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,
\(LCM (p, q, r) = \frac {p\times q \times r\times \text{HCF}(p,q, r)}{\text{HCF}(p,q) \times \text{HCF}(q,r) \times \text{HCF}(p,r)}\)
\(HCF (p, q, r) = \frac {p\times q \times r\times \text{LCM}(p,q, r)}{\text{LCM}(p,q)\times \text{LCM}(q,r) \times \text{LCM}(p,r)}\)
\(HCF (p, q, r) = \frac {p\times q \times r\times \text{LCM}(p,q, r)}{\text{LCM}(p,q)\times \text{LCM}(q,r) \times \text{LCM}(p,r)}\)
Common Mistakes to avoid:
1. Prime factorisation: Always use prime factors before finding \(HCF/LCM\)
2. \(HCF \): Take the lowest power of common prime factors.
3. \(LCM\): Take the highest power of all prime factors.
4. For application problems:
(i) \(HCF\): Used when splitting, dividing or arranging into the largest equal groups. For example, words like maximum capacity, largest size, equal groups, maximum length.
Order error in \(HCF\) remainder problems: First subtract the remainders before calculating the \(HCF\).
(ii) \(LCM\): Used when finding a common minimum time, distance or cycle where events coincide. For example, words like minimum time, least number, starting together, bells ringing simultaneously.
Order error in \(LCM\) remainder problems: First find the \(LCM\) then add\adjust the remainder at the end.
5. For irrationality proofs: Assume \(\Rightarrow\) Simplify \(\Rightarrow\) Contradiction \(\Rightarrow\) Conclude.
Exam Tips & Tricks:
1. Always double-check that \(HCF \le\) smallest given number and
\(LCM \ge \) largest given number.
2. In application problems, write the answers with units like \(12\ liters\) or \(36\ seconds\) not just \(12\) or \(36\).
3. In every proof for irrationality, highlight the phrase: "where \(p\) and \(q\) are co-prime integers and \(q\neq 0\).
4. Never skip the theoretical justification. Write, "We know that if a prime \(p\) divides \(a^2\), then \(p\) divides \(a\).