i)
Answer:The Plane Exercise 28.13 Question 18(i)
Answer: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.
The RD Sharma Class 12th Exercise 28.13 material is the prescribed book to refer to the solved sums of this exercise.
The Career360 website gives access for everyone to view and download the RD Sharma solution books for free of cost.
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.
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.
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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As per latest 2024 syllabus. Maths formulas, equations, & theorems of class 11 & 12th chapters