Answer :

(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.

Answer :

(i): $5 \in A$

(ii): $8 \notin A$

(iii): $ 0 \notin A$

(iv): $4 \in A$

(v): $2 \in A$

(vi): $10 \notin A$

Answer :

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

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

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

Therefore, the given set can be written in roster from as $ \mathrm{A}={-2 , -1,0,1,2,3,4,5,6} $

The natural numbers less than $6$ are $1, 2, 3, 4, 5$

So, the elements of this set are $1,2,3,4$, and $5$ only.

Therefore, the given set can be written in roster from as $ \mathrm{B}={1,2,3,4,5} $

(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={17,26,35,44,53,62,71,80} $

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

$\underline{2 \mid 60}$

$ \dfrac{\dfrac{2 \mid 30}{3 \mid 15}}{\dfrac{5 \mid 5}{\mid 1}} $

$ \therefore \ \ 60=2 \times 2 \times 3 \times 5 $

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

Therefore, this set can be written in roster form as $\mathrm{D}={2,3,5}$

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

There are $12$ letters in the word TRIGONOMETRY, out of which the letters, $T,$ $R$, and $O$ are repeated. And we write the repeated letters once only.

Therefore, this set can be written in roster form as

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

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

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

Therefore, this set can be written in roster form as $ F={B, E, T, R} $

Answer :

(i): ${3,6,9,12}={x: x=3 n, n \in N$ and $1 \leq n \leq 4}$

(ii): ${2,4,8,16,32}$

It can be seen that $2=2^{1}, 4=2^{2}, 8=2^{3}, 16=2^{4}$, and $32=2^{5}$

$\therefore \ \ \ \ {2,4,8,16,32}={x: x=2^{n}, n {\in} N$ and $1 \leq n \leq 5}$

(iii): ${5,25,125,625}$

It can be seen that $5=5^{1}, 25=5^{2}, 125=5^{3}$, and $625=5^{4}$

$\therefore \ \ \ \ {5,25,125,625}={x: x=5^{n}, n \in N$ and $1 \leq n \leq 4}$

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

It is a set of all even natural numbers.

$\therefore \ \ \ \ {2,4,6 \ldots}={x: x$ is an even natural number $}$

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

It can be seen that $1=1^{2}, 4=2^{2}, 9=3^{2} \ldots 100=10^{2}$

$\therefore \ \ \ \ {1,4,9 \ldots 100}={x: x=n^{2}, n \in N$ and $1 \leq n \leq 10}$

Answer :

(i): $\mathrm{A}={\mathrm{x}: \mathrm{x}$ is an odd natural number $}={1,3,5,7,9 \ldots \ldots}$

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

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

$ \therefore \ \ \mathrm{B}={0, 1,2,3,4} $

(iii): $\mathrm{C}=\left{\mathrm{x}: \mathrm{x}\right.$ is an integer; $\left.\mathrm{x}^2 \leq 4\right}$

It can be seen that

$ \begin{aligned} & (-1)^2=1 \leq 4 \\ \\ & (-2)^2=4 \leq 4 \\ \\ & (-3)^2=9>4 \\ \\ & \mathrm{0}^2=0 \leq 4 \\ \\ & 1^2=1 \leq 4 \\ \\ & 2^2=4 \leq 4 \\ \\ & 3^2=9>4 \\ \\ & \therefore \ \ C={-2,-1,0,1,2} \end{aligned} $

(iv): $\mathrm{D}={\mathrm{x}: \mathrm{x}$ is a letter in the word LOYAL $}={\mathrm{L}, \mathrm{O}, \mathrm{Y}, \mathrm{A}}$

(v): $\mathrm{E}={\mathrm{x}: \mathrm{x}$ is a month of a year not having 31 days $}$ $={\text{February, April, June, September, November }}$

(vi): $\mathrm{F}={\mathrm{x}: \mathrm{x}$ is a consonant in the English alphabet which precedes $k}$ $={\mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{f}, \mathrm{g}, \mathrm{h}, \mathrm{j}}$

Answer :

(i): All the elements of this set are natural numbers as well as the divisors of $6 $

Therefore, : matches with $(c)$

(ii): It can be seen 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 MATHEMATICS.

Therefore, (iii) matches with $(d)$

(iv): All the elements of this set are odd natural numbers less than $10$

Therefore, (iv) matches with $(b)$