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Assertion (A): is an irrational number. Reason (R): The square root of every prime number is irrational. Which option is correct?

Class 10Real NumbersNumber Systems
  • ABoth A and R are true, and R is the correct explanation of A
  • BBoth A and R are true, but R is NOT the correct explanation of A
  • CA is true but R is false
  • DA is false but R is true

Step-by-Step Solution

is irrational (A true). The square root of any prime is irrational (R true), and 2 is prime, so R directly explains why is irrational.
Answer: A) Both A and R are true, and R is the correct explanation of A

Key idea: Sum/product of rational and irrational

**Useful results**:

1. The sum or difference of a rational and an irrational is IRRATIONAL.
Proof: if (rational + irrational) = rational, then irrational = rational − rational = rational. Contradiction.

2. The product of a NON-ZERO rational and an irrational is IRRATIONAL.

3. The sum/product of two irrationals can be RATIONAL or IRRATIONAL.
Example: (irrational); (rational).
Example: (rational); (irrational).

These are the rules you cite when proving expressions like ' is irrational'.

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