This chapter builds the key idea of irrational numbers, how to simplify surds, and how to solve equations involving square roots. It is important for both board exams and competitive tests because many questions reduce to rationalization, using identities like , and checking domain conditions to avoid extraneous roots.
15
Minutes
15
Questions
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Marking
Q1. The value of is equal to:
Q2. If satisfies , then equals:
Q3. Assertion (A): is irrational.
Reason (R): For any two irrational numbers and , the difference is necessarily irrational.
Which option is correct?
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.
Q4. Let . Then equals:
Q5. The value of satisfying (where the expression is defined) is:
Q6. Find the value of .
Q7. If is a rational number such that is also rational, then the value of is:
Q8. Evaluate .
Q9. Assertion (A): For all , .
Reason (R): .
Choose the correct option.
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.
Q10. Let be a non-negative integer such that is rational. Then equals:
Q11. Find the value of .
Q12. If for real , then equals:
Q13. If for real , then equals:
Q14. Consider the following statements for the equation .
Assertion (A): The equation has no real solution.
Reason (R): After squaring, we get or , and for both values the right-hand side becomes negative, so the original equation cannot hold.
Assertion (A) is true, but Reason (R) is false.
Assertion (A) is false, but Reason (R) is true.
Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A).
Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).
Q15. If for real , then equals: