# NCERT Solutions For Class 11 Maths Chapter 1 Exercise 1.1

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## Chapter 1 Ex.1.1 Question 1

(i) The collection of all months of a year beginning with the letter J.

(ii) The collection of ten most talented writers of India.

(iii) A team of eleven best-cricket batsmen of the world.

(iv) The collection of all boys in your class.

(v) The collection of all natural numbers less than $$100.$$

(vi) A collection of novels written by the writer Munshi Prem Chand.

(vii) The collection of all even integers.

(viii) The collection of questions in this Chapter.

(ix) A collection of most dangerous animals of the world.

### Solution

(i) The collection of all months of a year beginning with the letter J is a well-defined collection of objects because one can definitely identify a month that belongs to this collection.

Hence, this collection is a set.

(ii) The collection of ten most talented writers of India is not a well-defined collection because the criteria for determining a writer’s talent may vary from person to person.

Hence, this collection is not a set.

(iii) A team of eleven best cricket batsmen of the world is not a well-defined collection because the criteria for determining a batsman’s talent may vary from person to person.

Hence, this collection is not a set.

(iv) The collection of all boys in your class is a well-defined collection because you can definitely identify a boy who belongs to this collection.

Hence, this collection is a set.

(v) The collection of all natural numbers less than $$100$$ is a well-defined collection because one can definitely identify a number that belongs to this collection.

Hence, this collection is a set.

(vi) A collection of novels written by the writer Munshi Prem Chand is a well-defined collection because one can definitely identify a book that belongs to this collection.

Hence, this collection is a set.

(vii) The collection of all even integers is a well-defined collection because one can definitely identify an even integer that belongs to this collection.

Hence, this collection is a set.

(viii) The collection of questions in this chapter is a well-defined collection because one can definitely identify a question that belongs to this chapter.

Hence, this collection is a set.

(ix) The collection of most dangerous animals of the world is not a well-defined collection because the criteria for determining the dangerousness of an animal can vary from person to person.

Hence, this collection is not a set.

## Chapter 1 Ex.1.1 Question 2

Let $$A = {1, 2, 3, 4, 5, 6}.$$ Insert the appropriate symbol $$∈$$ or $$∉$$ in the blank spaces:

(i) $$5\,\dots\,A$$

(ii) $$8\,\dots\,A$$

(iii) $$0\,\dots\,A$$

(iv) $$4\,\dots\,A$$

(v) $$2\,\dots\,A$$

(vi) $$10\,\dots\,A$$

### Solution

(i) $$5 \in A$$

(ii) $$8 \notin A$$

(iii) $$0 \notin A$$

(iv) $$4 \in A$$

(v) $$2 \in A$$

(vi) $$10 \notin A$$

## Chapter 1 Ex.1.1 Question 3

Write the following sets in roster form:

(i) $$A =$${$$x:x$$ is an integer and $$-3 \le x < 7$$}.

(ii) $$B =$${$$x:x$$ is a natural number less than $$6$$}.

(iii) $$C =$${$$x:x$$ is a two-digit natural number such that the sum of its digits is $$8$$}

(iv) $$D =$${$$x:x$$ is a prime number which is divisor of $$60$$}.

(v) $$E =$$ The set of all letters in the word $$\rm{TRIGONOMETRY.}$$

(vi) $$F =$$ The set of all letters in the word $$\rm{BETTER.}$$

### Solution

(i) $$A =$$ {$$x:x$$ is an integer and $$-3 \le x < 7$$}

The elements of this set are $$-3, –2, –1, 0, 1, 2, 3, 4, 5,$$ and $$6$$ only.

Therefore, the given set can be written in roster form as

$A = \left\{ { - 3,\;-2,\;-1,\;0,\;1\;,2,\;3,\;4,\;5,\;6} \right\}$

(ii) $$B =$${$$x:x$$ is a natural number less than $$6$$}

The elements of this set are $$1,\; 2,\; 3,\; 4,\;$$ and $$5$$ only.

