# NCERT Solutions For Class 11 Maths Chapter 1 Exercise 1.2

## Chapter 1 Ex.1.2 Question 1

Which of the following are examples of the null set

(i) Set of odd natural numbers divisible by $$2$$

(ii) Set of even prime numbers

(iii) {$$x:x$$ is a natural number, $$x < 5$$ and $$x > 7$$}

(iv) {$$y:y$$ is a point common to any two parallel lines}

### Solution

(i) A set of odd natural numbers divisible by $$2$$ is a null set because no odd number is divisible by $$2.$$

(ii) A set of even prime numbers is not a null set because $$2$$ is an even prime number.

(iii) {$$x:x$$ is a natural number, $$x < 5$$ and $$x > 7$$} is a null set because a number cannot be simultaneously less than $$5$$ and greater than $$7.$$

(iv) {$$y:y$$ is a point common to any two parallel lines} is a null set because parallel lines do not intersect. Hence, they have no common point.

## Chapter 1 Ex.1.2 Question 2

Which of the following sets are finite or infinite?

(i) The set of months of a year

(ii) $$\left\{{1, 2, 3 \dots} \right\}$$

(iii) $$\left\{{1, 2, 3 \dots 99, 100} \right\}$$

(iv) The set of positive integers greater than $$100$$

(v) The set of prime numbers less than $$99$$

### Solution

(i) The set of months of a year is a finite set because it has $$12$$ elements.

(ii) $$\left\{{1, 2, 3 \dots} \right\}$$ is an infinite set as it has infinite number of natural numbers.

(iii) $$\left\{{1, 2, 3 \dots 99, 100} \right\}$$ is a finite set because the numbers from $$1$$ to $$100$$ are finite.

(iv) The set of positive integers greater than $$100$$ is an infinite set because positive integers greater than $$100$$ are infinite in number.

(v) The set of prime numbers less than $$99$$ is a finite set because prime numbers less than $$99$$ are finite in number.

## Chapter 1 Ex.1.2 Question 3

State whether each of the following set is finite or infinite:

(i) The set of lines which are parallel to the $$x$$-axis

(ii) The set of letters in the English alphabet

(iii) The set of numbers which are multiple of $$5$$

(iv) The set of animals living on the earth

(v) The set of circles passing through the origin $$(0,0)$$

### Solution

(i) The set of lines which are parallel to the $$x$$-axis is an infinite set because lines parallel to the $$x$$-axis are infinite in number.

(ii) The set of letters in the English alphabet is a finite set because it has $$26$$ elements.

(iii) The set of numbers which are multiple of $$5$$ is an infinite set because multiples of $$5$$ are infinite in number.

(iv) The set of animals living on the earth is a finite set because the number of animals living on the earth is finite (but it is quite a large number).

(v) The set of circles passing through the origin $$(0,0)$$ is an infinite set because infinite number of circles can pass through the origin.

## Chapter 1 Ex.1.2 Question 4

In the following, state whether $$A = B$$ or not:

(i) $$A = \left\{ {a,b,c,d} \right\} \qquad B = \left\{ {d,c,b,a} \right\}$$

(ii) $$A = \left\{ {4,8,12,16} \right\} \qquad B = \left\{ {8,4,16,18} \right\}$$

(iii) $$A = \left\{ {2,4,6,8,10} \right\} \quad B =$$ {$$x:x$$ is positive even integer and $$x \le 10$$ }

(iv) $$A =$$ {$$x:x$$ is a multiple of $$10$$} B = {$$10,15,20,25,30,\dots$$}

### Solution

(i) $$A = \left\{ {a,b,c,d} \right\}\qquad B = \left\{ {d,c,b,a} \right\}$$

The elements of sets $$A$$ and $$B$$ are same.

Therefore, $$A = B$$

(ii) $$A =$$ {$$4, 8, 12, 16$$}; $$\qquad B = {8, 4, 16, 18}$$

We see that $$12 \in A$$ but $$12 \notin B$$.

Therefore, $$A \ne B$$

(iii) $$A = \left\{ {2,4,6,8,10} \right\} \qquad B = \left\{ {x:x{\text { is positive even integer and }}x \le 10} \right\}$$

We see that $$B = \left\{ {2,4,6,8,10} \right\}$$

Therefore, $$A = B$$

(iv) $$A = \left\{ {x:x\;\;{\text{ is a multiple of }}10} \right\} \qquad B = \left\{ {10,15,20,25,30...} \right\}$$

We see that $$15 \in B$$ but $$15 \notin A$$.

Therefore, $$A \ne B$$

## Chapter 1 Ex.1.2 Question 5

Are the following pair of sets equal? Give reasons.

