EXERCISE 1.4                                                                                                     PAGE:20

Question 1. Classify the following numbers as rational or irrational:
(i)

(ii)

(iii)

(iv)

(v)

Solution :

(i)

We know that

which is also an irrational number.

Therefore, we conclude thatis an irrational number.

(ii)

= 3

Therefore, we conclude thatis a rational number.

(iii)

We can cancelin the numerator and denominator, asis the common number in numerator as well as denominator, to get

Therefore, we conclude thatis a rational number.

(iv)

We know that.

We can conclude that, when 1 is divided by, we will get an irrational number.

Therefore, we conclude thatis an irrational number.

(v)

We know that

We can conclude that will also be an irrational number.

Therefore, we conclude that is an irrational number.

Question 2. Simplify each of the following expressions

Solution :
(i) (3 + √3)(2 + √2)
= 2(3 + √3) + √2(3 + √3)
= 6 + 2√3 + 3√2 + √6
Thus, (3 + √3)(2 + √2) = 6 + 2√3 + 3√2 + √6
(ii) (3 + √3)(3 – √3) = (3)2 – (√3)2
= 9 – 3 = 6
Thus, (3 + √3)(3 – √3) = 6
(iii) (√5 + √2)2 = (√5)2 + (√2)2 + 2(√5)(√2)
= 5 + 2 + 2√10 = 7 + 2√10
Thus, (√5 + √2 )2 = 7 + 2√10
(iv) (√5 – √2)(√5 + √2) = (√5)2 – (√2)2 = 5 – 2 = 3
Thus, (√5 – √2) (√5 + √2) = 3

Question 3. Recall, π is defined as the ratio of the circumference (say c) of a circle to its diameter (say d). That is π = . This seems to contradict the fact that n is irrational. How will you resolve this contradiction?
Solution : When we measure the length of a line with a scale or with any other device, we only get an approximate ational value, i.e. c and d both are irrational.
∴  is irrational and hence π is irrational.
Thus, there is no contradiction in saying that it is irrational.

Question 4. Represent on the number line.
Solution :
Draw a line segment AB = 9.3 units and extend it to C such that BC = 1 unit.
Find mid point of AC and mark it as O.
Draw a semicircle taking O as centre and AO as radius. Draw BD ⊥ AC.
Draw an arc taking B as centre and BD as radius meeting AC produced at E such that BE = BD = units.

Question 5. Rationalise the denominator of the following

Solution :