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Question

If tan1x , tan1y , tan1z are in A.P. and x, y, z are also in A.P. (y being not equal to 0, 1, or -1) , then :

A
x, y, z are in G.P.
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B
(xy)2+(yz)2+(2zx)2=0
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C
x=2y=z
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D
none of these
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Solution

The correct option is D x, y, z are in G.P.
Since tan1(x) , tan1(y) and tan1(z) are in A.P

2tan1(y)=tan1(x)+tan1(z)

tan1(2y1y2)=tan1(x+z1xz)

2y1y2=x+z1xz

Now 2y=x+z
Since x, y,z are in A.P
2y1y2=2y1xz

1xz=1y2

xz=y2
Hence x,y,z form a G.P.

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