Class 10 Exam  >  Class 10 Notes  >  Mathematics (Maths) Class 10  >  Short Answer Questions: Real Numbers

Class 10 Maths Chapter 1 Question Answers - Real Numbers - 1

Class 10 Maths Chapter 1 Question Answers - Real Numbers - 1

Q1: Prove that √3 is irrational.

Let 3 be rational in the simplest form of P/q.
i.e., p and q are integers having no common factor other than 1 and q ≠ 0.
Now,  3 = p/q  
Squaring both sides, we have
⇒ (√3)2 = (p/q)2
⇒ 3 = p2/q2  
⇒ 3q2 = p............(1)
Since 3q2 is divisible by 3
∴ p2 is also divisible by 3
⇒ p is divisible by 3 ..........(2)
Let p = 3c for some integer ‘c’.
Substituting p = 3c in (1), we have:
⇒ 3q = (3c)2
⇒ 3q2 = 9c2
⇒ q2  = 3c2
3cis divisible by 3
∴ q2 is divisible by 3
⇒ q is divisible by 3   ...(3)
From (2) and (3)
3 is a common factor of ‘p’ and ‘q’. But this contradicts our assumption that p and q are having no common factor other than 1.
∴ Our assumption that 3 is rational is wrong.
Thus, 3 is an irrational.

Q2: The decimal representation of 6/1250 will terminate after how many places of decimal?

1. Simplify the fraction to its lowest form.

2. Now we can either convert the lowest fraction into decimals or a quicker way to do this is to convert the denominator to a multiple of 10 so that it is easier to convert it into decimals.
3. So we multiply with and divide by 2to convert the denominator to 104. 
24 2

Class 10 Maths Chapter 1 Question Answers - Real Numbers - 1 

Therefore, this representation will terminate after 4 decimal places.

Question for Short Answer Questions: Real Numbers
Try yourself:Q.2. If HCF of a and b is 12 and product of these numbers is 1800. Then what is LCM of these numbers?
View Solution
 

Q3: If ‘p’ is prime, prove that √p is irrational.

Let p be rational in the simplest form a/b, where p is prime.
∴ a and b are integers having no common factor other than 1 and b ≠ 0.
⇒ Now, p = a/b
⇒ Squaring both sides, we have
Class 10 Maths Chapter 1 Question Answers - Real Numbers - 1
⇒ pb= a2 ...(1)
 Since pb2 is divisible by p, a2 is also divisible by p.
⇒ a is also divisible by p    ...(2)
Let a = pc for some integer c.
⇒ Substituting a = pc in (1), we have
pb2 = (pc)2
⇒ pb= p2c2
⇒ b2  = pc2
pc2 is divisible by p,
∴ b2 is divisible by p
⇒ b is divisible by p    ...(3)
From (2) and (3),
⇒ p is a common factor of ‘a’ and ‘b’. But this contradicts our assumption that a and b are co-prime.
∴ Our assumption that p is rational is wrong. Thus, p is irrational if p is prime.

Q4: Find the HCF of 18 and 24 using prime factorisation.

Using factor tree method, we have:
Class 10 Maths Chapter 1 Question Answers - Real Numbers - 1
Class 10 Maths Chapter 1 Question Answers - Real Numbers - 1
∴ 18 = 2 × 3 × 3 = 2 × 32
24 = 2 × 2 × 2 × 3 = 23 × 3
HCF = Product of common prime factors with lowest powers.
⇒ HCF (18, 24) = 3 × 2  = 6


Q5: Find the LCM of 10, 30 and 120.

Class 10 Maths Chapter 1 Question Answers - Real Numbers - 1            Class 10 Maths Chapter 1 Question Answers - Real Numbers - 1             Class 10 Maths Chapter 1 Question Answers - Real Numbers - 1  
∴ 10 = 2 × 5 = 21 × 51
30 = 2 × 3 × 5 = 21 × 31 × 51
120 = 2 × 2 × 2 × 3 × 5 = 23 × 31 × 51
LCM = Product of each prime factor with highest powers
⇒ LCM of 10, 30 and 120 = 23 × 3 × 5 = 120.

Q6: Find the LCM and HCF of 1296 and 5040 by prime factorisation method.

