Properties of ITF

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If $\cos^{- 1} x + \cos^{- 1} 2 x + \cos^{- 1} 3 x = π$ and $a x^{3} + b x^{2} + c x = 0$ satisfies the equation then the value of $\frac{b - a - c}{100}$ is

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Solution

$\cos^{- 1} x + \cos^{- 1} 2 x + \cos^{- 1} 3 x = π \Rightarrow \cos^{- 1} 2 x + \cos^{- 1} 3 x = π - \cos^{- 1} x = \cos^{- 1} (- x) \Rightarrow 6 x^{2} - \sqrt{1 - 4 x^{2}} \sqrt{1 - 9 x^{2}} = - x \Rightarrow 6 x^{2} + x = \sqrt{1 - 4 x^{2}} \sqrt{1 - 9 x^{2}} \Rightarrow 36 x^{4} + 12 x^{3} + x^{2} = 1 - 13 x^{2} + 36 x^{4} \Rightarrow 12 x^{3} + 14 x^{2} - 1 = 0$
$\Rightarrow a = 12, b = 14, c = - 1 ∴ \frac{b - a - c}{100} = \frac{3}{100} = 0.03$

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Properties of ITF

Remember concepts with our Masterclasses.

If cos−1x+cos−12x+cos−13x=πand ax3+bx2+cx=0 satisfies the equation then the value of b−a−c100 is

cos−1x+cos−12x+cos−13x=π ⇒cos−12x+cos−13x=π-cos−1x=cos-1-x ⇒6x2-1-4x21-9x2=-x ⇒6x2+x=1-4x21-9x2 ⇒36x4+12x3+x2=1-13x2+36x4 ⇒12x3+14x2-1=0⇒a=12,b=14,c=-1 ∴b-a-c100=3100=0.03

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If $\cos^{- 1} x + \cos^{- 1} 2 x + \cos^{- 1} 3 x = π$ and $a x^{3} + b x^{2} + c x = 0$ satisfies the equation then the value of $\frac{b - a - c}{100}$ is