11th Class Mathematics Sample Paper Mathematics Olympiad - Sample Paper-1

  • question_answer In a triangle ABC, If \[\mathbf{cotA},\mathbf{cotB},\mathbf{cotC}\]are in A.P. than\[\mathbf{a},\mathbf{b},\mathbf{c}\]are in-

    A) G.P.                                         

    B) A. P.           

    C) H.P.                                         

    D) None of these

    Correct Answer: B

    Solution :

    [b] \[\therefore cotA,cotB,cotC\text{ }be\text{ }in\text{ }A.\text{ }P.\] Then \[2.cotB=cotA+cotC\] \[\Rightarrow 2\sqrt{\frac{s(s-b)}{(s-a)(s-c)}}=\sqrt{\frac{s(s-a)}{(s-b)(s-c)}}+\sqrt{\frac{s(s-c)}{(s-a)(s-b)}}\]\[\Rightarrow 2\frac{s(s-b)}{\Delta }=\frac{s(s-a)}{\Delta }+\frac{s(s-c)}{\Delta }\] \[\Rightarrow 2s\left( s-b \right)=s\left( s-a \right)+s\left( s-c \right)~\] \[\Rightarrow 2\left( s-b \right)=\left( s-a \right)+\left( s-c \right)~\]     \[\Rightarrow 2s-2b=2s-\left( a+c \right)\] \[\Rightarrow 2b=a+c\] Hence \[a,b,c\]be in A.P.


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