# 11.E: Exercises for Chapter 11 - Mathematics

## Calculational Exercises

1. Consider ( mathbb{R}^3 )with two orthonormal bases: the canonical basis ( e = (e_1 , e_2 , e_3) ) and the basis (f = (f_1 , f_2 , f_3) ), where
[ f1 = frac{1}{sqrt{3}}(1,1,1), f_2 = frac{1}{sqrt{6}}(1,-2,1), f_3 = frac{1}{sqrt{2}}(1,0,-1). ]

Find the canonical matrix, A, of the linear map (T in cal{L}(mathbb{R}^3) ) with eigenvectors (f_1 , f_2 , f_3 ) and eigenvalues 1, 1/2, −1/2, respectively.

2. For each of the following matrices, verify that (A) is Hermitian by showing that (A = A^*) ,ﬁnd a unitary matrix (U) such that (U^{−1}AU) is a diagonal matrix, and compute (exp(A)).
[ (a)~ A = left[ egin{array}{cc} 4 & 1-i 1+i & 5 end{array} ight] ~~ (b)~A = left[ egin{array}{cc} 3 & -i i & 3 end{array} ight] ~~ (c)~A = left[ egin{array}{cc} 6 & 2+2i 2-2i & 4 end{array} ight]]
[ (d)~A = left[ egin{array}{cc} 0 & 3+i 3-i & -3 end{array} ight] ~~ (e)~ A = left[ egin{array}{ccc} 5 & 0 & 0 0 & -1 & -1+i 0 & -1-i & 0 end{array} ight] ~~ (f)~ A = left[ egin{array}{ccc} 2 & frac{i}{sqrt{2}} & frac{-i}{sqrt{2}} frac{-i}{sqrt{2}} & 2 & 0 frac{i}{sqrt{2}} & 0 & 2 end{array} ight] ]

3. For each of the following matrices, either ﬁnd a matrix (P) (not necessarily unitary) such that (P^{−1}AP) is a diagonal matrix, or show why no such matrix exists.

[ (a)~ A = left[ egin{array}{ccc} 19 & -9 & -6 25 & -11 & -9 17 & -9 & -4 end{array} ight] ~~ (b)~A = left[ egin{array}{ccc} -1 & 4 & -2 -3 & 4 & 0 -3 & 1 & 3 end{array} ight] ~~ (c)~A = left[ egin{array}{ccc} 5 & 0 & 0 1 & 5 & 0 0 & 1 & 5 end{array} ight]]

[ (d)~ A = left[ egin{array}{ccc} 0 & 0 & 0 0 & 0 & 0 3 & 0 & 1 end{array} ight] ~~ (e)~A = left[ egin{array}{ccc} -i & 1 & 1 -i & 1 & 1 -i & 1 & 1 end{array} ight] ~~ (f)~A = left[ egin{array}{ccc} 0 & 0 & i 4 & 0 & i 0 & 0 & i end{array} ight]]

4. Let (r in mathbb{R}) and let (T in cal{L}(mathbb{C}^2)) be the linear map with canonical matrix

[ T = left( egin{array}{cc} 1 & -1 -1 & r end{array} ight) ]
(a) Find the eigenvalues of (T) .
(b) Find an orthonormal basis of ( mathbb{C}^2) consisting of eigenvectors of (T) .
(c) Find a unitary matrix (U) such that (UT U^*) is diagonal.

5. Let (A) be the complex matrix given by:
[ A = left[ egin{array}{ccc} 5 & 0 & 0 0 & -1 & -1+i 0 & -1-i & 0 end{array} ight]]
(a) Find the eigenvalues of (A).
(b) Find an orthonormal basis of eigenvectors of (A).
(c) Calculate (|A| = sqrt{A^*A}).
(d) Calculate (e^A) .

6. Let (θ in mathbb{R}), and let (T in cal{L}(mathbb{C}^2) )have canonical matrix
[ M(T ) = left( egin{array}{cc} 1 & e^{i heta} e^{-i heta} & -1 end{array} ight). ]
(a) Find the eigenvalues of (T) .
(b) Find an orthonormal basis for (mathbb{C}^2) that consists of eigenvectors for (T) .