Class 10 Maths Chapter 1 Question Answers - Real Numbers - 1  and     Class 10 Maths Chapter 1 Question Answers - Real Numbers - 1
∴ 5040 = 2 × 2 × 2 × 2 × 3 × 3 × 5 × 7
= 24 × 32 × 5 × 7 and
⇒ 1296 =  2 × 2 × 2 × 2 × 3 × 3 × 3 × 3
= 24 × 34
∴ LCM = Product of each prime factor with highest powers
= 24 × 34 × 5 × 7
= 16 × 81 × 5 × 7 = 45360
HCF = Product of common prime factors with lowest powers
= 24 × 32
= 16 × 9 = 144

Q7: Show that  3√2 is irrational.

Let 3√2 be a rational number
∴ p/q = 3√2 where p and q are prime to each other and q ≠ 0.
∴ p/3q = √2   ...(1)
Since, p is integer and 3q is also integer (3q≠ 0).
∴ p/3q is a rational number.
From (1), √2 is a rational number.
But this contradicts the fact that √2 is irrational. Therefore, our assumption that 3√2 is rational is incorrect.
Hence, 3√2 is irrational.

Q8: Show that 2 - √3 is an irrational number.

Let  2 - √3 is rational.
∴ It can be expressed as p/q where p and q are integers (prime to each other) such that q ≠ 0.
∴  2 - √3   =  p/q  
Class 10 Maths Chapter 1 Question Answers - Real Numbers - 1     ...(1)

∵ p is an integer}
∴ q is an integer}
⇒ p/q is a rational number.
∴ Class 10 Maths Chapter 1 Question Answers - Real Numbers - 1 is a rational number. ...(2)
From (1) and (2), √3 is a rational number. This contradicts the fact that √3 is an irrational number.
∴ Our assumption that (2 -√3) is a rational number is not correct. Thus, (2 - √3) is irrational.

Q9: State whether Class 10 Maths Chapter 1 Question Answers - Real Numbers - 1  is a rational number or not.

Class 10 Maths Chapter 1 Question Answers - Real Numbers - 1 = 1.23333..... is a non-terminating repeating decimal.
Class 10 Maths Chapter 1 Question Answers - Real Numbers - 1 is a rational number.
3/4 is in the form of p/q, where q ≠ 0  [Here 4 ≠ 0]
∴ 3/4 is a rational number.
Since the sum of two rational numbers is a rational number.
Therefore, Class 10 Maths Chapter 1 Question Answers - Real Numbers - 1  is a rational number.

Q10: The LCM of two numbers is 45 times their HCF. If one of the numbers is 225 and sum of their LCM and HCF is 1150, find the other number.

One of the numbers = 225
Let the other number = x
Also LCM = 45 (HCF)                 ...(1)
And LCM + HCF =1150
⇒ (45 HCF) + HCF = 1150
⇒ 46 HCF = 1150
⇒ HCF = 1150/46 = 25
From (1),
LCM = 45 × 25
∴ LCM × HCF = (45 × 25) × 25
Now,  LCM × HCF = Product of the numbers
∴  x × 225 = (45 × 25) × 25
Class 10 Maths Chapter 1 Question Answers - Real Numbers - 1

= 125
Thus, the required number is 125.

Q11: Three different containers contain 496 litres, 403 litres and 713 litres of a mixture. What is the capacity of the biggest container that can measure all the different quantities exactly?

For the capacity of the biggest container, we have to find the HCF.

HCF: By Long Division method

First find the HCF of two numbers, 496 and 403
Class 10 Maths Chapter 1 Question Answers - Real Numbers - 1
The HCF of 496 and 403 = 31
Now find the HCF of 31 and 713
Class 10 Maths Chapter 1 Question Answers - Real Numbers - 1
HCF of 713 and 31 is 31
So, the maximum capacity is 31 liters.


Q12:  Prove that (5+32) is an irrational number.

Sol: 

Let (5+3√2) is a rational number.
∴ (5+3√2) =  a/b [where ‘a’ and ‘b’ are co-prime integers and b ≠ 0

Class 10 Maths Chapter 1 Question Answers - Real Numbers - 1
Class 10 Maths Chapter 1 Question Answers - Real Numbers - 1
Class 10 Maths Chapter 1 Question Answers - Real Numbers - 1

‘a’ and ‘b’ are integers,
∴  Class 10 Maths Chapter 1 Question Answers - Real Numbers - 1 is a rational number.
⇒ √2 is a rational number.
But this contradicts the fact that √2 is an irrational number.
∴ Our assumption that (5+3√2) is a rational is incorrect.
⇒ (5+3√2) is an irrational number.

