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JEE Main & Advanced Sample Paper JEE Main Sample Paper-11

  • question_answer
    The expansion of \[{{(1+x)}^{n}}\]has 3 consecutive terms with coefficients in the ratio 1 : 2 : 3 and can be written in the form\[^{\text{n}}{{\text{C}}_{\text{k}}}{{\text{:}}^{\text{n}}}{{\text{C}}_{\text{k+1}}}{{\text{:}}^{\text{n}}}{{\text{C}}_{\text{k+2}}}\text{.}\] The sum of all possible values of (n + k) is

    A)  18                                         

    B)  21

    C)  28                                         

    D)  32

    Correct Answer: A

    Solution :

     \[\frac{^{n}{{C}_{k}}}{^{n}{{C}_{k+1}}}=\frac{1}{2}\Rightarrow \frac{n!}{k!(n-k)!}\frac{(k+1)!(n-k-1)!}{n!}=\frac{1}{2}\] or\[\frac{k+1}{n-k}=\frac{1}{2}\] \[2k+2=n-k\] \[n-3k=2\]           ?.(1) Similarly, \[\frac{^{n}{{C}_{k+1}}}{^{n}{{C}_{k+2}}}=\frac{2}{3}\] \[\frac{n!}{(k+1)!(n-k-1)!}.=\frac{(k+2)!(n-k-2)!}{n!}=\frac{2}{3}\] \[\frac{k+2}{n-k-1}=\frac{2}{3}\] \[3k+6=2n-2k-2\] \[2n-5k=8\]                                                        ?.(2) From (1) and (2) n = 14 and k = 4 \[\therefore \]\[\therefore \,\,n+k=18\]


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