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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.

a²- b² = (a + b) (a - b) .......(1)

straight a plus straight b equals fraction numerator 3 minus square root of 5 over denominator 3 plus square root of 5 end fraction plus fraction numerator 3 plus square root of 5 over denominator 3 minus square root of 5 end fraction equals space fraction numerator 3 minus square root of 5 over denominator 3 plus square root of 5 end fraction cross times fraction numerator 3 minus square root of 5 over denominator 3 minus square root of 5 end fraction equals fraction numerator 3 minus square root of 5 over denominator 3 plus square root of 5 end fraction cross times fraction numerator 3 minus square root of 5 over denominator 3 minus square root of 5 end fraction plus fraction numerator 3 minus square root of 5 over denominator 3 plus square root of 5 end fraction cross times fraction numerator 3 plus square root of 5 over denominator 3 plus square root of 5 end fraction equals space fraction numerator left parenthesis 3 minus square root of 5 right parenthesis squared over denominator 9 minus 5 end fraction plus fraction numerator left parenthesis 3 plus square root of 5 right parenthesis squared over denominator 9 minus 5 end fraction equals space fraction numerator left parenthesis 9 plus 5 minus 6 square root of 5 right parenthesis plus left parenthesis 9 plus 5 plus 6 square root of 5 right parenthesis over denominator 4 end fraction equals space 28 over 4 equals space 7 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 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

straight a minus straight b space equals space fraction numerator 3 minus square root of 5 over denominator 3 plus square root of 5 end fraction minus fraction numerator 3 plus square root of 5 over denominator 3 minus square root of 5 end fraction equals space fraction numerator 3 minus square root of 5 over denominator 3 plus square root of 5 end fraction cross times space fraction numerator 3 minus square root of 5 over denominator 3 minus square root of 5 end fraction minus space fraction numerator 3 plus square root of 5 over denominator 3 minus square root of 5 end fraction cross times space fraction numerator 3 plus square root of 5 over denominator 3 plus square root of 5 end fraction equals fraction numerator left parenthesis 3 minus square root of 5 right parenthesis squared over denominator 9 minus 5 end fraction minus fraction numerator left parenthesis 3 plus square root of 5 right parenthesis squared over denominator 9 minus 5 end fraction equals space fraction numerator left parenthesis 3 minus square root of 5 right parenthesis squared minus left parenthesis 3 plus square root of 5 right parenthesis squared over denominator 4 end fraction equals space minus fraction numerator 12 square root of 5 over denominator 4 end fraction equals negative 3 square root of 5 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 space space space space space space space space space space space space... left parenthesis 3 right parenthesis

From (1), (2) and (3),

straight a squared minus straight b squared equals space left parenthesis 7 right parenthesis left parenthesis negative 3 square root of 5 right parenthesis equals negative 21 square root of 5

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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.

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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.