For the functions $f(\theta)=\alpha \tan ^{2} \theta+\beta \cot ^{2} \theta$, and $\mathrm{g}(\theta)=\alpha \sin ^{2} \theta+\beta \cos ^{2} \theta, \alpha>\beta>0$, let $\min _{0<\theta<\frac{\pi}{2}} f(\theta)=\max _{0<\theta<\pi} \mathrm{g}(\theta)$. If the first term of a G.P. is $\left(\frac{\alpha}{2 \beta}\right)$, its common ratio is $\left(\frac{2 \beta}{\alpha}\right)$ and the sum of its first 10 terms is $\frac{\mathrm{m}}{\mathrm{n}}, \operatorname{gcd}(\mathrm{m}, \mathrm{n})=1$, then $\mathrm{m}+\mathrm{n}$ is equal to

JEE Main · 2026 · 06 Apr 2026 · Shift 1 · Sequences and Series

For the functions

f (θ) = α tan^{2} θ + β cot^{2} θ

, and

g (θ) = α sin^{2} θ + β cos^{2} θ, α > β > 0

, let

min_{0 < θ < \frac{π}{2}} f (θ) = max_{0 < θ < π} g (θ)

. If the first term of a G.P. is

(\frac{α}{2 β})

, its common ratio is

(\frac{2 β}{α})

and the sum of its first 10 terms is

\frac{m}{n}, gcd (m, n) = 1

, then

m + n

is equal to

Answer: 1279

By AM--GM,

α tan^{2} θ + β cot^{2} θ \geq 2 α β

, so

min_{0 < θ < π /2} f (θ) = 2 α β

. Writing

g (θ) = (α - β) sin^{2} θ + β

with

α > β > 0

, its greatest value on

(0, π)

occurs at

sin^{2} θ = 1

, so

max g (θ) = α

. Equating,

2 α β = α

, hence

α = 4 β

. The G.P. then has first term

\frac{α}{2 β} = 2

and common ratio

\frac{2 β}{α} = \frac{1}{2}

, so

S_{10} = \frac{2 ( 1 - ( 1/2 ) ^{10} )}{1 - \frac{1}{2}} = \frac{1023}{256}

. As

g cd (1023, 256) = 1

m + n = 1023 + 256 = 1279

JEE Main · 2026 · 06 Apr 2026 · Shift 1 · Integer answer