The roots of equation x2 + 2(a - 3)x + 9 = 0 lie between - 6 and 1 and 2, h1, h2, ..... h20, [a] are in H.P where [a] denotes the integral part of a and 2, a1, a2, .........a20, [a] are in A.P then

[a] = 6 , h 18 = 14/3 , a 3 = 18/7

The roots of equation x2 + 2(a - 3)x + 9 = 0 lie between - 6 and 1 and 2, h1, h2, ..... h20, [a] are in H.P where [a] denotes the integral part of a and 2, a1, a2, .........a20, [a] are in A.P then

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The roots of equation x² + 2(a - 3)x + 9 = 0 lie between - 6 and 1 and 2, h₁, h₂, ..... h₂₀, [a] are in H.P where [a] denotes the integral part of a and 2, a₁, a₂, .........a₂₀, [a] are in A.P then

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[a] = 6

h₁₈ = 14/3

a₃ = 18/7

a₃h₁₈ = 11

answer is A, B, C.

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$(1) D > 0 (2) - 6 < \frac{- b}{2 a} < 1 (3) f (- 6) f (1) > 0 6 \leq a < 6 .25 \Rightarrow [a] = 6 \Rightarrow 2, h, h_{2}, \dots \dots . h_{2} o, [a]$
are in H.P 2,h, h₂,......h₂o,6 are in H.P
$\frac{1}{h_{18}} = \frac{1}{2} + 18 [\frac{\frac{1}{6} - \frac{1}{2}}{21}] \Rightarrow h_{18} = \frac{14}{3}$
$\Rightarrow$ 2, a₁, a₂, ....a₂₀, 6 are in A.P
$\Rightarrow a_{3} = 2 + 3 (\frac{6 - 2}{21}) = \frac{18}{7}$

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