Therefore, the given set can be written in roster form as

$B = \left\{ {1,2,3,4,5} \right\}$

(iii) $$C =$$ {$$x:x$$ is a two-digit natural number such that the sum of its digits is $$8$$}

The elements of this set are $$17, 26, 35, 44, 53, 62, 71$$, and $$80$$ only.

Therefore, this set can be written in roster form as

$C = \left\{ {17,\,26,\,35,\,44,\,53,\,62,\,71,\,80} \right\}$

(iv) $$D =$$ {$$x:x$$ is a prime number which is a divisor of $$60$$}

\begin{align}2\left| \!{\underline {\, {60} \,}} \right. \\2\left| \!{\underline {\, {30} \,}} \right. \\3\left| \!{\underline {\, {15} \,}} \right. \\5\end{align}

Hence, $$60 = 2 \times 2 \times 3 \times 5$$

The elements of this set are $$2, \,3,$$ and $$5$$ only.

Therefore, this set can be written in roster form as

$D = \left\{ {2,3,5} \right\}$.

(v) $$E =$$ The set of all letters in the word $$\rm{TRIGONOMETRY}$$

There are $$12$$ letters in the word $$\rm{TRIGONOMETRY}$$, out of which letters $$\rm{T,\, R,}$$ and $$\rm{O}$$ are repeated.

Therefore, this set can be written in roster form as

$E = \left\{ {T,\,R,\,I,\,G,\,O,\,N,\,M,\,E,\,Y} \right\}$

(vi) $$F =$$ The set of all letters in the word $$\rm{BETTER}$$

There are $$6$$ letters in the word $$\rm{BETTER},$$ out of which letters $$\rm{E}$$ and $$\rm{T}$$ are repeated.

Therefore, this set can be written in roster form as

$F = \left\{ {B,\,E,\,T,\,R} \right\}$

## Chapter 1 Ex.1.1 Question 4

Write the following sets in the set-builder form:

(i) $$\left\{ {3,\,6,\,9,\,12} \right\}$$

(ii) $$\left\{ {2,\,4,\,8,\,16,\,32} \right\}$$

(iii) $$\left\{ {5,\,25,\,125,\,625} \right\}$$

(iv) $$\left\{ {2,\,4,\,6, \ldots } \right\}$$

(v) $$\left\{ {1,\,4,\,9, \ldots,\, 100} \right\}$$

### Solution

(i) $$\left\{ {3,6,9,12} \right\}$$

We see that

\begin{align}3 &= 3 \times 1\\6 &= 3 \times 2\\9& = 3 \times 3\\12 &= 3 \times 4\end{align}

Therefore,

$\left\{ {3,6,9,12} \right\} = \left\{ {x:x = 3n,\;n \in N\;{\text{ and }}\;1 \le n \le 4} \right\}$

(ii) $$\left\{ {2,4,8,16,32} \right\}$$

We see that

$2 = {2^1},\;4 = {2^2},\;8 = {2^3},\;16 = {2^4},\;32 = {2^5}$

Therefore,

$\left\{ {2,4,8,16,32} \right\} = \left\{ {x:x = {2^n},\;n \in N\;{\text{ and }}\;1 \le n \le 5} \right\}$

(iii) $$\left\{ {5,\,25,\,125,\,625} \right\}$$

We see that

$5 = {5^1},\;25 = {5^2},\;125 = {5^3},\;625 = {5^4}$

Therefore,

$\left\{ {5,25,125,625} \right\} = \{ x:x = {5^n},\;n \in N{\text{ and }}1 \le n \le 4\}$

(iv) $$\left\{ {2,4,6 \ldots } \right\}$$

We see that it is a set of all even natural numbers.

