Evaluate: (i) `|(x^2+x y+y^2,x+y ),(x^2-x y+y^2,x-y)|` (ii) `|(1,(log)_b a),((log)_a b,1)|`

Explanation

(i)`|(x^2+x y+y^2,x+y ),(x^2-x y+y^2,x-y)|`

= `(x^{2}+x y+y^{2})(x-y)-(x+y)(x^{2}-x y+y^{2}) `

= `(x^{3}-y^{3})-(x^{3}+y^{3}) `

=`x^{3}-y^{3}-x^{3}-y^{3}=-2 y^{3}`

(ii) `|(1,(log)_b a),((log)_a b,1)|`

=`1 \times 1-\log _{b} a \times \log _{a} b `

As `log_a b=1/(log_b a)`

`=1-1`

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