## Proof-Writing Exercises

1. Prove or give a counterexample: The product of any two self-adjoint operators on a ﬁnite-dimensional vector space is self-adjoint.

2. Prove or give a counterexample: Every unitary matrix is invertible.

3. Let (V) be a ﬁnite-dimensional vector space over ( mathbb{F}), and suppose that (T in cal{L}(V)) satisﬁes (T^2 = T) . Prove that (T) is an orthogonal projection if and only if (T) is self-adjoint.

4. Let (V) be a ﬁnite-dimensional inner product space over ( mathbb{C}), and suppose that (T in cal{L}(V)) has the property that (T^* = −T ). (We call T a skew Hermitian operator on (V).)
(a) Prove that the operator (iT in cal{L}(V)) deﬁned by ((iT )(v) = i(T (v))), for each (v in V) , is Hermitian.
(b) Prove that the canonical matrix for (T) can be unitarily diagonalized.
(c) Prove that (T) has purely imaginary eigenvalues.

5. Let (V) be a ﬁnite-dimensional vector space over (mathbb{F}), and suppose that (S, T in cal{L}(V)) are positive operators on (V) . Prove that (S + T) is also a positive operator on (T) .

6. Let (V) be a ﬁnite-dimensional vector space over (mathbb{F}), and let (T in cal{L}(V))be any operator on (V) . Prove that (T) is invertible if and only if 0 is not a singular value of (T) .

## NCERT Solutions for Class 12 Maths Exercise 11.2

NCERT Solutions for Class 12 Maths Chapter 11 Exercise 11.2 of 3-D Three Dimensional Geometry updated for academic session 2021-2022 for CBSE and state boards. All the contents are available in Hindi and English Medium free to use online or download in PDF file format free without any login or password.

## Class 12 Maths Chapter 11 Exercise 11.2 Solution for 2021-2022

### CBSE NCERT Solutions for Class 12 Maths Exercise 11.2

The NCERT Solutions for Class 12 Maths Chapter 11 Exercise 11.2 updated for session 2021-2022 are given here in Hindi and English Medium free to use. Videos based on Hindi Medium and English Medium of Exercise 11.2 are available on the page below free to download or use online.

### Class 12 Maths Chapter 11 Exercise 11.2 Solutions in Videos

#### Elements of 3-D Geometry

A line given in space can extend in two opposite directions, thus it contains two-direction cosines. For a given line in space to have a unique set of direction cosines, we must take the given line as a directed line. These unique direction cosines are denoted by L, M and N. If the given line in space does not pass through the origin, to find its cosine direction, we draw a line through the origin and are parallel to the given line.

##### How to find a Line in 3-D

Now take one of the lines directed from the origin and find its direction cosine, because the two parallel lines have the same set of direction cosines. Any three numbers that are proportional to the direction corners of a line are called the direction relationships of the line. If l, m, n are directions and a, b, c are the direction relations of a line, then a = λl, b = λm and c = λn, for any nonzero real number λ.

###### Angle between two lines

Explain that L1 and L2 are two lines that pass through the origin and the direction relation a1, b1, c1 and a2, b2, c2 respectively. Explain that P is a point on L1 and Q is a point on L2. Consider the directed lines OP and OQ. Let A be the acute angle between OP and OQ. Now remember that the directed line segments OP and OQ are vectors with components A1, B1, C1 and A2, B2, C2 respectively. Therefore, the angle A between them is given by the formula CosA.

One of the most important chapters for the board exam point of view in Chapter 11 – Three Dimensional Geometry. The NCERT Solutions Class 12 Maths Exercise 11.1 Chapter 11 talks about and tries to explain the concepts of the chapter with detail and ease.

The NCERT Solutions Class 12 Maths Exercise 11.1 Chapter 11 consists of many topics. All the topics have designed and explained in proper detail and the solutions given are solved by subject matter experts for your better understanding.

## NCERT Solutions for Class 12 Maths Exercise 11.1

NCERT Solutions for Class 12 Maths Chapter 11 Exercise 11.1 of 3-D Three Dimensional Geometry in Hindi Medium as well as English Medium free to download in PDF or use online. All the contents are updated for academic session 2021-2022 for CBSE and state board students who are following NCERT Books for their exams.

