1 a 0 b I. value of [(𝑎+𝑏) 2 − 𝑎2 − 𝑏²] / [(𝑎+𝑏) 2 - ( 𝑎−𝑏)²] II. value of 1 / 2(𝑎𝑏3 + 𝑎𝑏)

Question

1 > a > 0 > b I. value of [(𝑎+𝑏) 2 − 𝑎2 − 𝑏²] / [(𝑎+𝑏) 2 - ( 𝑎−𝑏)²] II. value of 1 / 2(𝑎𝑏3 + 𝑎𝑏)

Accepted Answer

- Simplify Quantity I: - Numerator: $(a+b)^2-a^2-b^2=(a^2+2ab+b^2)-a^2-b^2=2ab$. - Denominator: $(a+b)^2-(a-b)^2=(a^2+2ab+b^2)-(a^2-2ab+b^2)=4ab$. - Hence Quantity I $=\dfrac{2ab}{4ab}=\dfrac{1}{2}$ (since $ab eq0$). - Evaluate sign of Quantity II: - Interpret II as $ frac{1}{2}(ab^3+ab)= frac{1}{2}ab(b^2+1)$. - Given $a>0$ and $b<0$, we have $ab<0$, while $b^2+1>0$, so $ frac{1}{2}ab(b^2+1)<0$. - Comparison: - Quantity I $= frac{1}{2}>0$ and Quantity II $<0$, therefore Quantity I $>$ Quantity II. FINAL_ANSWER: b

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