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NCERT Solutions for miscellaneous exercise chapter 11 class 12 Three Dimensional Geometry are discussed here. These NCERT solutions are created by subject matter expert at Careers360 considering the latest syllabus and pattern of CBSE 2023-24. Three Dimensional Geometry is the 11th chapter of NCERT Class 12 Mathematics. The chapter covers a few concepts from lines and planes in three dimensions. Concepts like equations of a line, plane and distance between them etc. A the end of the chapter the miscellaneous exercise comes. Here the NCERT solutions for Class 12 Maths chapter 11 miscellaneous exercise is given. The NCERT book class 12 maths chapter 11 miscellaneous exercise solutions covers questions related to all the concepts of the chapter. These Class 12 Maths chapter 11 miscellaneous solutions are designed by mathematics expert faculties.
Miscellaneous exercise class 12 chapter 11 are designed as per the students demand covering comprehensive, step by step solutions of every problem. Practice these questions and answers to command the concepts, boost confidence and in depth understanding of concepts. Students can find all exercise enumerated in NCERT Book together using the link provided below.
Answer:
We can assume the line joining the origin, be OA where and the point and PQ be the line joining the points and .
Then the direction ratios of the line OA will be and that of line PQ will be
So to check whether line OA is perpendicular to line PQ then,
Applying the relation we know,
Therefore OA is perpendicular to line PQ.
Answer:
Given that are the direction cosines of two mutually perpendicular lines.
Therefore, we have the relation:
.........................(1)
.............(2)
Now, let us assume be the new direction cosines of the lines which are perpendicular to the line with direction cosines.
Therefore we have,
Or,
......(3)
So, l,m,n are the direction cosines of the line.
where, ........................(4)
Then we know that,
So, from the equation (1) and (2) we have,
Therefore, ..(5)
Now, we will substitute the values from the equation (4) and (5) in equation (3), to get
Therefore we have the direction cosines of the required line as;
Question:3 Find the angle between the lines whose direction ratios are a, b, c and b – c, c – a, a – b.
Answer:
Given direction ratios and .
Thus the angle between the lines A is given by;
a
Thus, the angle between the lines is
Question:4 Find the equation of a line parallel to x-axis and passing through the origin.
Answer:
Equation of a line parallel to the x-axis and passing through the origin is itself x-axis .
So, let A be a point on the x-axis.
Therefore, the coordinates of A are given by , where .
Now, the direction ratios of OA are
So, the equation of OA is given by,
or
Thus, the equation of the line parallel to the x-axis and passing through origin is
Answer:
Direction ratios of AB are
and Direction ratios of CD are
So, it can be noticed that,
Therefore, AB is parallel to CD.
Thus, we can easily say the angle between AB and CD which is either .
Question:6 If the lines and are perpendicular, find the value of k.
Answer:
Given both lines are perpendicular so we have the relation;
For the two lines whose direction ratios are known,
We have the direction ratios of the lines, and are and respectively.
Therefore applying the formula,
or
For, the lines are perpendicular.
Question:7 Find the vector equation of the line passing through (1, 2, 3) and perpendicular to the plane
Answer:
Given that the plane is passing through the point so, the position vector of the point A is and perpendicular to the plane whose direction ratios are and the normal vector is
So, the equation of a line passing through a point and perpendicular to the given plane is given by,
, where
Question:8 Find the equation of the plane passing through (a, b, c) and parallel to the plane .
Answer:
Given that the plane is passing through and is parallel to the plane
So, we have
The position vector of the point is,
and any plane which is parallel to the plane, is of the form,
. .......................(1)
Therefore the equation we get,
Or,
So, now substituting the value of in equation (1), we get
.................(2)
So, this is the required equation of the plane .
Now, substituting in equation (2), we get
Or,
Question:9 Find the shortest distance between lines and .
Answer:
Given lines are;
and
So, we can find the shortest distance between two lines and by the formula,
...........................(1)
Now, we have from the comparisons of the given equations of lines.
So,
and
Now, substituting all values in equation (3) we get,
Hence the shortest distance between the two given lines is 9 units.
Question:10 Find the coordinates of the point where the line through (5, 1, 6) and (3, 4,1) crosses the YZ-plane.
