1.

Let `alphaa n dbeta`be any two positive values of `x`for which `2cosx ,|cosx|,`and `1-3cos^2x`are in G.P. The minimum value of `|alpha-beta|`is`pi/3`(b) `pi/4`(c) `pi/2`(d) none of these

Answer» As `2cosx, |cosx| and 1-3cos^2x` are in G.P.
`:. |cosx|^2 = 2cosx(1-3cos^2x)`
`=>cos^2x = 2cosx-6cos^3x`
`=>6cos^3x+cos^2x-2cosx = 0`
`=>cosx(6cos^2x+cosx-2) = 0`
`=>cosx(6cos^2x+4cosx-3cosx-2) = 0`
`=>cosx(2cosx-1)(3cosx+2) = 0`
`=>cosx = 0 or 2cosx-1 = 0 or 3cosx+2 = 0`
`=> cosx = 0 or cos x = 1/2 or cosx = -2/3`
`=> x = pi/2 or x = pi/3 or x = cos^-1(2/3)`
But, as `alpha and beta` are positive, so we will take inly positive values.
`:. alpha = pi/2 and beta = pi/3`
`|alpha-beta| = pi/2-pi/3 = pi/6.`
So, option `d` is the correct option.


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