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Three successive terms of a G.P. will form the sides of a triangle if the common ratio r satisfies the inequalityA. `(sqrt(3)-1)/(2)ltrlt(sqrt(3)+1)/(2)`B. `(sqrt(5)-1)/(2)ltrlt(sqrt(5)+1)/(2)`C. `(sqrt(2)-1)/(2)ltrlt(sqrt(2)+1)/(2)`D. none of these |
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Answer» Correct Answer - B Let the lengths of the sides of the triangle be `a,ar,ar^(2)`. We have the following three cases : CASE I When r=1 In this case, the lengths of sides of the triangle are a,a,a i.e. the triangle is equilateral. CASE II When `rgt1` In this case, the length of the largest side is `ar^(2)`. Therefore, the triangle will be formed, if `a+argtar^(2)` `rArr" "r^(2)-r-1lt0` `rArr" "(1-sqrt(5))/(2)ltrlt(1+sqrt(5))/(2)` `rArr" "rlt(1+sqrt(5))/(2)" "[becausergt1]` . . ..(i) CASE III When `rlt1` In this case, the length of the largest side is a. So, the triangle will be formed, if `ar+ar^(2)gta` `rArr" "r^(2)+r-1gt0` `rArr" "rlt(-1-sqrt(5))/(2)or,rgt(-1+sqrt(5))/(2)` `rArr" "(sqrt(5)-1)/(2)ltrlt1" "[becauserlt1]` . . . (ii) From (i) and (ii), we obtain : `(sqrt(5)-1)/(2)ltrlt(sqrt(5)+1)/(2)` |
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