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Class 10 Maths Chapter 1 Practice Question Answers - Real Numbers

Q1. Use Euclid’s division lemma to show that the square of any positive integer is either of form 3m or 3m + 1 for some integer m.

Class 10 Maths Chapter 1 Practice Question Answers - Real Numbers  View Answer

Ans:
Let x be any positive integer and y = 3.
By Euclid’s division algorithm;
x =3q + r (for some integer q ≥ 0 and r = 0, 1, 2 as r ≥ 0 and r < 3)
Therefore,
x = 3q, 3q + 1 and 3q + 2
As per the given question, if we take the square on both the sides, we get;
x2 = (3q)2 = 9q2 = 3.3q2
Let 3q2 = m
Therefore,
x2 = 3m ………………….(1)
x2 = (3q + 1)2
= (3q)2 + 12 + 2 × 3q × 1
= 9q2 + 1 + 6q
= 3(3q2 + 2q) + 1
Substituting 3q2+2q = m we get,
x2 = 3m + 1 ……………………………. (2)
x2 = (3q + 2)2
= (3q)2 + 22 + 2 × 3q × 2
= 9q2 + 4 + 12q
= 3(3q2 + 4q + 1) + 1
Again, substituting 3q2 + 4q + 1 = m, we get,
x2 = 3m + 1…………………………… (3)
Hence, from eq. 1, 2 and 3, we conclude that the square of any positive integer is either of form 3m or 3m + 1 for some integer m.

Q2. Express each number as a product of its prime factors:
(i) 140
(ii) 156
(iii) 3825
(iv) 5005
(v) 7429

Class 10 Maths Chapter 1 Practice Question Answers - Real Numbers  View Answer

Ans:
(i) 140
Using the division of a number by prime numbers method, we can get the product of prime factors of 140.
Therefore, 140 = 2 × 2 × 5 × 7 × 1 = 22 × 5 × 7

(ii) 156
Using the division of a number by prime numbers method, we can get the product of prime factors of 156.
Hence, 156 = 2 × 2 × 13 × 3 = 22 × 13 × 3

(iii) 3825
Using the division of a number by prime numbers method, we can get the product of prime factors of 3825.
Hence, 3825 = 3 × 3 × 5 × 5 × 17 = 32 × 52 × 17

(iv) 5005
Using the division of a number by prime numbers method, we can get the product of prime factors of 5005.
Hence, 5005 = 5 × 7 × 11 × 13 = 5 × 7 × 11 × 13

(v) 7429
Using the division of a number by prime numbers method, we can get the product of prime factors of 7429.
Hence, 7429 = 17 × 19 × 23 = 17 × 19 × 23

Q3. Given that HCF (306, 657) = 9, find LCM (306, 657).

Class 10 Maths Chapter 1 Practice Question Answers - Real Numbers  View Answer

Ans: As we know that,
HCF × LCM = Product of the two given numbers
So,
9 × LCM = 306 × 657
LCM = (306 × 657)/9 = 22338
Therefore, LCM(306,657) = 22338

Q4. Prove that 3 + 2√5 is irrational.

Class 10 Maths Chapter 1 Practice Question Answers - Real Numbers  View Answer

Ans: Let 3 + 2√5 be a rational number.
Then the co-primes x and y of the given rational number where (y ≠ 0) is such that:
3 + 2√5 = x/y
Rearranging, we get,
2√5 = (x/y) – 3
√5 = 1/2[(x/y) – 3]
Since x and y are integers, thus, 1/2[(x/y) – 3] is a rational number.
Therefore, √5 is also a rational number. But this confronts the fact that √5 is irrational.
Thus, our assumption that 3 + 2√5 is a rational number is wrong.
Hence, 3 + 2√5 is irrational.

Q.5: Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion:
(i) 13/3125
(ii) 17/8
(iii) 64/455
(iv) 15/1600 

Class 10 Maths Chapter 1 Practice Question Answers - Real Numbers  View Answer

Ans:
Note: If the denominator has only factors of 2 and 5 or in the form of 2m × 5n then it has a terminating decimal expansion.
If the denominator has factors other than 2 and 5 then it has a non-terminating repeating decimal expansion.

(i) 13/3125
Factoring the denominator, we get,
3125 = 5 × 5 × 5 × 5 × 5 = 55
Or
= 20 × 55
Since the denominator is of the form 2m × 5n then, 13/3125 has a terminating decimal expansion.

