The chapter “Real Numbers” builds the foundation for understanding rational and irrational numbers, their decimal behavior (terminating vs. recurring), and key tools like Euclid’s algorithm and properties of surds. These concepts frequently appear in board exams and are also essential for competitive exams because they directly test reasoning with gcd-lcm, rationality conditions, and number classification.
15
Minutes
15
Questions
1 / -0
Marking
Q1. If and , what is the value of ?
Q2. Express the repeating decimal as a fraction in lowest terms.
Q3. Using the Euclidean algorithm, find .
Q4. Assertion (A): For positive integers , if is rational then both and are rational.
Reason (R): Both and are algebraic numbers.
Both A and R are true and R is the correct explanation of A
Both A and R are true but R is not the correct explanation of A
A is true but R is false
Both A and R are false
Q5. For which positive integers is the number a rational number?
All perfect squares
only
No positive integer
of the form
Q6. Find the HCF of 495 and 385 by applying the Euclidean algorithm.
11
55
5
35
Q7. Two positive integers have HCF and LCM . If their sum is , find the two integers.
Q8. Let and . The value of HCF is
Q9. Given HCF and LCM for positive integers , the number of ordered pairs possible is
Q10. Assertion (A): For a positive integer which is not a perfect square, is irrational.
Reason (R): If is irrational, then for any integer , is irrational.
Both A and R are true and R is the correct explanation of A.
Both A and R are true but R is not the correct explanation of A.
A is true but R is false.
A is false but R is true.
Q11. Let the HCF of and be and their LCM be . Find the value of .
Q12. Let be positive integers with and . How many ordered pairs of positive integers satisfy these conditions?
Q13. A rational number is expressed in lowest terms. What is the length of the repetend (period of the repeating block) in its decimal expansion?
Q14. Assertion (A): If a rational number (in lowest terms) has denominator not divisible by or , then its decimal expansion is purely recurring.
Reason (R): Because , there exists the least positive integer such that ; this is the length of the repeating period.
Both A and R are true, and R is the correct explanation of A.
Both A and R are true, but R is not the correct explanation of A.
A is true but R is false.
A is false but R is true.
Q15. Let and be distinct positive integers with . If is a rational number, which of the following must be true?
Both and are prime numbers.
The product is a perfect square, but at least one of or is not a perfect square.
It is impossible; is always irrational for distinct positive integers with .
Both and are perfect squares.