Скачать с ютуб CLASS 10 ll CBSE II NCERT ll REAL NUMBERS в хорошем качестве

CLASS 10 ll CBSE II NCERT ll REAL NUMBERS

Dear students , Here are some questions for you to practice. Practice will make you fast and perfect. 1. Prove that √2 is irrational. Hint: Use proof by contradiction. Assume √2 is rational and express it as a fraction in its simplest form. Then, square both sides and arrive at a contradiction. 2. Prove that √3 is irrational. Hint: Similar to proving √2 irrational, use proof by contradiction and focus on the prime factorization of the numerator and denominator. 3. Prove that √5 is irrational. Hint: Again, use proof by contradiction and analyze the prime factors after squaring both sides as explained in video. 4. If p is a prime number, prove that √p is irrational. Hint: Generalize the proof used for √2, √3, and √5 to any prime number 'p'. 5. Prove that 3 + 2√5 is irrational. Hint: Assume it's rational and try to express √5 as a rational number, leading to a contradiction as explained in the video. 6. Prove that 5 - √3 is irrational. Hint: Similar to the previous question, assume it's rational and isolate √3 to reach a contradiction as explained in the video. 7. Prove that the sum of a rational number and an irrational number is irrational. Hint: Use proof by contradiction. Assume the sum is rational and try to express the irrational number as a rational number. 8. Prove that the product of a non-zero rational number and an irrational number is irrational. Hint: Again, use proof by contradiction and express the irrational number as a rational number, leading to a contradiction. 9. Prove that √2 + √3 is irrational. Hint: Assume it's rational, square both sides, and try to isolate either √2 or √3 to get a contradiction. 10. Prove that √2 - √3 is irrational. Hint: Use proof by contradiction. Assume √2 - √3 is rational and express it as a fraction. Some more helpful tips: * The key to most of these proofs is to use the method of "proof by contradiction." * Focus on the prime factorization of numbers and how they relate to rational and irrational numbers. * Practice these proofs and try to understand the logic behind them. Please let me know if you need help with any of these questions or want more practice problems!