JEE 2023 — Mathematics PYQ

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Question

JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

Let $\vec{u} = \hat{i} - \hat{j} - 2\hat{k}$ , $\vec{v} = 2\hat{i} + \hat{j} - \hat{k}$ , $\vec{v} \cdot \vec{w} = 2$ and $\vec{v} \times \vec{w} = \vec{u} + \lambda \vec{v}$ . Then $\vec{u} \cdot \vec{w}$ is equal to:

Choose the correct answer:

Explanation

1. $\lambda$ ki value nikalna:

Di gayi equation ka dot product $\vec{v}$ ke saath lene par:

\vec{v} \cdot (\vec{v} \times \vec{w}) = \vec{v} \cdot (\vec{u} + \lambda \vec{v})

Kyuki $\vec{v} \cdot (\vec{v} \times \vec{w}) = 0$ hota hai, isliye:

0 = \vec{u} \cdot \vec{v} + \lambda |\vec{v}|^2

Values calculate karte hain:

$\vec{u} \cdot \vec{v} = (1)(2) + (-1)(1) + (-2)(-1) = 2 - 1 + 2 = 3$
$|\vec{v}|^2 = 2^2 + 1^2 + (-1)^2 = 4 + 1 + 1 = 6$

Inhe equation mein rakhne par:

0 = 3 + \lambda(6) \implies 6\lambda = -3 \implies \lambda = -\frac{1}{2}

2. $\vec{u} \cdot \vec{w}$ ki value nikalna:

Ab di gayi equation $\vec{v} \times \vec{w} = \vec{u} + \lambda \vec{v}$ ka dot product $\vec{w}$ ke saath lene par:

\vec{w} \cdot (\vec{v} \times \vec{w}) = \vec{w} \cdot (\vec{u} + \lambda \vec{v})

Kyuki $\vec{w} \cdot (\vec{v} \times \vec{w}) = 0$ hota hai, isliye:

0 = \vec{u} \cdot \vec{w} + \lambda (\vec{v} \cdot \vec{w})

Humein sawal mein $\vec{v} \cdot \vec{w} = 2$ aur humne $\lambda = -1/2$ nikala hai:

0 = \vec{u} \cdot \vec{w} + \left( -\frac{1}{2} \right)(2)

0 = \vec{u} \cdot \vec{w} - 1

\vec{u} \cdot \vec{w} = 1

Final Answer

Sahi option (C) 1 hai.

Choose the correct answer:

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