(i) $$A = \left\{ {2,3} \right\};{\rm{ }}B = \left\{ {x:x{\text{is solution of}}{x^2} + 5x + 6 = 0} \right\}$$

(ii) $$A = \left\{ {x:x{\text{ is a letter in the word }}FOLLOW} \right\}$$

$$B = \left\{ {y:y{\text{ is a letter in the word }}WOLF} \right\}$$

### Solution

(i) $$A = \left\{ {2,3} \right\};{\rm{ }}B = \left\{ {x:x{\text{ is solution of }}{x^2} + 5x + 6 = 0} \right\}$$

The equation $${x^2} + 5x + 6 = 0$$ can be solved as:

\begin{align}x\left( {x + 3} \right) + 2\left( {x + 3} \right) &= 0\\\left( {x + 2} \right)\left( {x + 3} \right) &= 0\\x = -2\;\;\;or\;\;\;x &= -3\end{align}

Hence,$$A = \left\{ {2,3} \right\};{\rm{ }}B = \left\{ { - 2, - 3} \right\}$$

Therefore, $$A \ne B$$

(ii) $$A = \left\{ {x:x{\text{ is a letter in the word }}FOLLOW} \right\}$$

$$B = \left\{ {y:y{\text{ is a letter in the word }}WOLF} \right\}$$

Hence,$$A = \left\{ {F,L,O,W} \right\};{\rm{ }}B = \left\{ {W,O,L,F} \right\}$$

The elements of sets $$A$$ and $$B$$ are same.

Therefore, $$A = B$$

## Chapter 1 Ex.1.2 Question 6

From the sets given below, select equal sets:

\begin{align}A &= \left\{ {2,4,8,12} \right\}{\rm{,}}\;\;\;B = \left\{ {1,2,3,4} \right\}{\rm{,}}\;\;\;C = \left\{ {4,8,12,14} \right\}\;\;\;D = \left\{ {3,1,4,2} \right\}\\E &= \left\{ {-1,1} \right\},\;\;\;\;\;\;\;\;\;F = \left\{ {0,a} \right\},\;\;\;\;\;\;\;\;G = \left\{ {1,-1} \right\},\;\;\;\;\;\;\;\;\;\;H = \left\{ {0,1} \right\}\end{align}

### Solution

\begin{align}A &= \left\{ {2,4,8,12} \right\}{\rm{,}}\;\;\;B = \left\{ {1,2,3,4} \right\}{\rm{,}}\;\;\;C = \left\{ {4,8,12,14} \right\}\;\;\;D = \left\{ {3,1,4,2} \right\}\\E &= \left\{ {-1,1} \right\},\;\;\;\;\;\;\;\;\;F = \left\{ {0,a} \right\},\;\;\;\;\;\;\;\;G = \left\{ {1,-1} \right\},\;\;\;\;\;\;\;\;\;\;H = \left\{ {0,1} \right\}\end{align}

We see that

$8 \in A,\;\;8 \notin B,\;\;8 \in C,\;\;8 \notin D,\;\;8 \notin E,\;\;8 \notin F,\;\;8 \notin G,\;\;8 \notin H$

Therefore,

$A \ne B,\;\;A \ne D,\;\;A \ne E,\;\;A \ne F,\;\;A \ne G,\;\;A \ne H$

Also, $$2 \in A,\;\;2 \notin C$$

Therefore, $$A \ne C$$

$3 \in B,\;\;3 \notin C,\;\;3 \notin E,\;\;3 \notin F,\;\;3 \notin G,\;\;3 \notin H$

Therefore, $$B \ne C,\;\;B \ne E,\;\;B \ne F,\;\;B \ne G,\;\;B \ne H$$

Again, $$12 \in C,\;\;12 \notin D,\;\;12 \notin E,\;\;12 \notin F,\;\;12 \notin G,\;\;12 \notin H$$

Therefore, $$C \ne D,\;\;C \ne E,\;\;C \ne F,\;\;C \ne G,\;\;C \ne H$$

Now,

$$4 \in D,\;\;4 \notin E,\;\;4 \notin F,\;\;4 \notin G,\;\;4 \notin H$$

Therefore, $$D \ne E,\;\;D \ne F,\;\;{\rm{ }}D \ne G,\;\;D \ne H$$

Similarly, $$E \ne F,\;\;E \ne G,\;\;E \ne H,\;\;F \ne G,\;\;F \ne H,\;\;G \ne H$$

The elements of sets $$B$$ and $$D$$ are same also, the elements of sets $$E$$ and $$G$$ are same.

Therefore, $$B = D$$ and $$E = G$$

Hence, sets, $$B = D$$ and $$E = G$$

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