11th Class Mathematics Sample Paper Mathematics Olympiad - Sample Paper-3

  • question_answer
    The interior angles of a polygon are in arithmetic progression. The smallest angle is \[\mathbf{12}{{\mathbf{0}}^{{}^\circ }}\] and the common difference is\[{{\mathbf{5}}^{{}^\circ }}\]. What will be the number of sides of the polygon

    A)   8                  

    B)  9              

    C)  10                               

    D)  7

    Correct Answer: B

    Solution :

    [b] Let 'a' and "d" be the 1st term & common difference in A.P. respectively. Given, \[a={{120}^{{}^\circ }}\And d={{5}^{{}^\circ }}\] \[S=\frac{n}{2}[2a+(n-1).d]\] By Geometry, Sum of angles of the polygon be \[\left( 2n-4 \right)\times {{90}^{{}^\circ }}\] \[\therefore (2n-4)\times {{90}^{{}^\circ }}=\frac{n}{2}[2\times {{120}^{{}^\circ }}+(n-1)5]\] \[\Rightarrow {{180}^{{}^\circ }}(2n-4)=240n+5{{n}^{2}}-5n\] 5n2 + 240n - 5n - 360n + 720 = 0 \[\Rightarrow 5{{n}^{2}}-125n+720=0\Rightarrow {{n}^{2}}-25n+144=0\]\[\Rightarrow {{n}^{2}}-16n-9n+144=0\Rightarrow (n-16)(n-9)=0\] \[n=9,16\] \[\because n=16\]              (It is not possible) \[\therefore n=9\] Hence, the no. of sides of the polygone, \[n=9.\] Hence, option [b] is correct

You need to login to perform this action.
You will be redirected in 3 sec spinner