# ICSE Solutions for Selina Concise Chapter 1 Rational and Irrational Numbers Class 9 Maths

### Exercise 1(A)

**1. Is zero a rational number ? Can it be written in the form P/q, where p and q are integers and q≠0 ?**

**Answer**

Yes, zero is a rational number.

As it can be written in the form of , where p and q are integers and q≠0 ?

⇒ 0 = 0/1

**2.**

**Are the following statement true or false ? Give reason for your answer.**

**(i) Every whole number is a natural number.**

**(ii) Every whole number is a rational number.**

**(iii) Every integer is a rational number.**

**(iv) Every rational number is a whole number.****Answer**

(i) False, zero is a whole number but not a natural number.

(ii) True, Every whole can be written in the form of, where p and q are integers and q≠0.

(iii) True, Every integer can be written in the form of , where p and q are integers and q≠0.

(iv) False.

Example:** ** is a rational number, but not a whole number.

**3. Arrange −5/7, 7/12,−2/3 and 11/18 in ascending order of their magnitudes.**

Also, find the difference between the largest and smallest of these rational numbers. Express this difference as a decimal fraction correct to one decimal place.

Also, find the difference between the largest and smallest of these rational numbers. Express this difference as a decimal fraction correct to one decimal place.

**Answer**

**4. Arrange 5/8,−3/16,−1/4 and 17/32 in descending order of their magnitudes.**

Also, find the sum of the lowest and largest of these fractions. Express the result obtained as a decimal fraction correct to two decimal places.

Also, find the sum of the lowest and largest of these fractions. Express the result obtained as a decimal fraction correct to two decimal places.

**Answer**

**5.1. Without doing any actual division, find which of the following rational numbers have terminating decimal representation : 7/16**

**Answer**

Given number is 7/16

Since 16 = 2×2× 2×2 = 2^{4} = 2^{4} × 5^{0}

I.e. 16 can be expressed as 2^{m} × 5^{n}

∴ 7/16 is convertible into the terminating decimal.

**5.2. Without doing any actual division, find which of the following rational numbers have terminating decimal representation : 23/125**

**Answer**

Given number is 23/125

Since 125 = 5×5×5 = 5^{3 = }2^{0} × 5^{3}

i.e. 125 can be expressed as 2^{m}×5^{n}

∴ 231/25 is convertible into the terminating decimal.

**5.3. Without doing any actual division, find which of the following rational numbers have terminating decimal representation : 9/14**

**Answer**

Given number is 9/14

Since 14 = 2×7 = 2^{1}× 7^{1}

i.e. 14 cannot be expressed as 2^{m} × 5^{n}

∴ 9/14 is not convertible into the terminating decimal.

**5.4. Without doing any actual division, find which of the following rational numbers have terminating decimal representation : 32/45**

**Answer**

Given number is 32/45

Since, 45 = 3×3×5 = 3^{2} × 5^{1}

i.e. 45 cannot be expressed as 2^{m} × 5^{n}

∴ 32/45 is not convertible into the terminating decimal.

**5.5. Without doing any actual division, find which of the following rational numbers have terminating decimal representation : 43/50**

**Answer**

Given number is 46/50

Since 50 = 2×5×5 = 2^{1} × 5^{2}

i.e. 50 can be expressed as 2^{m} × 5^{n}

∴ 43/50 is convertible into the terminating decimal.

**5.6. Without doing any actual division, find which of the following rational numbers have terminating decimal representation : 17/40**

**Answer**

Given number is 17/40

Since 40 = 2×2 × 2 × 5 = 2^{3} × 5^{1}

i.e. 40 can be expressed as 2^{m} × 5^{n}

∴ 17/40 is convertible into the terminating decimal.

**5.7. Without doing any actual division, find which of the following rational numbers have terminating decimal representation : 61/75**

**Answer**

Given number is 61/75

Since, 75 = 3×5×5 = 3^{1} × 5^{2}

i.e. 75 cannot be expressed as 2^{m} × 5^{n}

∴ 61/75 is not convertible into the terminating decimal.

**5.8. Without doing any actual division, find which of the following rational numbers have terminating decimal representation : 123/250**

**Answer**

Given number is 123/250

Since 250 = 2 ×5×5×5 = 2^{1} × 5^{3}

i.e. 250 can be expressed as 2^{m} × 5^{n.1}

∴ 123/250 is convertible into the terminating decimal.

