JEE Main 2026 Apr 5 Shift 2, Mathematics Q1: Arithmetic, Geometric And Harmonic Progressions

JEE Main2026Apr 5, Shift 2Mathematics

Let $α, β$ be the roots of the equation $x^{2} - x + p = 0$ and $γ, δ$ be the roots of the equation $x^{2} - 4 x + q = 0$ ; $p, q \in Z$ . If $α, β, γ, δ$ are in G.P., then $∣ p + q ∣$ equals:

A
16
B
32
C
34
D
38

Solution

Write the four terms of the GP as $α = a$ , $β = a r$ , $γ = a r^{2}$ , $δ = a r^{3}$ .

Vieta on the first quadratic: $α + β = 1 \Rightarrow a (1 + r) = 1$ . On the second: $γ + δ = 4 \Rightarrow a r^{2} (1 + r) = 4$ .

Dividing: $r^{2} = 4 \Rightarrow r = \pm 2$ . Take $r = - 2$ , $a = - 1$ (the choice $r = 2$ gives $a = 1/3$ , which makes $p \in / Z$ ).

Then $p = α β = a^{2} r = - 2$ and $q = γ δ = a^{2} r^{5} = (- 1)^{2} (- 2)^{5} = - 32$ .

$∴ ∣ p + q ∣ = ∣ - 2 - 32∣ = 34$ .

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JEE Main 2026 Apr 5 Shift 2, Mathematics Q1: Arithmetic, Geometric And Harmonic Progressions

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