Answer:
We know that the equation of the line that passes through the points and is given by the relation;
and the line passing through the points,
And any point on the line is of the form .
So, the equation of the YZ plane is
Since the line passes through YZ- plane,
we have then,
or and
So, therefore the required point is
Question : 11 Find the coordinates of the point where the line through (5, 1, 6) and (3, 4, 1) crosses the ZX-plane.
Answer:
We know that the equation of the line that passes through the points and is given by the relation;
and the line passing through the points,
And any point on the line is of the form .
So, the equation of ZX plane is
Since the line passes through YZ- plane,
we have then,
or and
So, therefore the required point is
Answer:
We know that the equation of the line that passes through the points and is given by the relation;
and the line passing through the points, .
And any point on the line is of the form.
This point lies on the plane,
or .
Hence, the coordinates of the required point are or .
Answer:
Given
two planes x + 2y + 3z = 5 and 3x + 3y + z = 0.
the normal vectors of these plane are
Since the normal vector of the required plane is perpendicular to the normal vector of given planes, the required plane's normal vector will be :
Now, as we know
the equation of a plane in vector form is :
Now Since this plane passes through the point (-1,3,2)
Hence the equation of the plane is
Question:14 If the points (1, 1, p) and (– 3, 0, 1) be equidistant from the plane then find the value of p.
Answer:
Given that the points and are equidistant from the plane
So we can write the position vector through the point is
Similarly, the position vector through the point is
The equation of the given plane is
and We know that the perpendicular distance between a point whose position vector is and the plane, and
Therefore, the distance between the point and the given plane is
nbsp; .........................(1)
Similarly, the distance between the point , and the given plane is
.........................(2)
And it is given that the distance between the required plane and the points, and is equal.
therefore we have,
or or
Answer:
So, the given planes are:
and
The equation of any plane passing through the line of intersection of these planes is
..............(1)
Its direction ratios are and = 0
The required plane is parallel to the x-axis.
Therefore, its normal is perpendicular to the x-axis.
The direction ratios of the x-axis are 1,0, and 0.
Substituting in equation (1), we obtain
So, the Cartesian equation is
Answer:
We have the coordinates of the points and respectively.
Therefore, the direction ratios of OP are
And we know that the equation of the plane passing through the point is
where a,b,c are the direction ratios of normal.
Here, the direction ratios of normal are and and the point P is .
Thus, the equation of the required plane is
Answer:
The equation of the plane passing through the line of intersection of the given plane in
,,,,,,,,,,,,,(1)
The plane in equation (1) is perpendicular to the plane, Therefore
Substituting in equation (1), we obtain
.......................(4)
So, this is the vector equation of the required plane.
The Cartesian equation of this plane can be obtained by substituting in equation (1).
Therefore we get the answer
Question:18 Find the distance of the point (– 1, – 5, – 10) from the point of intersection of the line and the plane .
Answer:
Given,
Equation of a line :
Equation of the plane
Let's first find out the point of intersection of line and plane.
putting the value of into the equation of a plane from the equation from line
Now, from the equation, any point p in line is
So the point of intersection is
SO, Now,
The distance between the points (-1,-5,-10) and (2,-1,2) is
Hence the required distance is 13.
Question:19 Find the vector equation of the line passing through (1, 2, 3) and parallel to the planes and .
Answer:
Given
A point through which line passes
two plane
And
it can be seen that normals of the planes are
since the line is parallel to both planes, its parallel vector will be perpendicular to normals of both planes.
So, a vector perpendicular to both these normal vector is
Now a line which passes through and parallels to is
So the required line is
Questio n: 20 Find the vector equation of the line passing through the point (1, 2, – 4) and perpendicular to the two lines:
and
Answer:
Given
Two straight lines in 3D whose direction cosines (3,-16,7) and (3,8,-5)
Now the two vectors which are parallel to the two lines are
and
As we know, a vector perpendicular to both vectors and is , so
A vector parallel to this vector is
Now as we know the vector equation of the line which passes through point p and parallel to vector d is
Here in our question, give point p = (1,2,-4) which means position vector of this point is
So, the required line is
Question:21 Prove that if a plane has the intercepts a, b, c and is at a distance of p units from the origin, then .
