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

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.

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.

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 = 2⁴ = 2⁴ × 5⁰

I.e. 16 can be expressed as 2^m × 5ⁿ

∴ 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 =} 2⁰ × 5³

i.e. 125 can be expressed as 2^m×5ⁿ

∴ 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 = 2¹× 7¹

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

∴ 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 = 3² × 5¹

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

∴ 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 = 2¹ × 5²

i.e. 50 can be expressed as 2^m × 5ⁿ

∴ 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 = 2³ × 5¹

i.e. 40 can be expressed as 2^m × 5ⁿ

∴ 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 = 3¹ × 5²

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

∴ 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 = 2¹ × 5³

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  = 2² + 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  = 3² + 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. 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 = 3² + 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 = x²

⇒ 3 + 2 + 2 x √3 x √2 = x²

⇒ x² 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 = x²

⇒ 9 + 2 – 2 x 3 x √2 = x²

^⇒  11 – x² = 6√2

⇒ 11 – x² is an irrational number.

⇒ x² 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 = x²(√5-2)2 = x²

⇒ 5 + 4 – 2x² ×√5 = x²

⇒ 9 – x² = 4√5

⇒ x² = 9 - 4√5

Here, x is a rational number.

⇒ 9 – x² is an irrational number.

⇒ x² 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. 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 – 3² = 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√32. √7- 2/√7+2 = a√7 + b3. √3-√2 = a√3 - b√24. 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: x²

Answer

6.2. If y =  (√5+2)/(√5-2); find : y²

Answer

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

Answer

6.4. If x = .....; find : x² + y² + xy.

Answer

x² + y² + xy

= 161 – 72√5 + 161 +72√5 + 1

= 322 + 1 = 323

7.1. If m = 1/(3-2√2), find m²

Answer

7.2. If n = 1/(3+2√2), find n²

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 x² - 1/x²

Answer

10. If x = 5 - 2√6, find x² + 1/x²

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 x² – y².

Answer

16. Simplify :√18/(5√18 + 3√72 - 2√162)

Answer