Videos related to Exercise 11.1 are also given in Hindi and English Medium free to use without any login or registration.

## Class 12 Maths Chapter 11 Exercise 11.1 Solution for 2021-2022

### CBSE NCERT Solutions for Class 12 Maths Exercise 11.1

Get here the NCERT Solutions for Class 12 Maths Chapter 11 Exercise 11.1 in PDF file format free to use. Solutions are updated for session 2021-2022 in Hindi and English Medium with videos also. All the contents are based on latest NCERT Books for Class 12 session 2021-2022.

### Class 12 Maths Chapter 11 Exercise 11.1 Solutions in Videos

#### Previous knowledge of 3-D Geometry

In Class XI, as we study analytical geometry in two dimensions and introduce three-dimensional geometry, we limit ourselves to Cartesian methods only. In class 12 Maths chapter 10, we have learned about some basic concepts of vectors. Now we will use vector algebra for three-dimensional geometry. The purpose of this approach to three-dimensional geometry is that it makes the study simple and elegant.

##### Three dimensional geometry in mathematics from class 12

In Mathematics Class 12, Chapter 11, we will study about the fundamentals of 3-D geometry like the direction cosine and direction relationship of a line joining two points and we will also discuss equations of lines and planes in different situations, the angle between two lines, two planes. , A line and a plane, the shortest distance between two oblique lines and the distance of a plane from a point. Most of the above results are obtained in vector form. However, we will also translate these results into Cartesian form, which sometimes presents a clear geometric and analytical picture of the situation.

###### Equation of a Line in Space

We have gone through the equation of lines in two dimensions in eleventh grade, now we will study vector and Cartesian equations of a line in space. A line is uniquely determined if (i) passes through a given point and is its given direction, or (ii) passes through two given points.

##### Which question is considered as important one from class 12 Maths exercise 11.1?

There are only 5 questions in exercise 11.1, based on simple concepts. Students should do all these questions to learn the concepts behind them. These concepts are required to solve questions in later on exercises.

##### What is the weightage of Exercise 11.1 of class 12 Maths?

The separate weightage of any exercise is not available but generally questions carrying one marks come from Exercise 11.1 of class 12 Maths.

## NCERT Exemplar: CBSE Class 11 Mathematics &ndash All Chapters

All chapters of Class 11 Maths NCERT Exemplar are available here for download in PDF format. NCERT Exemplar book is very important for the students preparing for CBSE Class 11 Maths exams and other competitive exams like JEE Main, WBJEE etc.

NCERT Exemplar for Class 11 Maths is available here for download in PDF format. With this article, students can download all the chapters of Class 11 Maths NCERT Exemplar book in PDF format. Students can download all the chapters of the book from the links given below in the table.

NCERT Exemplar Class 11 Maths book published by NCERT is a very useful book for self assessment. There are 16 chapters in NCERT Exemplar Class 11 Maths book. Every chapter of NCERT Exemplar Class 11 Maths book starts with a brief summary of the chapter followed by solved examples and unsolved exercises.

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Questions from NCERT Exemplar are frequently asked in CBSE Class 11 Maths exams and various engineering entrance examinations like JEE Main, JEE Advanced, WBJEEE, UPSEE.

Jagranjosh.com has provided NCERT Exemplar Class 11 Maths Problems (and Answer Key) in PDF format with which students can enhance their preparation level.

## 11.E: Exercises for Chapter 11 - Mathematics

NCERT Solutions for Class 11 Maths

Solutions for all the questions from Maths NCERT, Class 11th

Doubtnut facilitates NCERT maths class 11 PDF solutions and NCERT video tutorials. Get here all systematically solved exercises of Class 11 Maths NCERT book. The exercise wise solutions are ideal for students to prepare for board examination as well as various engineering competitive examination. It can be your best resource to quickly practice any topic of class 11 mathematics. All NCERT Solutions for Class 11 Maths are as per the latest CBSE board syllabus and NCERT guidelines. These solutions have been prepared by our expert mathematicians and board examination experts. Our professional tutors have followed the simplest approach towards solving the questions and designing video tutorial for Class 11 Maths. By following the given tutorials you will find it very easy to solve the math problems. The video tutorials have been prepared to keep in mind students need and ensures comprehensive learning of basic concepts.

