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Number Systems

Short Answer Type

141.

If find the value of $straight x squared plus 1 over straight x squared.$

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142. Let ‘a’ he a rational number and ‘b’ be an irrational number. Is ‘ab’ necessarily an irrational? Justify your answer with an example.

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143.

If $straight a equals fraction numerator 3 minus square root of 5 over denominator 3 plus square root of 5 end fraction space and space straight b space equals fraction numerator 3 plus square root of 5 over denominator 3 minus square root of 5 end fraction$ , find a² - b².

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144.
If $straight x equals 3 plus 2 square root of 2 comma$ check whether $straight x plus 1 over straight x$ is rational or irrational.

$x equals 3 plus 2 square root of 2 space space space space space space space space space space space space space space space space space.... left parenthesis 1 right parenthesis$

$1 over straight x equals fraction numerator 1 over denominator 3 plus 2 square root of 2 end fraction$
$equals fraction numerator 1 over denominator 3 plus 2 square root of 2 end fraction cross times fraction numerator 3 minus 2 square root of 2 over denominator 3 minus 2 square root of 2 end fraction equals fraction numerator 3 minus 2 square root of 2 over denominator left parenthesis 3 right parenthesis squared minus left parenthesis 2 square root of 2 right parenthesis squared end fraction equals space fraction numerator 3 minus 2 square root of 2 over denominator 9 minus 8 end fraction equals fraction numerator 3 minus 2 square root of 2 over denominator 1 end fraction equals space 3 minus 2 square root of 2 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space.... left parenthesis 2 right parenthesis$

Adding (1) and (2), we get

straight x plus 1 over straight x equals left parenthesis 3 plus 2 square root of 2 right parenthesis plus left parenthesis 3 minus 2 square root of 2 right parenthesis equals 6

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145.

If $fraction numerator 2 over denominator square root of 3 plus square root of 5 end fraction plus fraction numerator 5 over denominator square root of 3 minus square root of 5 end fraction equals straight a square root of 3 plus straight b square root of 5 comma$ find a and b.