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Worksheet
Real Numbers encompass all rational and irrational numbers, forming a complete and continuous number line essential for various mathematical concepts.
Real Numbers - Practice Worksheet
Strengthen your foundation with key concepts and basic applications.
This worksheet covers essential long-answer questions to help you build confidence in Real Numbers from Mathematics for Class X (Mathematics).
Questions
Explain the Fundamental Theorem of Arithmetic and its significance.
Think about how prime numbers are the building blocks of all numbers and why their uniqueness in factorization matters.
How does Euclid's division algorithm work, and how is it used to find the HCF of two numbers?
Remember, the key is to keep dividing the divisor by the remainder until the remainder is zero.
Prove that √2 is irrational using the Fundamental Theorem of Arithmetic.
Start by assuming √2 is rational and look for a contradiction in the prime factorization.
Explain why the product of a non-zero rational number and an irrational number is irrational.
Assume the opposite and show that it leads to a contradiction with the definition of irrational numbers.
Find the LCM and HCF of 12, 15, and 21 using the prime factorization method.
Break each number into its prime factors and use the highest powers for LCM and the lowest common powers for HCF.
Why can't the number 6ⁿ end with the digit 0 for any natural number n?
Consider the prime factors needed for a number to end with 0 and check if 6ⁿ has those factors.
Show that the sum of a rational number and an irrational number is irrational.
Assume the sum is rational and derive a contradiction.
Explain the concept of terminating and non-terminating repeating decimal expansions of rational numbers.
Look at the denominator's prime factors to determine the nature of the decimal expansion.
Prove that 3 + 2√5 is irrational.
Assume the number is rational and show that it leads to √5 being rational.
Find the HCF and LCM of 306 and 657, given that their HCF is 9.
Use the relationship HCF × LCM = Product of the two numbers to find the LCM.
Real Numbers - Mastery Worksheet
Advance your understanding through integrative and tricky questions.
This worksheet challenges you with deeper, multi-concept long-answer questions from Real Numbers to prepare for higher-weightage questions in Class X Mathematics.
Questions
Explain the Fundamental Theorem of Arithmetic and its significance in the study of numbers.
Think about how prime factorization is used in various proofs and calculations.
Prove that √5 is irrational using the Fundamental Theorem of Arithmetic.
Use proof by contradiction and the theorem that if a prime divides a², it divides a.
Compare and contrast the methods to find HCF and LCM using prime factorization and Euclid's division algorithm.
Consider the computational efficiency and the nature of the results each method provides.
Show that the product of three numbers is not necessarily equal to the product of their HCF and LCM.
Choose specific numbers to test the statement.
Explain why 7 × 11 × 13 + 13 is a composite number.
Factor out the common term to reveal the composite nature.
Prove that 3 + 2√5 is irrational.
Assume the opposite and reach a contradiction.
Find the HCF and LCM of 12, 15, and 21 using prime factorization.
Identify the common and all prime factors with their highest powers.
Explain the concept of irrational numbers and give two examples not mentioned in the textbook.
Think beyond square roots and π.
How does the Fundamental Theorem of Arithmetic help in proving the irrationality of numbers?
Consider the role of unique factorization in contradiction proofs.
Given that HCF (306, 657) = 9, find LCM (306, 657).
Use the formula connecting HCF, LCM, and the product of two numbers.
Real Numbers - Challenge Worksheet
Push your limits with complex, exam-level long-form questions.
The final worksheet presents challenging long-answer questions that test your depth of understanding and exam-readiness for Real Numbers in Class X.
Questions
Prove that the square root of any prime number is irrational using the Fundamental Theorem of Arithmetic.
Consider the uniqueness of prime factorization and how it applies to both sides of the equation.
Explain why the sum of a rational number and an irrational number is always irrational.
Think about the properties of rational and irrational numbers under addition.
Given two numbers, explain how to find their HCF and LCM using the Fundamental Theorem of Arithmetic.
Remember that the HCF is about commonality, and the LCM is about the union of all primes.
Why can't a number ending with the digit 0 be a perfect square unless it ends with an even number of zeros?
Consider the prime factorization of a number ending with zeros and the definition of a perfect square.
Demonstrate that the product of two irrational numbers can be either rational or irrational, with examples.
Think about multiplying an irrational number by itself versus by another irrational number.
Using Euclid's division lemma, show that the square of any positive integer is of the form 3m or 3m + 1.
Consider the possible remainders when an integer is divided by 3 and square each case.
Explain the significance of the Fundamental Theorem of Arithmetic in proving the irrationality of numbers.
Think about how unique factorization leads to contradictions in proofs by contradiction.
How does the decimal expansion of a rational number relate to its denominator's prime factors?
Consider how the denominator's prime factors affect the divisibility in the base 10 system.
Prove that there are infinitely many prime numbers.
Use proof by contradiction and consider the properties of prime numbers.
Discuss the application of the Fundamental Theorem of Arithmetic in cryptography.
Think about how prime factorization is used in public-key cryptography.
Explore the world of Polynomials, understanding their types, degrees, and operations to solve algebraic expressions and equations effectively.
Explore the methods to solve a pair of linear equations in two variables, including graphical, substitution, elimination, and cross-multiplication techniques.
Explore the world of quadratic equations, learning to solve them using various methods like factoring, completing the square, and the quadratic formula.
A chapter that explores sequences where each term after the first is obtained by adding a constant difference, focusing on their properties, nth term, and sum formulas.
