QJEEM19Ma2772: JEE Main 2019 Mathematics - Class 11 JEEM19Ma2772 Solution: Let α and β be the roots of the quadratic equation x 2 sin θ –…

Question

Verified Solution • 2025–26 Session moderate

Let

α

and

β

be the roots of the quadratic equation x² sin

θ

– x(sin

θ

cos

θ

+ 1) + cos

θ

= 0 (0 <

θ

< 45^o), and

α

<

β

. Then

\sum_{n = 0}^{\infty} (α^{n} + \frac{{(- 1)}^{n}}{β^{n}})

is equal to :

Accepted Answer

D = (1 + sin

θ

cos

θ

)²

-

4sin

θ

cos

θ

= (1

-

sin

θ

cos

θ

)²

\Rightarrow

roots are

β

= cosec

θ

and

α

= cos

θ

\sum_{n = 0}^{\infty} (α^{n} + {(- \frac{1}{β})}^{n}) = \sum_{n = 0}^{\infty} {(\cos θ)}^{n} + \sum_{n = 0}^{n} {(- \sin θ)}^{n}

= \frac{1}{1 - \cos θ} + \frac{1}{1 + \sin θ}

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