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SSC Quantitative Aptitude Algebra Question Bank Elementary Algebra (II)

  • question_answer
    If \[{{a}^{4}}+{{a}^{2}}{{b}^{2}}+{{b}^{4}}=8\]and \[{{a}^{2}}+ab+{{b}^{2}}=4,\] then the value of ab is

    A) \[-1~\]

    B) \[0\]

    C) \[2\]

    D) \[1\]

    Correct Answer: D

    Solution :

    [d] Given equations are \[{{a}^{4}}+{{a}^{2}}{{b}^{2}}+{{b}^{4}}=8\]             …(ii) \[{{a}^{2}}+ab+{{b}^{2}}=4\]                                …(ii) Squaring Eq. (ii), we get                         \[{{({{a}^{2}}+ab+{{b}^{2}})}^{2}}={{(4)}^{2}}\]                         \[{{a}^{4}}+{{a}^{2}}{{b}^{2}}+{{b}^{4}}+2{{a}^{3}}b+2a{{b}^{3}}\]                                     \[+2{{a}^{2}}{{b}^{2}}=16\]             \[\Rightarrow \]   \[8+2ab({{a}^{2}}+{{b}^{2}}+ab)=16\][using Eq.(i)] \[\Rightarrow \]   \[2ab({{a}^{2}}+{{b}^{2}}+ab)=8\]             \[\Rightarrow \]   \[2ab\times 4=8\]                       [using Eq. (ii)] \[\Rightarrow \]   \[ab=1\]


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