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    Which one of the following pairs is not correctly matched?
    Question

    Which one of the following pairs is not correctly matched?

    A.

    a+b=8, ab=10=>a3+b3=272a+b=8,\ ab=10 \Rightarrow a^3+b^3=272 \\​​

    B.

    ab=4, ab=2=>a3b3=80 a-b=4,\ ab=2 \Rightarrow a^3-b^3=80 \\​​

    C.

    a+b=15=>(a10)3+(b5)3=0a+b=15 \Rightarrow (a-10)^3+(b-5)^3=0 \\​​

    D.

    a+b=15, ab=14=>ab=13 a+b=15,\ ab=14 \Rightarrow a-b=13​​

    Correct option is B

    Formulas Used:}
    a3+b3=(a+b)33ab(a+b)a^3 + b^3 = (a+b)^3 - 3ab(a+b)​​
    a3b3=(ab)(a2+ab+b2)a^3 - b^3 = (a-b)(a^2 + ab + b^2)
    (ab)2=(a+b)24ab(a-b)^2 = (a+b)^2 - 4ab​​
    Solution:
    (a)
    a3+b3=833(10)(8)=512240=272Correct\begin{aligned}a^3 + b^3 &= 8^3 - 3(10)(8) \\&= 512 - 240 = 272 \\\text{Correct}\end{aligned}​​
    (b) 
    (ab)2=(a+b)24ab16=(a+b)28(a+b)2=24a2+ab+b2=242=22a3b3=(4)(22)=8880Incorrect\begin{aligned}(a-b)^2 &= (a+b)^2 - 4ab \\16 &= (a+b)^2 - 8 \\(a+b)^2 &= 24 \\a^2 + ab + b^2 &= 24 - 2 = 22 \\a^3 - b^3 &= (4)(22) = 88 \neq 80 \\\text{Incorrect}\end{aligned}​​
    (C) 
    x=a10, y=b5x+y=(a+b)15=0y=xx3+y3=x3+(x)3=0Correct\begin{aligned}x &= a-10,\ y=b-5 \\x+y &= (a+b)-15 = 0 \\y &= -x \\x^3 + y^3 &= x^3 + (-x)^3 = 0 \\\text{Correct}\end{aligned}​​
    (D) 
    (ab)2=22556=169ab=13Correct\begin{aligned}(a-b)^2 &= 225 - 56 = 169 \\a-b &= 13 \\\text{Correct}\end{aligned}​​
    Correct answer is (B).

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