CBSE Class 10 Mathematics Real Numbers Assignment Set H

Question. Find HCF and LCM of 378, 180 and 420 by prime factorization method. Is HCF X LCM of these numbers equal to the product of given three numbers?
Answer : 378 = 3³ x 2 x 7
180 = 3² x 2² x 5 
420 = 3 x 2² x 5 x 7
HCF = 3 X 2 = 6 
LCM = 3³ x 2² x 5 x 7 = 3780 
HCF x LCM = 3780 x 6 = 22,680
Product of numbers = 378 x 180 x 420 = 28576800
No HCF x LCM is not equal to product of three numbers

Question. A woman wants to organise her birthday party. She was happy on her birthday but there was a problem that she does not want to serve fast food to her guests because she is very health conscious. She as 15 apples and 40 bananas at home and decided to serve them. She want to distribute fruits among guests. She does not want to discriminate among guests so she decided to distribute equally among all. So
i. How many guests she can invite?
ii. How many apples and banana will each guest get?
Answer : (i)HCF of(15,40) = 5
Fruits will be distributed equally among 5 guests
(ii)Out of 15 apples each guest will get 15 /5 = 3 apples
Out of 40 banana each guest will get 40/5 = 8 bananas

Question. The sum of LCM and HCF of two numbers is 7380.If the LCM of these numbers is 7340 more than their HCF. Find the product of the two numbers
Answer : LCM + HCF = 7380
LCM – HCF = 7340
2LCM = 14720
LCM = 14720/2
LCM = 7360
LCM + HCF = 7380
7360+ HCF = 7380
HCF = 7380 – 7360
HCF = 20 
HCF x LCM = product of numbers
20 x 7360 = product of numbers
147200 = product of numbers

Question. A charitable trust donates 28 different books of Maths,16 different books of science and 12 different books of Social Science to the poor students. Each student is given maximum number of books of only one subject of his interest and each student got equal number of books
i. Find the number of books each student got.
ii. Find the total number of students who got books.
Answer : (i) HCF of 28,16 and 12 is 4
Therefore maximum number of books each student get is 4
(ii) Number of maths books 28/4 = 7
Number of science books 16/4 = 4
Number of social science = 12/4 = 3
Total books = 7 + 4 + 3 = 14

Question. A hall has a certain number of chairs. Guests want to sit in different groups like in pairs, triplets, quadruplets, fives and sixes etc. When organiser arranges chairs in such pattern like 2’s, 3’s ,4’s.5’s and 6’s then 1,2,3,4 and 5 chairs are left respectively. But when he arranges in 11’s no chair will be left
i. In the hall how many chairs are available?
a) 407
b) 143
c) 539
d) 209
ii. If one chair is added to the total number of chairs, how many chairs will be left when arranged in 11’s
Answer : (i) 539 chairs
(ii) if 1 chair is added as 539 is already divisible by 11 ,1 chair will be left

Question. Prove that √5 is an irrational number
Answer : Assume that √5 is a rational number
Therefore √5 = 𝑝𝑞 p and q are co primes and q ≠ 0 (1)
p = √5q
Squaring both the sides
p² = 5q²
Thus 5 is a factor of p²
Therefore 5 is a factor of p (1)
Let p = 5c where c is some integer, then we have
p² = 25c²
Substituting p² = 5q²
5q² = 25c²
q² = 5c² (1)
Thus 5 is a factor of q² and also 5 is also a factor of q
Thus 5 is a factor of both p and q .But this is a contradiction to the fact that p and q are co primes (1)
Thus our assumption is wrong that √5 is a rational number
Hence √5 𝑖𝑠 an irrational number