# CBSE Class 10 Mathematics Real Numbers Worksheet Set D

Read and download free pdf of CBSE Class 10 Mathematics Real Numbers Worksheet Set D. Students and teachers of Class 10 Real Numbers can get free printable Worksheets for Class 10 Real Numbers in PDF format prepared as per the latest syllabus and examination pattern in your schools. Standard 10 students should practice questions and answers given here for Real Numbers in Grade 10 which will help them to improve your knowledge of all important chapters and its topics. Students should also download free pdf of Class 10 Real Numbers Worksheets prepared by school teachers as per the latest NCERT, CBSE, KVS books and syllabus issued this academic year and solve important problems provided here with solutions on daily basis to get more score in school exams and tests

## Real Numbers Worksheet for Class 10

Class 10 Real Numbers students should refer to the following printable worksheet in Pdf in standard 10. This test paper with questions and answers for Grade 10 Real Numbers will be very useful for exams and help you to score good marks

### Class 10 Real Numbers Worksheet Pdf

Real Numbers

Q.- Insert a rational and an irrational number between 2 and 3.

Sol. If a and b are two positive rational numbers such that ab is not a perfect square of a rational number, then √ab is an irrational number lying between a and b. Also, if a,b are
rational numbers, then a+b/2 is a rational number between them.
∴ A rational number between 2 and 3 is 2+3 /2 = 2.5
An irrational number between 2 and 3 is √2×3=  √6

Q.-  Prove that
(i) √2 is irrational number
(ii) √3 is irrational number
Similarly √5, √7, √11…... are irrational numbers.

Sol. (i) Let us assume, to the contrary, that 2 is rational.
So, we can find integers r and s (≠ 0) such that .√2= r/s
Suppose r and s not having a common factor other than 1. Then, we divide by the common factor to get ,√2=a/b
where a and b are coprime.
So, b √2= a.

Squaring on both sides and rearranging, we get 2b2 = a2. Therefore, 2 divides a2. Now, by
Theorem it following that 2 divides a.
So, we can write a = 2c for some integer c.
Substituting for a, we get 2b2 = 4c2, that is,
b2 = 2c2.

This means that 2 divides b2, and so 2 divides b (again using Theorem with p = 2).
Therefore, a and b have at least 2 as a common factor.

But this contradicts the fact that a and b have no common factors other than 1.
This contradiction has arisen because of our incorrect assumption that √2 is rational.
So, we conclude that √2 is irrational.

(ii) Let us assume, to contrary, that √3 is rational. That is, we can find integers a and b
(≠ 0) such that √3=a/b

Suppose a and b not having a common factor other than 1, then we can divide by the common factor, and assume that a and b are coprime.
So, b √3= a .
Squaring on both sides, and rearranging, we get 3b2 = a2.
Therefore, a2 is divisible by 3, and by Theorem, it follows that a is also divisible by 3.
So, we can write a = 3c for some integer c.

Substituting for a, we get 3b2 = 9c2, that is,b2 = 3c2.

This means that b2 is divisible by 3, and so b is
also divisible by 3 (using Theorem with p = 3).
Therefore, a and b have at least 3 as a common factor.

But this contradicts the fact that a and b are coprime.
This contradicts the fact that a and b are coprime.

This contradiction has arisen because of our incorrect assumption that √3 is rational.
So, we conclude that √3 is irrational.

Q.- Using Euclid’s division algorithm, find the
H.C.F. of [NCERT]
(i) 135 and 225
(ii) 196 and 38220
(iii) 867 and 255

Sol.(i) Starting with the larger number i.e., 225, we get:
225 = 135 × 1 + 90
Now taking divisor 135 and remainder 90, we
get 135 = 90 × 1 + 45
Further taking divisor 90 and remainder 45,
we get 90 = 45 × 2 + 0
∴  Required H.C.F. = 45 (Ans.)

(ii) Starting with larger number 38220, we get:
38220 = 196 × 195 + 0
Since, the remainder is 0
=> H.C.F. = 196 (Ans.)

(iii) Given number are 867 and 255
=>  867 = 255 × 3 + 102    (Step-1)
255 = 102 × 2 + 51             (Step-2)
102 = 51 × 2 + 0                 (Step-3)
=> H.C.F. = 51  (Ans.)

Q.- Show that one and only one out of n; n + 2 or n + 4 is divisible by 3, where n is any positive integer.

Sol. Consider any two positive integers a and b such that a is greater than b, then according to Euclid’s division algorithm:
a = bq + r; where q and r are positive integers and 0 ≤ r < b
Let a = n and b = 3, then
a = bq + r => n = 3q + r; where 0 ≤ r < 3.
r = 0 => n = 3q + 0 = 3q
r = 1 => n = 3q + 1 and r = 2 =>  n = 3q + 2
If n = 3q; n is divisible by 3

If n = 3q + 1; then n + 2 = 3q + 1 + 2
= 3q + 3; which is divisible by 3
=> n + 2 is divisible by 3

If n = 3q + 2; then n + 4 = 3q + 2 + 4
= 3q + 6; which is divisible by 3
=> n + 4 is divisible by 3

Hence, if n is any positive integer, then one and only one out of n, n + 2 or n + 4 is divisible by 3.
Hence the required result.

Q.- Consider the number 6n, where n is a natural number. Check whether there is any value of ϵ N for which 6n is divisible by 7.

Sol. Since, 6 = 2 × 3; 6n = 2n × 3n
=> The prime factorisation of given number 6n
=> 6n is not divisible by 7. (Ans)

More question-

1. If 7 x 5 x 3 x 2 + 3 is composite number? Justify your answer

2. Show that any positive odd integer is of the form 4q + 1 or 4q + 3 where q is a positive integer

3. Show that 8n cannot end with the digit zero for any natural number n

4. Prove that 3√2 is irrational 5

5. Prove that √2 + √5 is irrational

6. Prove that 5 – 2 √3 is an irrational number

7. Prove that √2 is irrational

8. Use Euclid’s Division Algorithms to find the H.C.F of a) 135 and 225 (45)

b) 4052 and 12576 (4)

c) 270, 405 and 315 (45)

9. Using Euclid’s division algorithm, check whether the pair of numbers 50 and 20 are co-prime or not.

10. Find the HCF and LCM of 26 and 91 and verify that LCM X HCF = Product of two numbers (13,182)

11. Explain why 29 is a terminating decimal expansion 23 x 53

12. 163 will have a terminating decimal expansion. State true or false .Justify your answer. 150

13. Find HCF of 96 and 404 by prime factorization method. Hence, find their LCM. (4, 9696)

14. Using prime factorization method find the HCF and LCM of 72, 126 and 168 (6, 504)

15. If HCF (6, a) = 2 and LCM (6, a) = 60 then find a (20)

16. given that LCM (77, 99) = 693, find the HCF (77, 99) (11)

17. Find the greatest number which exactly divides 280 and 1245 leaving remainder 4 and 3 (138)

18. The LCM of two numbers is 64699, their HCF is 97 and one of the numbers is 2231. Find the other (2813)

19. Two numbers are in the ratio 15: 11. If their HCF is 13 and LCM is 2145 then find the numbers (195,143)

20. Express 0.363636………… in the form a/b (4/11)

21. Write the HCF of smallest composite number and smallest prime number

22. Write whether 2√45 + 3√20 on simplification give a rational or an irrational number 2√5 (6)

23. State whether 10.064 is rational or not. If rational, express in p/q form

24. Write a rational number between √2 and √3

25. State the fundamental theorem of arithmetic

26. The decimal expansion of the rational number 74 will terminate after ………. Places

23 . 54

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