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Find the number of solution(s) of following equations: (i) x^(2)-sinx-cosx+2=0 (ii) cos(x^(2)+x+1)=2^(x) + 2^(-x) (iii) sin^(2)x - 2sinx - x^(2)-3=0, x int [0,2pi] (iv) (sin (pix)/(2sqrt(2)))= x^(2)-2sqrt(2)x+3 |
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Answer» Solution :We have, (i) `x^(2)+2= sinx + cosx` The range of `x^(2) + 2` is `[2, infty]` and the range of `sinx + cosx` is `[-sqrt(2), sqrt(2)]` As there is no common value ini the ranges hence no solution EXISTS. (iii) `cos(x^(2)+ x+1) = 2^(x) + 2^(-x)` The range of `cos(x^(2)+x+1)` is `[-1,1]` Using A.M. `ge` G.M. `(2^(x) + 2^(-x))/(2) ge sqrt(2^(x). 2^(-x))` `RARR r^(x) + 2^(-x) ge 2` Hence, RHS `int [2, infty]` Since, there is no common value in the ranges, hence number of SOLUTIONS is zero. (iii) The equation can be written as `(sinx-1)^(2)=1` Hence no solution exists. (iv) `sin (pix)/(2sqrt(2))= (x-sqrt(2))^(2)+1` LHS `int [-1,1]` and RHS `int [1,infty]` Hence, solution is obtained at LHS = RHS=1 If (`x-sqrt(2))^(2) + 1=1 rArr x=sqrt(2)` Also at `x=sqrt(2)`, LHS `=sin(PI xx sqrt(2))/(2sqrt(2))=1` Hence, `x=1` is the only value satisfying the given equation and consequently number of solution is one. |
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