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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.
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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.
Hello aspirant,
The purpose of graphic design extends beyond the brand's look. Nevertheless, by conveying what the brand stands for, it significantly aids in the development of a sense of understanding between a company and its audience. The future in the field of graphic designing is very promising.
There are various courses available for graphic designing. To know more information about these courses and much more details, you can visit our website by clicking on the link given below.
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Thank you
Hope this information helps you.
hello,
Yes you can appear for the compartment paper again since CBSE gives three chances to a candidate to clear his/her exams so you still have two more attempts. However, you can appear for your improvement paper for all subjects but you cannot appear for the ones in which you have failed.
I hope this was helpful!
Good Luck
Hello dear,
If you was not able to clear 1st compartment and now you giving second compartment so YES, you can go for your improvement exam next year but if a student receives an improvement, they are given the opportunity to retake the boards as a private candidate the following year, but there are some requirements. First, the student must pass all of their subjects; if they received a compartment in any subject, they must then pass the compartment exam before being eligible for the improvement.
As you can registered yourself as private candidate for giving your improvement exam of 12 standard CBSE(Central Board of Secondary Education).For that you have to wait for a whole year which is bit difficult for you.
Positive side of waiting for whole year is you have a whole year to preparing yourself for your examination. You have no distraction or something which may causes your failure in the exams. In whole year you have to stay focused on your 12 standard examination for doing well in it. By this you get a highest marks as a comparison of others.
Believe in Yourself! You can make anything happen
All the very best.
Hello Student,
I appreciate your Interest in education. See the improvement is not restricted to one subject or multiple subjects and we cannot say if improvement in one subject in one year leads to improvement in more subjects in coming year.
You just need to have a revision of all subjects what you have completed in the school. have a revision and practice of subjects and concepts helps you better.
All the best.
Hi,
You just need to give the exams for the concerned two subjects in which you have got RT. There is no need to give exam for all of your subjects, you can just fill the form for the two subjects only.
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A multimedia specialist is a media professional who creates, audio, videos, graphic image files, computer animations for multimedia applications. He or she is responsible for planning, producing, and maintaining websites and applications.
Welding Engineer Job Description: A Welding Engineer work involves managing welding projects and supervising welding teams. He or she is responsible for reviewing welding procedures, processes and documentation. A career as Welding Engineer involves conducting failure analyses and causes on welding issues.
A quality controller plays a crucial role in an organisation. He or she is responsible for performing quality checks on manufactured products. He or she identifies the defects in a product and rejects the product.
A quality controller records detailed information about products with defects and sends it to the supervisor or plant manager to take necessary actions to improve the production process.
A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.
A QA Lead is in charge of the QA Team. The role of QA Lead comes with the responsibility of assessing services and products in order to determine that he or she meets the quality standards. He or she develops, implements and manages test plans.
A Structural Engineer designs buildings, bridges, and other related structures. He or she analyzes the structures and makes sure the structures are strong enough to be used by the people. A career as a Structural Engineer requires working in the construction process. It comes under the civil engineering discipline. A Structure Engineer creates structural models with the help of computer-aided design software.
The Process Development Engineers design, implement, manufacture, mine, and other production systems using technical knowledge and expertise in the industry. They use computer modeling software to test technologies and machinery. An individual who is opting career as Process Development Engineer is responsible for developing cost-effective and efficient processes. They also monitor the production process and ensure it functions smoothly and efficiently.
An AWS Solution Architect is someone who specializes in developing and implementing cloud computing systems. He or she has a good understanding of the various aspects of cloud computing and can confidently deploy and manage their systems. He or she troubleshoots the issues and evaluates the risk from the third party.
An Azure Administrator is a professional responsible for implementing, monitoring, and maintaining Azure Solutions. He or she manages cloud infrastructure service instances and various cloud servers as well as sets up public and private cloud systems.
Careers in computer programming primarily refer to the systematic act of writing code and moreover include wider computer science areas. The word 'programmer' or 'coder' has entered into practice with the growing number of newly self-taught tech enthusiasts. Computer programming careers involve the use of designs created by software developers and engineers and transforming them into commands that can be implemented by computers. These commands result in regular usage of social media sites, word-processing applications and browsers.
A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.
Individuals in the information security manager career path involves in overseeing and controlling all aspects of computer security. The IT security manager job description includes planning and carrying out security measures to protect the business data and information from corruption, theft, unauthorised access, and deliberate attack
An Automation Test Engineer job involves executing automated test scripts. He or she identifies the project’s problems and troubleshoots them. The role involves documenting the defect using management tools. He or she works with the application team in order to resolve any issues arising during the testing process.
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