CBSE Class 10 Mathematics Real Numbers Worksheet Set G

Fill in the blanks/tables with suitable information:

Question. (2+√5/3) is .................... number.
Answer : irrational

Question. The HCF of two numbers is 27 and their LCM is 162. If one of the numbers is 54, the other number is .................... .
Answer : 81

Question. If a = (2² × 3³ × 5⁴) and b = (2³ × 3² × 5) then HCF (a, b) = .................... .
Answer : 180

Question. A decimal number 0.8 can be expressed in its simplest form as .................... .
Answer : x = 8/9

Question. Product of two numbers is 18144 and their HCF is 6, then their LCM is .................... .
Answer : 324

Question. The decimal expression of the rational number 23/2² × 5 will terminate after ............. decimal place(s).
Answer : 2

Question. The HCF of smallest composite number and the smallest prime number is .................... .
Answer : 2

Question. If a and b are positive integers, then
HCF (a,b) x LCM (a,b) /ab = ....................
Answer : 1

Question. ................... is the H.C.F. of two consecutive even numbers.
Answer : 2

Question. If two positive integers p and q can be expressed as p = a²b³ and q = a⁴b; a, b being prime numbers, then LCM (p, q) is.....................
Answer : a⁴b³

Short Answer Type Questions

Question. There are 44 boys and 32 girls in a class. These students arranged in rows for a prayer in such a way that each row consists of only either boys or girls, and every row contains an equal number of students. Find the minimum number of rows in which all students can be arranged.
Answer : 44 = 2² x 11
32 = 2⁵
HCF = 2² = 4
Therefore, minimum number of rows in which all srudents can be arranged = 44/4 + 32/4 = 11 + 8 = 19 rows

Question. Find the largest number which divides 70 and 125 leaving reminder 5 and 8 respectively.
Answer : 70 – 5 = 65
125 – 8 = 117
65 = 5 x 13
117 = 3² x 13
HCF = 13
i.e., 13 is the largest number that will divide 65 and 117.

Question. The LCM of two number is 14 times their HCF. The sum of LCM and HCF is 600.If one number is 280, then find the other number.
Answer : HCF = x
LCM = 14 x HCF = 14x
LCM + HCF = 600
14x + x = 500
15x = 600
x = 40
HCF = 40 and LCM = 14 x 40 = 560
Since, LCM x HCF = product of the numbers
560 x 40 = 280 x second number
Second number = 80

Question. 144 Cartons of coke can and 90 cartons of Pepsi can are to be stacked in a canteen. If each stack is of the same height and is to contain cartons of the same drink. What would be the greater number of cartons each stack would have?
Answer : 144 = 2⁴ x 3²
90 = 2 x 3² x 5
HCF = 2 x 3² = 18 cartons

Question. Can two numbers have 15 as their HCF and 175 as their LCM? Give reasons
Answer : No, two numbers cannot have 15 as their HCF and 175 as LCM because,
HCF of the numbers must be a factor of the LCM. Therefore, LCM = k x HCF (k ∈ N)
175 = k x 15
k = 175/15 = 35/3 ∉ N

Question. Prove that √𝑛 is not a rational number if n is not a perfect square.
Answer : Let on the contrary say it is rational .
Then
√𝑛 = 𝑝/𝑞, 𝑞 ≠ 0 where p and q are coprime integers.
so 𝑛 = 𝑝2𝑞2
𝑝2 = 𝑛𝑞2
This shows p divides q
which is a contradiction.
Hence √𝑛 is irrational if n is not a perfect square.

Question. Two bells toll at intervals of 24 minutes and 36 minutes respectively. If they toll together at 9 am, after how many minutes do they toll together again, at the earliest?
Answer : 24 = 2³ x 3
36 = 2² x 3²
LCM = 23 x 3² = 8 x 9 = 72
After 72 minutes = 1 hr 12 minutes they toll together.

Question. Prove that the difference and quotient of ( 3 + 2√3) and ( 3 - 2√3) are irrational.
Answer : The difference of ( 3 + 2√3) and ( 3 - 2√3) is 4√3 which is irrational.
Dividing ( 3 + 2√3) by ( 3 - 2√3) we get -7 - 4√3 which is irrational

Question. Explain why 17 x 5 x 11 x 3 x 2 + 2 x11 is a composite number.
Answer : 17 x 5 x 11 x 3 x 2 + 2 x 11 = 2 x 11 (17 x 5 x 3 + 1)
= 2 x 11 (255 + 1)
= 2 x 11 x 256
= 2 x 11 x 2⁸
This number has more than 2 prime factors.
Therefore, 17 x 5 x 11 x 3 x 2 + 2 x 11 is a composite number.

Question. Find the largest number that will divide 398 , 436 and 542 leaving reminders 7,11 and 15 respectively.
Answer : 398 – 7 = 391
436 – 11 = 425
542 – 15 = 527
391 = 17 x 23
425 = 5² x 17
527 = 17 x 31
HCF = 17
i.e., 17 is the largest number that will divide 398, 436 and 542 leaving remainders 7, 11 and 15 respectively.

1. If 7x5x3x2 + 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. Prove that √2 + √5 is irrational

4. Prove that 5 - 2√3 is an irrational number

5. Prove that √2 is irrational

6. 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)

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

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

9. 163 will have a terminating decimal expansion. State true or false .Justify your answer.150

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

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

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

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

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

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

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

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

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

19. Write whether 2√45 + 3√20 on simplification give a rational or an irrational number 2√5

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

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

22. State the fundamental theorem of arithmetic

PREPARED BY: MAHABOOB PASHA IX – X BOYS

Please click the below link to access CBSE Class 10 Maths Real Numbers (6)