Q13: Prove that 3-√5 is an irrational number.

Sol: 

Let (3-√5) is a rational number.
∴ 3-√5 = p/q , such that p and q are co-prime integers and q ≠ 0.

Class 10 Maths Chapter 1 Question Answers - Real Numbers - 1
Class 10 Maths Chapter 1 Question Answers - Real Numbers - 1
Class 10 Maths Chapter 1 Question Answers - Real Numbers - 1

Since, p and q are integers,
∴ Class 10 Maths Chapter 1 Question Answers - Real Numbers - 1 is a rational number.
⇒ √5 is a rational number.
But this contradicts the fact that √5 is an irrational number.
∴ Our assumption that (3-√5)  is a rational number’ is incorrect.
⇒ (3-√5)  is an irrational number.

Q14: Show that there is no positive integer ‘p’ for which Class 10 Maths Chapter 1 Question Answers - Real Numbers - 1is rational.

Sol: 

If possible let there be a positive integer p for which Class 10 Maths Chapter 1 Question Answers - Real Numbers - 1 = a/b is equal to a rational i.e. where a and b are positive integers.

Class 10 Maths Chapter 1 Question Answers - Real Numbers - 1
Class 10 Maths Chapter 1 Question Answers - Real Numbers - 1
Class 10 Maths Chapter 1 Question Answers - Real Numbers - 1

Now

Class 10 Maths Chapter 1 Question Answers - Real Numbers - 1
Also,

Class 10 Maths Chapter 1 Question Answers - Real Numbers - 1
Since a, b are integer

Class 10 Maths Chapter 1 Question Answers - Real Numbers - 1 are rationals

⇒ (p + 1) and (p – 1) are perfect squares of positive integers, which is not possible (because any two perfect squares differ at least by 3). Hence, there is no positive integer p for which Class 10 Maths Chapter 1 Question Answers - Real Numbers - 1 is rational.

Q15: Prove that Class 10 Maths Chapter 1 Question Answers - Real Numbers - 1is irrational, where p and q are primes.

Sol: 

Let Class 10 Maths Chapter 1 Question Answers - Real Numbers - 1be rational
Let it be equal to ‘r’
i.e. Class 10 Maths Chapter 1 Question Answers - Real Numbers - 1
Squaring both sides, we have

Class 10 Maths Chapter 1 Question Answers - Real Numbers - 1

Class 10 Maths Chapter 1 Question Answers - Real Numbers - 1
⇒ Class 10 Maths Chapter 1 Question Answers - Real Numbers - 1 ...(i)
Since, p, q are both rationals
Also, r2 is rational (∵ r is rational)
∴ RHS of (i) is a rational number
⇒ LHS of (i) should be rational i.e.q  should be rational.
But q is irrational (∵ p is prime).
∴ We have arrived at a contradiction.
Thus, our supposition is wrong.
Hence, p+√q  is irrational.

The document Class 10 Maths Chapter 1 Question Answers - Real Numbers - 1 is a part of the Class 10 Course Mathematics (Maths) Class 10.
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FAQs on Class 10 Maths Chapter 1 Question Answers - Real Numbers - 1

1. What are real numbers and how are they classified?
Ans.Real numbers are the set of numbers that include all the rational and irrational numbers. They can be classified into various categories, including natural numbers, whole numbers, integers, rational numbers (fractions), and irrational numbers (numbers that cannot be expressed as a fraction).
2. How do you perform operations with real numbers?
Ans. Operations with real numbers include addition, subtraction, multiplication, and division. To perform these operations, one must follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
3. What is the significance of the number line in understanding real numbers?
Ans. The number line is a visual representation of real numbers, where each point corresponds to a real number. It helps in understanding the relative position of numbers, comparing them, and performing operations like addition and subtraction by moving left or right along the line.
4. Can you explain the difference between rational and irrational numbers?
Ans. Rational numbers are numbers that can be expressed as the quotient of two integers (where the denominator is not zero), such as 1/2 or 3. Conversely, irrational numbers cannot be expressed as a simple fraction, meaning their decimal representation is non-repeating and non-terminating, like √2 or π.
5. How do you solve problems involving real numbers in Class 10 exams?
Ans. To solve problems involving real numbers in Class 10 exams, it is essential to understand the properties of real numbers, practice different types of problems, and apply the appropriate mathematical operations. Familiarity with the concepts and regular practice can significantly enhance problem-solving skills.
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