**Exercise – 1(B)**

**1.1.**

**State, whether the following numbers is rational or not :**(2 + √2)^{2}**Answer**

(2 + √2)^{2 } = 2^{2} + 2 (2) (√2) + (√2)^{2}

= 4 + 4√2 + 2

= 6 + 4√2

Irrational number.

**1.2.**

**State, whether the following numbers is rational or not :**(3 – √3)^{2}**Answer**

(3 – √3)^{2 } = 3^{2} + 2(3) (√3) + (√3)^{2}

= 9 – 6√3 + 3

= 12 – 6√3 = 6 (2 – √3)

Irrational number.

**1.3.**

**State, whether the following numbers is rational or not :**(5 + √5) (5 – √5)**Answer**

(5 + √5)(5 – √5) = (5)^{2} – (√5)^{2}

= 25 – 5 = 20

Rational Number

**1.4.**

**State, whether the following numbers is rational or not :**(√3 – √2)^{2}**Answer**

(√3 – √2)^{2 } = (√3)^{2} – 2(√3)(√2) + (√2)^{2}

= 3 – 2√6 + 2

= 5 – 2√6

Irrational Number

**1.5.**

**State, whether the following numbers is rational or not:**

(32√2)²**Answer**

**1.6.**

(√76√2)

**State, whether the following number is rational or not :**(√76√2)

**Answer**

**2.1.**

**Find the square of :**3√5/5**Answer**

**2.2.**

**Find the square of :**√3 + √2**Answer**

(√3 + √2)^{2} = (√3)^{2} + 2(√3)(√2) + (√2)^{2}

= 3 + 2√6 + 2

= 5 + 2√6

**2.3.**

**Find the square of :**√5 – 2**Answer**

(√5 – 2)^{2} = (√5)^{2} – 2(√5)(2) + (2)^{2}

= 5 – 4√5 + 4

= 9 – 4√5

**2.4.**

**Find the square of :**3 + 2√5**Answer**

(3 + 2√5)^{2} = 3^{2} + 2(3)(2√5) + (2√5)^{2}

= 9 + 12√5 + 20

= 29 + 12√5

**3.1.**

**State, in each case, whether true or false :**

√2 + √3 = √5**Answer**

False

**3.2.**

**State, in each case, whether true or false :****2√4 + 2 = 6**

**Answer**

2√4 + 2 = 2×2 + 2 = 4 + 2 = 6 which is True.

**3.3.**

**State, in each case, whether true or false :****3√7 – 2√7 = √7**

**Answer**

3√7 – 2√7 = √7

True.

**3.4.**

**State, in each case, whether true or false :****2/7 is an irrational number.**

**Answer**

False Because 2/7= 0.28571427 = 0.285714

which is recurring and non-terminating and hence it is rational.

**3.5.**

**State, in each case, whether true or false :****5/11 is a rational number.**

**Answer**

True, because 5/11= 0.45 which is recurring and non-terminating

**3.6.**

**State, in each case, whether true or false :****All rational numbers are real numbers.**

**Answer**

True

**3.7.**

**state, in each case, whether true or false :****All real numbers are rational numbers.**

**Answer**

False

**3.8.**

**State, in each case, whether true or false :****Some real numbers are rational numbers.**

**Answer**

True

**4.1.**

**Given universal set = {****−6, −5(3/4), −√4 , −3/5, −3/8, 0 , 4/5 , 1, 1(2/3), √8 , 3.01 , 𝛑 , 8.47}****Answer**

Given Universal set is

**4.2 .**

**Given universal set = {****−6, −5(3/4), −√4 , −3/5, −3/8, 0 , 4/5 , 1, 1(2/3), √8 , 3.01 , 𝛑 , 8.47}****From the given set, find : Set of irrational numbers **

**Answer**

**4.3. Given universal set = { -6, -5 3/4, -√4, -3/5, -3/8, 0, 4/5, 1, 1 2/3, √8, 3.01, π, 8.47 }`**

**From the given set, find: set of integers**

**Answer**

Given Universal set is

{-6, -5 3/4, -√4, -3/5, -3/8, 0, 4/5, 1, 1 2/3, √8, 3.01, π, 8.47}

We need to find the set of integers.

Set of integers consists of zero, the natural numbers and their additive inverses.

