JEE Advanced 2024 — Paper 1 — MATH-III solution

JEE Advanced2024Paper 1MATH-III

Let $O P = \frac{α - 1}{α} \hat{i} + \hat{j} + \hat{k}$ , $O Q = \hat{i} + \frac{β - 1}{β} \hat{j} + \hat{k}$ and $O R = \hat{i} + \hat{j} + \frac{1}{2} \hat{k}$ be three vectors, where $α, β \in R - {0}$ and $O$ denotes the origin. If $(O P \times O Q) \cdot O R = 0$ and the point $(α, β, 2)$ lies on the plane $3 x + 3 y - z + l = 0$ , then the value of $l$ is _______.

Your answer

Solution

The scalar triple product $(O P \times O Q) \cdot O R$ equals the determinant

$\frac{α - 1}{α} 11 1 \frac{β - 1}{β} 1 11 \frac{1}{2} = 0$

Expanding along the first row and simplifying (multiply by $2 α β$ to clear fractions), the condition reduces to

$α + β + 1 = 0$

Now substitute $(α, β, 2)$ into $3 x + 3 y - z + l = 0$ :

$3 (α + β) - 2 + l = 0 ⟹ 3 (- 1) - 2 + l = 0 ⟹ l = 5$

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