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Question

If x R and the numbers 5(1x)+5(x+1), a/2, (25x+25x) form an A. P. then a must lie in the interval ............ .

A
[12,]
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B
[5,5]
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C
[12,12]
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D
[8,]
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Solution

The correct option is A [12,]
Since 51+x+51x,a/2,25x+25x are in A. P., we have
2a/2=51+x+51x+25x+25x,
Now put 5x=t so that t>0, we then have
a=5t+5/t2+1/t2=+ive ...(1)
=(t2+1/t2)+5(t+1/t)2+5.2=12
because A.M.G.M.
Also when a12 then (1) becomes
a=(t+1t)22+5(t+1t)
or y2+5y(a+2)=0
where y=t+1t2asA.M.G.M.
y=5±25+4a+82
y=5±33+4a25±33+482
=5±92=952=2y2
y=t+1t=5x+5x2
which we know is true and gives a[12,)

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