x r2 If [x + +3=5, then [x2 + +1=? 5, a) 25 b) 27 c) 23 d) 68
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Answer:
Since 5
1+x
+5
1−x
,a/2,25
x
+25
−x
are in A. P., we have
2a/2=5
1+x
+5
1−x
+25
x
+25
−x
,
Now put 5
x
=t so that t>0, we then have
a=5t+5/t
2
+1/t
2
=+ive ...(1)
=(t
2
+1/t
2
)+5(t+1/t)≥2+5.2=12
because A.M.≥G.M.
Also when a≥12 then (1) becomes
a=(t+
t
1
)
2
−2+5(t+
t
1
)
or y
2
+5y−(a+2)=0
where y=t+
t
1
≥2asA.M.≥G.M.
∴y=
2
−5±
25+4a+8
y=
2
−5±
33+4a
≥
2
−5±
33+48
=
2
−5±9
=
2
9−5
=2∴y≥2
∴y=t+
t
1
=5
x
+5
−x
≥2
which we know is true and gives a∈[12,∞)
Step-by-step explanation:
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