The set of integers is Z.

Z = {…-3, -2, -1, 0, 1, 2, 3,….}

Here, the set of integers is U ∩ Z = {-6, √4, 0, 1}

**4.4.**

**Given universal set = {****−6, −5(3/4), −√4 , −3/5, −3/8, 0 , 4/5 , 1, 1(2/3), √8 , 3.01 , 𝛑 , 8.47}****From the given set, find : Set of non - negative integers**

**Answer**Given universal set = { −6, −5(3/4), −√4 , −3/5, −3/8, 0 , 4/5 , 1, 1(2/3), √8 , 3.01, 𝛑 , 8.47}

From the given set, find : Set of non - negative integers

We need to find the set of non-negative integers.

Set of non-negative integers consists of zero and the natural numbers.

The set of non-negative integers is Z^{+} and Z^{+ }= { 0, 1, 2, 3,……}

Here, the set of integers is U ∩ Z^{+ }= {0, 1}

**5. Use method of contradiction to show that √3 and √5 are irrational numbers.**

**Answer**

**6.1.**

**Prove that the following number is irrational:**√3 + √2**Answer**

3 + √2

Let √3 + √2 be a rational number.

⇒ √3 + √2 = x

Squaring on both the sides, we get

(√3 + √2)^{2} = x^{2}

⇒ 3 + 2 + 2 x √3 x √2 = x^{2}

⇒ x^{2} is an irrational number.

⇒ x is an irrational number.

But we have assume that x is a rational number.

∴ we arrive at a contradiction.

So, our assumption that √3 + √2 is a rational number is wrong.

∴ √3 + √2 is an irrational number.

**6.2.**

**Prove that the following number is irrational:**3 -√2**Answer**

3 – √2

Let 3 – √2 be a rational number.

⇒ 3 – √2 = x

Squaring on both the sides, we get

(3 – √2)^{2} = x^{2}

⇒ 9 + 2 – 2 x 3 x √2 = x^{2}

^{⇒ } 11 – x^{2} = 6√2

⇒ 11 – x^{2} is an irrational number.

⇒ x^{2} is an irrational number.

⇒ x is an irrational number.

But we have assume that x is a rational number.

∴ we arrive at a contradiction.

So, our assumption that 3 – √2 is a rational number is wrong.

∴ 3 – √2 is an irrational number.

**6.3.**

**Prove that the following number is irrational:**√5 – 2**Answer**

√5 – 2

Let √5 – 2 be a rational number.

⇒ √5 – 2 = x

Squaring on both the sides, we get

(√5−2)2 = x^{2}(√5-2)2 = x^{2}

⇒ 5 + 4 – 2x^{2} ×√5 = x^{2}

⇒ 9 – x^{2 }= 4√5

⇒ x^{2} = 9 - 4√5

Here, x is a rational number.

⇒ 9 – x^{2} is an irrational number.

⇒ x^{2} is an irrational number.

⇒ x is an irrational number.

But we have assume that x is a rational number.

∴ we arrive at a contradiction.

So, our assumption that √5 – 2 is a rational number is wrong.

∴ √5 – 2 is an irrational number.

**7. Write a pair of irrational numbers whose sum is irrational.**

**Answer**

√3 + 5 and √5 – 3 are irrational numbers whose sum is irrational.

(√3 + 5) + (√5 – 3) = √3 + √5 + 5 – 3 = √3 + √5 + 2 which is irrational.

**8. Write a pair of irrational numbers whose sum is rational.**

**Answer**

√3 + 5 and 4 – √3 are two irrational numbers whose sum is rational.

(√3 + 5) + (4 – √3) = √3 + 5 + 4 – √3 = 9

**9. Write a pair of irrational numbers whose difference is irrational.**

**Answer**

√3 + 2 and √2 – 3 are two irrational numbers whose difference is irrational.

(√3 + 2) – (√2 – 3) = √3 – √2 + 2 + 3 = √3 – √2 + 5 which is irrational.

**10.Write a pair of irrational numbers whose difference is rational.**

**Answer**

√5 – 3 and √5 + 3 are irrational numbers whose difference is rational.

( √5 – 3 ) – ( √5 + 3 ) = √5 – 3 – √5 – 3 = -6 which is rational.