The maths experts at Doubtnut have solved the entire exercise questions from your NCERT class 11 textbooks and prepared video tutorials. We’ve covered all the chapters of class 11 maths. The solutions are classified in the chapter wise manner for your ease of learning. Listed below are all the chapter links. Click on the name of the chapter which you want to study. You will be directed it’s exercise page in which you can find all the detailed, step-wise solutions for every question given in your class 11th Math NCERT textbook. The video tutorials are filled with amazing math tips and tricks to make it easier for you to meet the answer.

Along with video tutorials, we’ve also provided the NCERT solutions in PDF format. The mathematics subject matter experts have hand-tailored the NCERT solutions for class 11 Maths PDF format. It is to give you a competitive edge to ace the examination with flying colours. You can also install the DOUBTNUT App to download the NCERT class 11 Math’s PDF in your mobile or tablet.

Question 1.
The length and the breadth of a rectangular piece of land is 500 m and 300 m respectively. Find
(i) its area
(ii) the cost of the land, if 1 m 2 of the land cost ₹ 10,000.
Solution:

Here, length = 500 m, breadth = 300 m
(i) Area = length × breadth = (500 × 300) m 2 = 1,50,000 m 2
(ii) Cost of land at the rate of ₹ 10,000 per 1 m 2 = ₹ (10,000 × 1,50,000) = ₹ 1,50,00,00,000.

Question 2.
Find the area of a square park whose perimeter is 320 m.
Solution:

Question 3.
Find the breadth of a rectangular plot of land, if its area is 440 m 2 and the length is 22 m. Also, find its perimeter.
Solution:

Question 4.
The perimeter of a rectangular sheet is 100 cm. If the length is 35 cm, find its breadth. Also find the area.

Solution:

Question 5.
The area of a square park is the same as a rectangular park. If the side of the square park is 60 m and the length of the rectangular park is 90 m, find the breadth of the rectangular park.

Solution:

Hence, the breadth of the rectangular park is 40 m.

Question 6.
A wire is in the shape of a rectangle. Its length is 40 cm and breadth is 22 cm. If the same wire is rebent in the shape of a square, what will be the measure of each side? Also, find which shape encloses more area?
Solution:

Question 7.
The perimeter of a rectangle is 130 cm. If the breadth of the rectangle is 30 cm, find its length. Also, find the area of the rectangle.

Solution:

Question 8.
A door of length 2 m and breadth 1 m is fitted in a wall. The length of the wall is 4.5 m and the breadth is 3.6 m (figure). Find the cost of whitewashing the wall, if the rate of whitewashing the wall is ₹ 20 per m 2 .
Solution:

We hope the NCERT Solutions for Class 7 Maths Chapter 11 Perimeter and Area Ex 11.1 help you. If you have any query regarding NCERT Solutions for Class 7 Maths Chapter 11 Perimeter and Area Ex 11.1, drop a comment below and we will get back to you at the earliest.

You can check the solved exercises for Maths Class 11 Chapter 1 PDF below. Also, you can download NCERT Solutions for Sets in PDF to practice offline. If you don’t want to download the NCERT Solutions for Class 11 Maths Sets Chapter 1 PDF (Sets Class 11 NCERT Solutions PDF), then you can also access the solutions through the images given below:

### Why Refer To Embibe’s NCERT Solutions For Class 11 Maths Sets?

At Embibe, you can find the solutions for the chapter – Sets. All the NCERT solutions have been solved by the academic experts of Embibe. The solutions have been explained in such a manner that any 11th Class student can understand. The experts have explained the solutions in a step-by-step manner taking care of the CBSE guidelines. Class 11 Maths Chapter 1 Sets notes are prepared by experts to help all students in their studies.

Every chapter in the NCERT Textbook contains intext questions and exercise questions. A lot of students tend to ignore these questions. NCERT questions are not only important for the board exams but also for the competitive examinations. Exams like JEE Mains and NEET ask questions from NCERT Textbooks. Therefore, solving these NCERT Questions will help you in developing a strong foundation on the chapter Sets. You can access Embibe’s NCERT Solutions for Class 11 Maths Sets PDF for free without any signup or login.

