COMPLETE
MATHEMATICS
BY- HARSH
, CLASS 10 COMPLETE MATHEMATICS THEORY + IMPORTANT QUES
CHAPTER 1 REAL NUMBERS
•FUNDAMENTAL THEOREM OF ARITHMETIC :- Every composite number can be expressed as a product o
this factorisation is unique, apart from the order in which the prime occur.
EXAMPLE:- 6 = 3*2: 120 = 2*2*2*3*5
•HCF = product of smallest power of each common prime factor in the numbers.
•LCM = product of greatest power of each prime factor involved in the numbers.
•For any two positive integers ‘a’ and ‘b’, HCF(a,b)*LCM(a,b) = product of ‘a’ and ‘b’.
•Let ‘p’ be a prime number. IF ‘p’ DIVIDES ‘a²’ THEN ‘p’ ALSO DIVIDES ‘a’.
IMPORTANT QUESTIONS:-
1. Express 360 as a product of its prime factors.
2. Find the prime factorization of 5005.
3. Find the HCF and LCM of 120 and 144 using the prime factorization method.
4. The HCF of two numbers is 23 and their LCM is 1449. If one of the numbers is 161, find the other.
5. Find the HCF and LCM of 17, 23, and 29.
6. Explain why 7 × 11 × 13 + 13 is a composite number.
7. Check whether 6^n can end with the digit 0 for any natural number n.
8. There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while Ravi ta
minutes for the same. Suppose they both start at the same point and at the same time, and go in the same direct
how many minutes will they meet again at the starting point?
9. Three bells ring at intervals of 12, 15, and 18 minutes respectively. If they start ringing together, after what time w
ring together?
10. Determine the largest number that divides 245 and 1029 leaving remainder 5 in each case.
, CLASS 10 COMPLETE MATHEMATICS THEORY + IMPORTANT QUES
CHAPTER 1 REAL NUMBERS
• PROVING √n IS IRRATIONAL NUMBER
-Assumption of √n as a rational number which can be represented in the form a/b in which b ≠ 0, a, b are integers a
-Represent √n = a/b and square both the sides.
-As a result, n = a²/b² ⇒ nb² = a²
-So, a² is divisible by ‘n’ also a is divisible by ‘n’.
-For some integer c, show a = nc, and square both sides.
-Substitute nb² = a² in the above result.
-So, b² is also divisible by n and also b is divisible by n.
-So, a, b are not co-primes since they have a common factor n.
-Assumption was wrong hence, √n is irrational number.
IMPORTANT QUESTIONS
1. Prove that √2 is irrational.
2. Prove that √3 is irrational.
3. Prove that √5 is irrational.
4. Prove that √7 is irrational.
, CLASS 10 COMPLETE MATHEMATICS THEORY + IMPORTANT QUES
CHAPTER 1 REAL NUMBERS
•x - √n IS IRRATIONAL NUMBER
-Assumption of x - √n as rational number which can be represented in the form a/b
in which b ≠ 0, a & b are integers and co-primes. (Here x is also an integer).
-Represent x - √n = a/b => x - a/b = √n
-Since x, a & b are integers therefore x - a/b is rational and so √n is also rational.
-But we know that √n is irrational.
-∴ Our assumption was wrong that x - √n is rational number.
-Hence, x - √n is irrational number.
IMPORTANT QUESTIONS
1. Prove that 3 + 2√5 is irrational.
2. Prove that 5 - √3 is irrational.
3. Prove that 4 + √2 is irrational.
4. Prove that √3 + √5 is irrational.
MATHEMATICS
BY- HARSH
, CLASS 10 COMPLETE MATHEMATICS THEORY + IMPORTANT QUES
CHAPTER 1 REAL NUMBERS
•FUNDAMENTAL THEOREM OF ARITHMETIC :- Every composite number can be expressed as a product o
this factorisation is unique, apart from the order in which the prime occur.
EXAMPLE:- 6 = 3*2: 120 = 2*2*2*3*5
•HCF = product of smallest power of each common prime factor in the numbers.
•LCM = product of greatest power of each prime factor involved in the numbers.
•For any two positive integers ‘a’ and ‘b’, HCF(a,b)*LCM(a,b) = product of ‘a’ and ‘b’.
•Let ‘p’ be a prime number. IF ‘p’ DIVIDES ‘a²’ THEN ‘p’ ALSO DIVIDES ‘a’.
IMPORTANT QUESTIONS:-
1. Express 360 as a product of its prime factors.
2. Find the prime factorization of 5005.
3. Find the HCF and LCM of 120 and 144 using the prime factorization method.
4. The HCF of two numbers is 23 and their LCM is 1449. If one of the numbers is 161, find the other.
5. Find the HCF and LCM of 17, 23, and 29.
6. Explain why 7 × 11 × 13 + 13 is a composite number.
7. Check whether 6^n can end with the digit 0 for any natural number n.
8. There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while Ravi ta
minutes for the same. Suppose they both start at the same point and at the same time, and go in the same direct
how many minutes will they meet again at the starting point?
9. Three bells ring at intervals of 12, 15, and 18 minutes respectively. If they start ringing together, after what time w
ring together?
10. Determine the largest number that divides 245 and 1029 leaving remainder 5 in each case.
, CLASS 10 COMPLETE MATHEMATICS THEORY + IMPORTANT QUES
CHAPTER 1 REAL NUMBERS
• PROVING √n IS IRRATIONAL NUMBER
-Assumption of √n as a rational number which can be represented in the form a/b in which b ≠ 0, a, b are integers a
-Represent √n = a/b and square both the sides.
-As a result, n = a²/b² ⇒ nb² = a²
-So, a² is divisible by ‘n’ also a is divisible by ‘n’.
-For some integer c, show a = nc, and square both sides.
-Substitute nb² = a² in the above result.
-So, b² is also divisible by n and also b is divisible by n.
-So, a, b are not co-primes since they have a common factor n.
-Assumption was wrong hence, √n is irrational number.
IMPORTANT QUESTIONS
1. Prove that √2 is irrational.
2. Prove that √3 is irrational.
3. Prove that √5 is irrational.
4. Prove that √7 is irrational.
, CLASS 10 COMPLETE MATHEMATICS THEORY + IMPORTANT QUES
CHAPTER 1 REAL NUMBERS
•x - √n IS IRRATIONAL NUMBER
-Assumption of x - √n as rational number which can be represented in the form a/b
in which b ≠ 0, a & b are integers and co-primes. (Here x is also an integer).
-Represent x - √n = a/b => x - a/b = √n
-Since x, a & b are integers therefore x - a/b is rational and so √n is also rational.
-But we know that √n is irrational.
-∴ Our assumption was wrong that x - √n is rational number.
-Hence, x - √n is irrational number.
IMPORTANT QUESTIONS
1. Prove that 3 + 2√5 is irrational.
2. Prove that 5 - √3 is irrational.
3. Prove that 4 + √2 is irrational.
4. Prove that √3 + √5 is irrational.