**11. Write a pair of irrational numbers whose product is irrational.**

**Answer**

Consider two irrational numbers (5 + √2) and (√5 – 2)

Thus, the product, (5 + √2) × (√5 – 2) = 5√5 – 10 + √10 – 2√2 is irrational.

**12.Write a pair of irrational numbers whose product is rational.**

**Answer**

Consider √2 as an irrational number.

√2×√2= √4= 2 which is a rational number.

**13.1.**

**Write in ascending order:**3√5 and 4√3**Answer**

**13.2.**

**Write in ascending order :**23√5 and33√2253 and323**Answer**

**13.3.**

**Write in ascending order :**6√5, 7√3, and 8√2**Answer**

**14.1.**

**Write in descending order:**23√6 and 3√2**Answer**

**14.2.**

**Write in descending order:**7√3 and 3√7**Answer**

**15.1. Compare:**

**Answer**

**15.2.**

**Compare : √24 and √35****Answer**

**16. Insert two irrational numbers between 5 and 6.**

**Answer**

We know that 5 = √25 and 6 = √36

Thus consider the numbers.

√25 < √26 < √27 < √28 < √29 < √30 < √31 < √32 < √33 < √34 < √35 < √36.

Therefore, any two irrational numbers between 5 and 6 is √27 and √28.

**17. Insert five irrational numbers between 2√5 and 3√3.**

**Answer**

We know that 2√5 = √4×54×5 = √20 and 3√3 = √20

Thus, We have, √20 < √21 < √22 < √23 < √24 < √25 < √26 < √27.

So, any five irrational numbers between 2√5 and 3√3 are :

√21, √22, √23, √24, and √26.

**18. Write two rational numbers between √2 and √3.**

**Answer**

**19. Write three rational numbers between √3 and √5.**

**Answer**

**20.1.**

**Answer**

**20.2. Simplify : ∜243/ ∜3**

**Answer**

**20.3.**

**Simplify :**(3 + √2)(4 + √7)**Answer**

(3 + √2)(4 + √7)

= 3×4 + 3×√7 + 4×√2 + √2×√7

= 12 + 3√7 + 4√2 + √14

**20.4.**

**Simplify :**(√3 – √2 )^{2}**Answer**

(√3 – √2 )^{2}

= ( √3 )^{2} + ( √2 )^{2} – 2×√3×√2

= 3 + 2 – 2√6

= 5 – 2√6

### Exercise 1(C)

**1.1.**

**State, with reason, of the following is surd or not :**√180**Answer**

√180 = √2×2×5×3×32×2×5×3×3 = 6√5 which is irrational.

∴ √180 is a surds.

**1.2.**

**State, with reason, of the following is surd or not :**4√27**Answer**

**1.3 . State, with reason, of the following is surd or not :**

**Answer**

**1.4. State, with reason, of the following is surd or not : ∛64**

**Answer**

**1.5.**

3√25×3√40

**State, with reason, of the following is surd or not :**3√25×3√40

**Answer**

**1.6.**

**State, with reason, of the following is surd or not :**…..**Answer**

**1.7.**

**State, with reason, of the following is surd or not :**√π**Answer**

√π not a surds as π is irrational.

**1.8.**

**State, with reason, of the following is surd or not :**√3+√2**Answer**

√3+√2 is not a surds because 3 + √2 is irrational.

**2.1.**

**Write the lowest rationalising factor of :**5√2**Answer**

5√2×5√2 = 5×2 = 10 which is rational.

∴ lowest rationalizing factor is √2

**2.2.**

**Write the lowest rationalising factor of :**√24**Answer**

**2.3.**

**Write the lowest rationalising factor of :**√5 – 3**Answer**

(√5 – 3)(√5 + 3) = (√5)^{2} – (3)^{2} = 5 – 9 = -4

∴ lowest rationalizing factor is (√5 + 3)

**2.4.**

**Write the lowest rationalising factor of :**7 – √7**Answer**

7 – √7

(7 – √7)(7 + √7) = 49 – 7 = 42

Therefore, lowest rationalizing factor is (7 + √7).

