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CEE Kerala Engineering CEE Kerala Engineering Solved Paper-2008

  • question_answer
    Let a, b, c be in AP. If \[0<a,b,c<1,x=\sum\limits_{n=0}^{\infty }{{{a}^{n}}},\] \[y=\sum\limits_{n=0}^{\infty }{{{b}^{n}}}\]and\[z=\sum\limits_{n=0}^{\infty }{{{c}^{n}}},\]then

    A)  \[2y=x+z\]      

    B)  \[2x=y+z\]

    C)  \[2z=x+y\]      

    D)         \[2xz=xy+yz\]

    E)  \[z=\frac{2xy}{x+y}\]

    Correct Answer: D

    Solution :

    Since, \[x=\sum\limits_{n=0}^{\infty }{{{a}^{n}}}\] \[\therefore \]  \[x=1+a+{{a}^{2}}+....\infty \] \[\Rightarrow \]               \[x=\frac{1}{1-a}\] \[\Rightarrow \]               \[(1-a)x=1\] \[\Rightarrow \]               \[a=\frac{x-1}{x}\] Similarly,\[b=\frac{y-1}{y}\]and \[c=\frac{z-1}{z}\] Since, a, b and c are in AP. \[\therefore \]  \[b=\frac{a+c}{2}\] \[\Rightarrow \]               \[\frac{y-1}{y}=\frac{\frac{x-1}{x}+\frac{z-1}{z}}{2}\] \[\Rightarrow \]               \[2xz(y-1)=y[z(x-1)+x(z-1)]\] \[\Rightarrow \]               \[2xyz-2xz=xyz-yz+xyz-xy\] \[\Rightarrow \]               \[-2xz=-yz-xy\] \[\Rightarrow \]               \[2xz=xy+yz\]


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