(ii) 17/8
Factoring the denominator, we get,
8 = 2× 2 × 2 = 23
Or
= = 23 × 50
Since the denominator is of the form 2m × 5n then, 17/8 has a terminating decimal expansion.

(iii) 64/455
Factoring the denominator, we get,
455 = 5 × 7 × 13
Since the denominator is not in the form of 2m × 5n, therefore 64/455 has a non-terminating repeating decimal expansion.

(iv) 15/1600
Factoring the denominator, we get,
1600 = 26 × 52
Since the denominator is in the form of 2m × 5n, 15/1600 has a terminating decimal expansion.

Q.6: The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If they are rational, and of the form, p/q what can you say about the prime factors of q?
(i) 43.123456789
(ii) 0.120120012000120000. . .

Class 10 Maths Chapter 1 Practice Question Answers - Real Numbers  View Answer

Ans:
(i) 43.123456789
Since it has a terminating decimal expansion, it is a rational number in the form of p/q and q has factors of 2 and 5 only.
(ii) 0.120120012000120000. . .
Since it has a non-terminating and non-repeating decimal expansion, it is an irrational number.

Q.7: Check whether 6n can end with the digit 0 for any natural number n.

Class 10 Maths Chapter 1 Practice Question Answers - Real Numbers  View Answer

Ans: If the number 6n ends with the digit zero (0), then it should be divisible by 5, as we know any number with a unit place as 0 or 5 is divisible by 5.
Prime factorization of 6n = (2 × 3)n
Therefore, the prime factorization of 6n doesn’t contain the prime number 5.
Hence, it is clear that for any natural number n, 6n is not divisible by 5 and thus it proves that 6n cannot end with the digit 0 for any natural number n.

Q.8: What is the HCF of the smallest prime number and the smallest composite number?

Class 10 Maths Chapter 1 Practice Question Answers - Real Numbers  View Answer

Ans:The smallest prime number = 2
The smallest composite number = 4
Prime factorisation of 2 = 2
Prime factorisation of 4 = 2 × 2
HCF(2, 4) = 2
Therefore, the HCF of the smallest prime number and the smallest composite number is 2.

Q.9: Using Euclid’s Algorithm, find the HCF of 2048 and 960.

Class 10 Maths Chapter 1 Practice Question Answers - Real Numbers  View Answer

Ans:
2048 > 960
Using Euclid’s division algorithm,
2048 = 960 × 2 + 128
960 = 128 × 7 + 64
128 = 64 × 2 + 0
Therefore, the HCF of 2048 and 960 is 64.

Q.10: Find HCF and LCM of 404 and 96 and verify that HCF × LCM = Product of the two given numbers.

Class 10 Maths Chapter 1 Practice Question Answers - Real Numbers  View Answer

Ans: Prime factorisation of 404 = 2 × 2 × 101
Prime factorisation of 96 = 2 × 2 × 2 × 2 × 2 × 3 = 25 × 3
HCF = 2 × 2 = 4
LCM = 25 × 3 × 101 = 9696
HCF × LCM = 4 × 9696 = 38784
Product of the given two numbers = 404 × 96 = 38784
Hence, verified that LCM × HCF = Product of the given two numbers.

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

1. What is a real number in class 10?
Ans. In class 10, a real number is a number that can be either a rational number or an irrational number. It includes numbers such as whole numbers, integers, fractions, decimals, and square roots of non-perfect squares.
2. What is the difference between a rational number and an irrational number?
Ans. A rational number is a number that can be expressed as a fraction, where both the numerator and denominator are integers. An irrational number, on the other hand, cannot be expressed as a fraction and has non-repeating, non-terminating decimal expansions.
3. How do you classify a number as a real number?
Ans. A number can be classified as a real number if it can be plotted on the number line. This means that it should either be a rational number or an irrational number. Real numbers encompass all possible numbers, including positive and negative numbers, fractions, and decimals.
4. Give an example of a rational number and an irrational number.
Ans. An example of a rational number is 7/3, which can be expressed as a fraction. An example of an irrational number is √2, which cannot be expressed as a fraction and has a non-repeating, non-terminating decimal expansion.
5. How are real numbers used in everyday life?
Ans. Real numbers are used in various aspects of everyday life, such as measuring distances, calculating prices, keeping track of time, and estimating quantities. They are essential in fields like science, engineering, finance, and statistics, where precise measurements and calculations are required.
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