Answer:
The equation of plane having a, b and c intercepts with x, y and z-axis respectively is given by
The distance p of the plane from the origin is given by
Hence proved
Question:22 Distance between the two planes: 2x + 3y + 4z = 4 and 4x + 6y + 8z = 12 is
(A) 2 units (B) 4 units (C) 8 units (D)
Answer:
Given equations are
and
Now, it is clear from equation (i) and (ii) that given planes are parallel
We know that the distance between two parallel planes is given by
Put the values in this equation
we will get,
Therefore, the correct answer is (D)
Question:23 The planes: 2x – y + 4z = 5 and 5x – 2.5y + 10z = 6 are
(A) Perpendicular (B) Parallel (C) intersect y-axis (D) passes through
Answer:
Given equations of planes are
and
Now, from equation (i) and (ii) it is clear that given planes are parallel to each other
Therefore, the correct answer is (B)
Twenty-three questions are there in the Class 12 Maths chapter 11 miscellaneous solutions. Miscellaneous exercise chapter 11 Class 12 are important to understand the concepts explained in the chapter. The NCERT syllabus for Class 12 Maths chapter 11 miscellaneous exercise covers all the concepts of the chapter.
Also Read| Three Dimensional Geometry Class 12th Notes
To practise the complete chapter the miscellaneous exercise chapter 11 Class 12 is useful
To practice for the board exam the NCERT solutions for Class 12 Maths chapter 11 miscellaneous exercise is helpful.
Total four exercises are covered in three-dimensional geometry.
One dimension is represented by a single axis (say x-axis), in two dimensions we have two axes (say x and y-axis) and three-dimension in space is is represented by three axes (say x,y and z-axis)
23 questions are solved in the miscellaneous exercise of class 12 chapter
Three exercises are solved before miscellaneous exercise of Cass 12 Maths chapter three dimensional geometry
NCERT syllabus is required for the preparation of CBSE board exams and some of the state board exams like Kerala state board. Also, NCERT is considered as the bible for the preparation of NEET exams (physics, chemistry and biology). Also, the NCERT syllabus is helpful for JEE main exam (Physics, Chemistry and Mathematics)
To practice the concepts covered in the chapter.
Miscellaneous exercises give a good variety of problems covering the entire chapter.
Yes, sometimes the type of questions discussed in miscellaneous exercises are asked for both board exams and competitive exams as well.
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Hello there! Thanks for reaching out to us at Careers360.
Ah, you're looking for CBSE quarterly question papers for mathematics, right? Those can be super helpful for exam prep.
Unfortunately, CBSE doesn't officially release quarterly papers - they mainly put out sample papers and previous years' board exam papers. But don't worry, there are still some good options to help you practice!
Have you checked out the CBSE sample papers on their official website? Those are usually pretty close to the actual exam format. You could also look into previous years' board exam papers - they're great for getting a feel for the types of questions that might come up.
If you're after more practice material, some textbook publishers release their own mock papers which can be useful too.
Let me know if you need any other tips for your math prep. Good luck with your studies!
It's understandable to feel disheartened after facing a compartment exam, especially when you've invested significant effort. However, it's important to remember that setbacks are a part of life, and they can be opportunities for growth.
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I hope this information helps you.
Hi,
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hello mahima,
If you have uploaded screenshot of your 12th board result taken from CBSE official website,there won,t be a problem with that.If the screenshot that you have uploaded is clear and legible. It should display your name, roll number, marks obtained, and any other relevant details in a readable forma.ALSO, the screenshot clearly show it is from the official CBSE results portal.
hope this helps.
Hello Akash,
If you are looking for important questions of class 12th then I would like to suggest you to go with previous year questions of that particular board. You can go with last 5-10 years of PYQs so and after going through all the questions you will have a clear idea about the type and level of questions that are being asked and it will help you to boost your class 12th board preparation.
You can get the Previous Year Questions (PYQs) on the official website of the respective board.
I hope this answer helps you. If you have more queries then feel free to share your questions with us we will be happy to assist you.
Thank you and wishing you all the best for your bright future.
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