**2.5.**

**Write the lowest rationalising factor of :**√18 – √50**Answer**

∴ lowest rationalizing factor is √2

**2.6.**

**Write the lowest rationalising factor of :**√5 – √2**Answer**

√5 – √2

(√5 – √2)(√5 + √2) = (√5)^{2} – (√2)^{2} = 3

Therefore lowest rationalizing factor is √5 + √2

**2.7.**

**Write the lowest rationalising factor of :**√13 + 3**Answer**

(√13 + 3)(√13 – 3) = (√13)^{2} – 3^{2} = 13 – 9 = 4

Its lowest rationalizing factor is √13 – 3.

**2.8.**

**Write the lowest rationalising factor of :**15 – 3√2**Answer**

**2.9.**

**Write the lowest rationalising factor of :**3√2 + 2√3**Answer**

3√2 + 2√3

= ( 3√2 + 2√3 )( 3√2 – 2√3 )

= ( 3√2)^{2} – (2√3)^{2}

= 9×2 – 4×3

= 18 – 12

= 6

Its lowest rationalizing factor is 3√2 – 2√3.

**3.1.**

**Rationalise the denominators of :**3√5**Answer**

3√5×√5√5 = 3√5/5

**3.2.**

**Rationalise the denominators of :**2√3/√5**Answer**

**3.3.**

**Rationalise the denominators of :**1/√3−√2**Answer**

**3.4.**

**Rationalise the denominators of :****3/√5+√2****Answer**

**3.5.**

**Rationalise the denominators of :**2−√3/2+√3**Answer**

**3.6.**

**Rationalise the denominators of :**√3+1/√3−1**Answer**

**3.7.**

**Rationalise the denominators of :**√3−√2/√3+√2**Answer**

**3.8.**

**Rationalise the denominators of :**√6−√5/√6+√5**Answer**

**3.9.**

**Rationalise the denominators of : (2**√5 + 3√2)/(2√5 - 3√2)**Answer**

**4.**

**Find the values of ‘a’ and ‘b’ in each of the following :****1. 2+√3/2-√3 = a + b√3**

**2. √7- 2/√7+2 = a√7 + b**

**3. √3-√2 = a√3 - b√2**

**4. 5+3√2/5-3√2 = a + b√2****Answer**

**1.**

**2.**

**3.**

**4.**

**5.1.**

**Simplify:**22/(2√3+1) + 17/(2√3−1)**Answer**

**5.2.**

**Simplify :**√2/(√6-√2) - √3/(√6+√2)**Answer**

**6.1.**

**If x = (**√5-2)/(√5+2)**find:**x^{2}**Answer**

**6.2. If y**

**=****(**√5+2)/(√5-2)**; find :**y^{2}**Answer**

**6.3.**

**If x =****(**√5-2)/(√5+2) and y =**(**√5+2)/(√5-2)**; find:**xy**Answer**

**6.4.**

**If x = .....****; find :**x^{2}+ y^{2}+ xy.**Answer**

x^{2} + y^{2} + xy

= 161 – 72√5 + 161 +72√5 + 1

= 322 + 1 = 323

**7.1. If m = 1/(3-2√2), find m**

^{2}**Answer**

**7.2. If n = 1/(3+2√2), find n**

^{2}**Answer**

**8.1. If x = 2√3 + 2√2 , find : 1/x**

**Answer**

**8.2. If x = 2√3 + 2√2 , find :**

**x**+ 1/**x****Answer**

**8.3. If x = 2√3 + 2√2 , find : (**

**x**+ 1/**x)**^{2}**Answer**

**9. If x = 1 – √2, find the value of**

**x**- 1/^{2}**x**^{2}**Answer**

**10. If x = 5 - 2√6, find**

**x**+ 1/^{2 }**x**^{2}**Answer**

**11.**

**Show that : ……****Answer**

**12.**

**Rationalise the denominator of :**1/(√3−√2+1)**Answer**

**13.1.**

**If √2 = 1.4 and √3 = 1.7, find the value of :**1/(**√**3-**√2)****Answer**

**13.2.**

**If √2 = 1.4 and √3 = 1.7, find the value of :**1/(3+2**√2)****Answer**

**13.3.**

**Simplify**: 2−√3/√3**Answer**

**14.**

**Evaluate:**(4-√5)/(4+√5) + (4+√5)/(4-√5)**Answer**

**15. If (2+√5)/(2−√5) = x and y = (2-√5)/(2+√5); find the value of x**

^{2}– y^{2}.**Answer**

**16.**

**Simplify**:√18/(5√18 + 3√72 - 2√162)**Answer**