JEE Main 2026 — Jan 22, Shift 2 — Mathematics solution

JEE Main2026Jan 22, Shift 2Mathematics

Let a vector $a = 2 \hat{i} - \hat{j} + λ \hat{k}$ , $λ > 0$ , make an obtuse angle with the vector $b = - λ^{2} \hat{i} + 42 \hat{j} + 42 \hat{k}$ and an angle $θ$ , $\frac{π}{6} < θ < \frac{π}{2}$ , with the positive z-axis. If the set of all possible values of $λ$ is $(α, β) - {γ}$ , then $α + β + γ$ is equal to ______.

Your answer

Solution

Angle with z-axis: $cos θ = \frac{a \cdot k ^}{∣ a ∣} = \frac{λ}{3 + λ ^{2}}$ .

$\frac{π}{6} < θ < \frac{π}{2} \Rightarrow 0 < cos θ < \frac{3}{2}$ , so $0 < \frac{λ}{3 + λ ^{2}} < \frac{3}{2}$ .

Squaring: $4 λ^{2} < 3 (3 + λ^{2}) \Rightarrow λ^{2} < 9$ . With $λ > 0$ : $λ \in (0, 3)$ … (1).

Obtuse angle: $a \cdot b < 0$ gives $- 2 λ^{2} - 42 + 42 λ < 0 \Rightarrow λ^{2} - 4 λ + 4 > 0 \Rightarrow (λ - 2)^{2} > 0$ , so $λ \neq = 2$ … (2).

Combining: $λ \in (0, 3) - {2}$ , so $α = 0$ , $β = 3$ , $γ = 2$ and $α + β + γ = 5$ .

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