Therefore,

$\left\{ {2,4,6 \ldots } \right\} = \left\{ {x:x\;\;{\text{is an even natural number}}} \right\}$

(v) $$\left\{ {1,4,9 \ldots 100} \right\}$$

We see that

$1 = {1^2},\;4 = {2^2},\;9 = {3^2},\; \ldots 100 = {10^2}$

Therefore,

$\left\{ {1,4,9 \ldots 100} \right\} = \{ x:x = {n^2},\;n \in N{\text{ and }}1 \le n \le 10\}$

## Chapter 1 Ex.1.1 Question 5

List all the elements of the following sets:

(i) $$A =$$ {$$x:x$$ is an odd natural number}

(ii) $$B =$$ {$$x:x$$ is an integer,$$-\frac{1}{2}<x<\frac{9}{2}$$}

(iii) $$C =$$ {$$x:x$$ is an integer,$${{x}^{2}}\le 4$$}

(iv) $$D =$$ {$$x:x$$ is a letter in the word “$$\rm{LOYAL}$$”}

(v) $$E =$$ {$$x:x$$ is a month of a year not having $$31$$ days}

(vi) $$F =$$ {$$x:x$$ is a consonant in the English alphabet which proceeds $$k$$}.

### Solution

(i) $$A =$$ {$$x:x$$ is an odd natural number}

$A = \left\{ {1,3,5,7,9 \ldots } \right\}$

(ii) $$B =$${$$x:x$$ is an integer; $$- \frac{1}{2} < x < \frac{9}{2}$$}

We see that

$$- \frac{1}{2} = - 0.5$$ and $$\frac{9}{2} = 4.5$$

Therefore,

$B = \left\{ {0,\,1,\,2,\,3,\,4} \right\}$

(iii) $$C =$$ {$$x:x$$ is an integer; $${x^2} \le 4$$}

We see that

\begin{align}{\left( 0 \right)^2} &= 0,\;\;\;\;i.e.,\;{\left( 0 \right)^2} < 4\\{\left( {-1} \right)^2} &= 1,\;\;\;\;i.e.,\;{\left( {-1} \right)^2} < 4\\{\left( {-2} \right)^2} &= 4,\;\;\;\;i.e.,\;{\left( {-2} \right)^2} = 4\\{\left( {-3} \right)^2} &= 9,\;\;\;\;i.e.,\;{\left( {-3} \right)^2} > 9\end{align}

Therefore,

$C = \left\{ {-2,\, -1,\, 0, \,1,\, 2} \right\}$

(iv) $$D =$$($$x:x$$ is a letter in the word “$$\rm{LOYAL}$$”)

$D = \left\{ {L,O,Y,A} \right\}$

(v) $$E =$${$$x:x$$ is a month of a year not having $$31$$ days}

$E = \left\{ {February, April, June, September, November} \right\}$

(vi) $$F =$${$$x:x$$ is a consonant in the English alphabet which precedes $$k$$}

$F = \left\{ {b,c,d,f,g,h,j} \right\}$

## Chapter 1 Ex.1.1 Question 6

Match each of the set on the left in the roster form with the same set on the right described in set-builder form:

 (i) $$\left\{ {1, 2, 3, 6} \right\}$$ (a) {$$x:x$$ is a prime number and a divisor of $$6$$} (ii) $$\left\{ {2, 3} \right\}$$ (b) {$$x:x$$ is an odd natural number less than $$10$$} (iii) $$\left\{ \rm{M, A, T, H, E, I, C, S} \right\}$$ (c) {$$x:x$$ is natural number and divisor of $$6$$} (iv) $$\left\{{1, 3, 5, 7, 9} \right\}$$ (d) {$$x:x$$ is a letter of the word $$\left\{\rm{MATHEMATICS} \right\}$$}

### Solution

(i) All the elements of this set are natural numbers as well as the divisors of $$6$$. Therefore, (i) matches with (c).

(ii) We see that $$2$$ and $$3$$ are prime numbers. They are also the divisors of $$6.$$

Therefore, (ii) matches with (a).

(iii) All the elements of this set are letters of the word $$\rm{MATHEMATICS}$$. Therefore, (iii) matches with (d).

(iv) All the elements of this set are odd natural numbers less than $$10$$. Therefore, (iv) matches with (b).

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