JEE Main 2026 Apr 6 Shift 1, Mathematics Q24: Arithmetic, Geometric And Harmonic Progressions

JEE Main2026Apr 6, Shift 1Mathematics

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 $0 < θ < π /2 min f (θ) = 0 < θ < π /2 max g (θ)$ . 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 $\frac{m}{n}$ , $g cd (m, n) = 1$ , then $m + n$ is equal to __________.

Your answer

Solution

By AM-GM, $f (θ) = α tan^{2} θ + β cot^{2} θ \geq 2 α β$ , with equality when $tan^{2} θ = β / α$ , which is attainable on $(0, π /2)$ . So $min f = 2 α β$ .

Write $g (θ) = (α - β) sin^{2} θ + β$ . Since $α > β$ , $g$ is maximized as $sin^{2} θ \to 1$ , giving $max g = α$ .

Equating: $2 α β = α \Rightarrow 4 α β = α^{2} \Rightarrow α = 4 β$ (since $α \neq = 0$ ).

Then first term $= \frac{α}{2 β} = 2$ , common ratio $= \frac{2 β}{α} = \frac{1}{2}$ .

$S_{10} = 2 \cdot \frac{1 - ( 1/2 ) ^{10}}{1 - 1/2} = 4 (1 - \frac{1}{1024}) = \frac{4 \cdot 1023}{1024} = \frac{1023}{256}$ .

So $m = 1023$ , $n = 256$ , and $m + n = 1279$ .

Concept behind this question

Arithmetic, Geometric And Harmonic ProgressionsNotes, formulas and examples →

Practice more Mathematics

Concept-wise practice with instant solutions on Fundamenthol.

Start practicing →

JEE Main 2026 Apr 6 Shift 1, Mathematics Q24: Arithmetic, Geometric And Harmonic Progressions

Concept behind this question

More previous year questions on Arithmetic, Geometric And Harmonic Progressions

Practice more Mathematics