Real Numbers — Class 10 solved problems

26 problems from this chapter, each solved step by step.

Prove that √5 is irrational and find whether 3/13 is a terminating or non-terminating repeating decimal.
√5 is irrational. (3)/(13) is a non-terminating repeating decimal.
Find whether 3/13 is rational or irrational without performing long division
The number (3)/(13) is a rational number.
Prove that √5 is irrational and hence show that 3 + 2 √5 is irrational.
Therefore, 3 + 2√5 is irrational.
Prove that √3 is irrational.
√3 is irrational.
4. [HCF/LCM reasoning] Two bells ring at intervals of 18 minutes and 24 minutes respectively. If they ring together at 9:00 a.m., at what time will they next ri
The bells will next ring together at 10:12 a.m.
हिंदी में जवाब दो: Two bells ring at intervals of 18 minutes and 24 minutes respectively. If they ring together at 9:00 a.m., at what time will they next ring t
The bells will next ring together at 10:12 a.m.
Two bells ring at intervals of 18 minutes and 24 minutes respectively. If they ring together at 9:00 a.m., at what time will they next ring together?
The bells will next ring together at 10:12 a.m.
Prove that the square root of 3 is irrational.
√3 is irrational.
Express each number as a product of its prime factors: (i) 140
140 = 2² × 5 × 7
February 14 is celebrated as International Book Giving Day and many countries in the world celebrate this day. Some people in India also started celebrating thi
(i) 48 books are arranged in each stack. (ii) 7 stacks are used to arrange all the Mathematics books. (iii) (a) The total number of stacks used is 14. (iii) (b)
A school has invited 42 Mathematics teachers, 56 Physics teachers and 70 Chemistry teachers to attend a Science workshop. Find the minimum number of tables requ
The minimum number of tables required is 12.
Check whether there is any natural number 'n' for which (14)^n ends with the digit '0' or '5'.
No, there is no natural number 'n' for which (14)^n ends with the digit '0' or '5'.
Two positive integers m and n are expressed as m = p^5 q² and n = p³ q^4, where p and q are prime numbers. The LCM of m and n is:
p^5 q^4
(a) If √2 is given as an irrational number, then prove that (5 - 2√2) is an irrational number. OR (b) Check whether 6^n can end with the digit 0 for any natural
(a) 5 - 2√2 is an irrational number. (b) 6^n cannot end with the digit 0 for any natural number n.
Prove that (3 + 2√5) is an irrational number, given that √5 is an irrational number.
Therefore, 3 + 2√5 is an irrational number.
Find the greatest number which divides 65 and 117 completely.
13
The HCF and LCM of two numbers are 99 and 765, respectively. If one of the numbers is 45, what is the other number?
The other number is 1683.
The HCF and LCM of two numbers are 11 and 770, respectively. If one of the numbers is 45, what is the other number?
The other number is approximately 188.22. However, since HCF and LCM are defined for integers, and the result is not an integer, there might be an error in the
The HCF and LCM of two numbers are 9 and 360, respectively. If one of the numbers is 45, what is the other number?
The other number is 72.
Consider the numbers 4^n, where n is a natural number. Check whether there is any value of n for which 4^n ends with the digit zero.
No, there is no value of n for which the number 4^n ends with the digit zero.
Prove that √3 is irrational.
√3 is an irrational number.
Prove that the square root of 5 is irrational.
The square root of 5 is irrational.
Find the HCF of 96 and 404 by the prime factorisation method. Hence, find their LCM.
The HCF of 96 and 404 is 4, and their LCM is 9696.
Find the HCF and LCM of 6, 72 and 120, using the prime factorisation method.
The HCF of 6, 72 and 120 is 6, and the LCM is 360.
Show that 5 - √3 is irrational.
Therefore, 5 - √3 is an irrational number.
Prove that the square root of 2 is irrational.
√2 is irrational.

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