 2nd Chapter: Relations and Functions 3rd Chapter: Trigonometric Functions 4th Chapter: Principle of Mathematical Induction 5th Chapter: Complex Numbers and Quadratic Equations 6th Chapter: Linear Inequalities 7th Chapter: Permutations and Combinations 8th Chapter: Binomial Theorem 9th Chapter: Sequences and Series 10th Chapter: Straight Lines 11th Chapter: Conic Sections 12th Chapter: Introduction to Three Dimensional Geometry 13th Chapter: Limits and Derivatives 14th Chapter: Mathematical Reasoning 15th Chapter: Statistics 16th Chapter: Probability

### NCERT Solutions For Class 11 Maths Chapter 1 – Sets: Chapter Summary

A collection of objects which is well defined is considered a set. Set is defined by an existing rule, with which it is possible to conclude whether an object is a part of a set and whether a collection constitutes a set. Problems and solutions in this set tell us how to ascertain whether a collection is set, different types of sets, algebra of sets, and power sets.

This chapter will help students to build a strong base for Mathematics. It will help the students to know how sets are represented using symbols and Venn diagrams. Practical questions included in the chapter will help the students to understand the concepts in a better manner. The list of topics and sub-topics that will be taught under Maths Chapter 1 Sets are given below:

 Exercise Topics 1.1 Introduction 1.2 Sets and their Representations 1.3 The Empty Set 1.4 Finite and Infinite Sets 1.5 Equal Sets 1.6 Subsets 1.6.1 Subsets of set of real numbers 1.6.2 Intervals as subsets of R 1.7 Power Set 1.8 Universal Set 1.9 Venn Diagrams 1.10 Operations on Sets 1.10.1 Union of Sets 1.10.2 Intersection of Sets 1.10.3 Difference of Sets 1.11 Complement of a Set 1.12 Practical Problems on Union and Intersection of Two Sets

### FAQs Regarding NCERT Solutions for Class 11 Maths Sets

The frequently asked questions regarding CBSE Class 11 Maths Chapter 1 Sets are given below:

Q1. From where can I download CBSE Class 11 Maths Chapter 1 Solutions?
A.

A.You can solve CBSE Class 11 Maths Chapter 1 questions for free on Embibe.

 NCERT SOLUTIONS FOR CLASS 11 MATHS CHAPTER 2 NCERT SOLUTIONS FOR CLASS 11 MATHS

Now you are provided with all the necessary information regarding Class 11 Maths NCERT Solutions Chapter 1. In order to reap the benefits, first, try to solve the questions on your own. If you are unable to solve only then access the NCERT Solutions. Understand how the experts have arrived at the answer and then again solve it. Students of Class 11 can take free Class 11 Sets Mock Tests on Embibe. Taking these mock tests will definitely help students with their exam preparation.

We hope this detailed article on CBSE NCERT Solutions for Class 11 Maths Chapter 1 (Sets Class 11 PDF) helps you. Refer to the solutions that we have provided in this article. If you have any queries, feel free to ask in the comment section below and we will get back to you at the earliest.

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## Chapter 6 Class 11 Linear Inequalities

Get solutions of all questions and examples of Chapter 6 Class 11 Linear Inequalities of the NCERT Book. All questions are divided the NCERT wise - into exercises, examples and miscellaneous.

And also the Concept wise way, where you can pick a topic and start studying the questions one by one, from easy to difficult.

We have studied Linear Equations before, in this chapter we will discuss linear inequalities, where instead of the = sign, we have <, >, &ge, &le symbols. Like ( x + y &le 3)

The topics in this chapter include

• Solving Linear Inequality
• Solving Linear Inequality in the number line
• Making Linear Inequality from statements, and solving
• Solving a pair of linear inequalities
• Solving a pair of Linear Inequalities in the number line
• Making a pair of Linear Inequalities from statements, and solving
• Graphical Solution of a Linear Inequality (Like x + y > 5)
• Graphical Solution of a pair of Linear Inequalities (Like x + y > 5 and x + 2y < 3)

Check out the questions in the exercises below, or start from doing the chapter concept wise way