    Question

# Which of the following is an irrational number? i) 2√3 ii) 0.143¯¯¯¯¯¯32 iii) 5.¯¯¯¯¯¯46 iv) √5

A
Only iv)
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B
ii) and iii)
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C
i) and iv)
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D
All of the above
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Solution

## The correct option is C i) and iv)i) 2√3 is product of 2 and √3. We know that √3 is an irrational number and cannot be expressed in the form of pq where p and q are integers and q≠0. When we multiply a rational number with an irrational number, the result is an irrational number. Therefore, 2√3 is an irrational number. ii) 0.143¯¯¯¯¯¯32 is a recurring decimal Let x=0.143¯¯¯¯¯¯32. In this case we see that 143 does not repeat itself but the block 32 repeats itself. Since two digits are repeating, we multiply x by 1000 to get 1000x=143.323232323232...100000x=14332.3232.. Subtracting the two equations we get, 99000x=14159x=1415999000 which is in the form of pq where p and q are integers and q≠0. Hence it is a rational number. iii) 5.¯¯¯¯¯¯46 is again a recurring decimal. Let x=5.4646464646¯46100x=546.¯¯¯¯¯¯4699x=541 Therefore, x=54199 which is in the form of pq where p and q are integers and q≠0.Hence is a rational number. iv) √5 We know that √5 is an irrational number since it cannot be represented in the form of pq where p and q are integers and q≠0. Its value is 2.23606797749979... which is a non-terminating and non-recurring decimal which is the very definition of an irrational number.  Suggest Corrections  0      Similar questions
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