Real Numbers:
The chapter 'Real Number' is a completely scorable chapter and it carries six marks in board exam. It covers the concepts like 'Fundamental Theorem of Arithmetic, Relationaship between HCF and LCM, About Irrationals and Theorems on numbers.
Most possible variation of question types that we can expect in board exam as per previous year question paper are discussed below.
| Total Marks \(5\) - \(6\) | |
| Variation \(1\) | Variation \(2\) |
Total Mark \(= 6\)
|
Total Mark \(= 5\)
|
To prepare well for the board examination, it is necessary to understand the following concepts clearly.
- Fundamental Theorem of Arithmetic - Prime factorization method, HCF and LCM
- Relationship between HCF and LCM, Word problems on HCF and LCM
- Irrationals, Proving Irrationality
- Problems on composite numbers
| Important Concept (Learning Outcomes) | Expected Question Type | Concept dealt with |
|
Fundamental Theorem of Arithmetic - Prime factorization method, HCF and LCM
|
Sec A -MCQ,
Sec B
|
|
|
Relationship between HCF and LCM, Word problems on HCF and LCM
|
Sec A, Assertion Reasoning | |
|
Irrationals, Proving Irrationality
|
Sec A, Sec B | |
|
Problems on composite numbers
|
Sec C |
Let us recall the concepts in Real Numbers:
1. HCF: Product of the smallest power of each common prime factor in the numbers.
2. LCM: Product of the greatest power of each prime factors involved in the numbers.
3. For any two integers \(a\) and \(b\), \(HCF(a,b) \times LCM(a,b) = a \times b.\)
4. Fundamental theorem of Arithmetic: Every Composite number can be expressed (factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occurs.
5. If \(p\) is a prime and \(p\) divides \(a^2\), then \(p\) divides \(a\), where \(a\) is a positive integer.
6. \(HCF ( p, q, r) \times LCM (p, q, r) \neq p \times q \times r \), where \(p, q, r\) are positive integers,
\(LCM (p, q, r) = \frac {p\times q \times r\times \text{HCF}(p,q, r)}{\text{HCF}(p,q) \times \text{HCF}(q,r) \times \text{HCF}(p,r)}\)
\(HCF (p, q, r) = \frac {p\times q \times r\times \text{LCM}(p,q, r)}{\text{LCM}(p,q)\times \text{LCM}(q,r) \times \text{LCM}(p,r)}\)
\(HCF (p, q, r) = \frac {p\times q \times r\times \text{LCM}(p,q, r)}{\text{LCM}(p,q)\times \text{LCM}(q,r) \times \text{LCM}(p,r)}\)