Let $\vec{a} = \hat{i} + \alpha\hat{j} + \beta\hat{k}$, $\alpha, \beta \in \mathbb{R}$. Let a vector $\vec{b}$ be such that the angle between $\vec{a}$ and $\vec{b}$ is $\frac{\pi}{4}$ and $|\vec{b}|^2 = 6$. If $\vec{a} \cdot \vec{b} = 3\sqrt{2}$, then the value of $(\alpha^2 + \beta^2)|\vec{a} \times \vec{b}|^2$ is equal to

JEE Main 2024 · MathsMultiple choiceSingle correctMediumMulti-step

JEE Main 30 January 2024, Shift 2, Maths Q19

Question 19 of 90 in this shift, Maths question 19 of 30, Section A.

Let

\vec{a} = \hat{i} + \alpha\hat{j} + \beta\hat{k}

\alpha, \beta \in \mathbb{R}

. Let a vector

\vec{b}

be such that the angle between

\vec{a}

and

\vec{b}

\frac{\pi}{4}

and

|\vec{b}|^2 = 6

. If

\vec{a} \cdot \vec{b} = 3\sqrt{2}

, then the value of

(\alpha^2 + \beta^2)|\vec{a} \times \vec{b}|^2

is equal to

(1) $75$
(2) $90$ Official answer
(3) $85$
(4) $95$

Official answer

Option 2

NTA final key.

Chapter: Vector Algebra
Topic: Cross Product
Idea tested: Lagrange's Identity

Same idea in other shifts

Asked 8× in all

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25 Jul 2021, Shift 2 · Q78If $|\vec{a}|=2$ , $|\vec{b}|=5$ and $|\vec{a}\times\vec{b}|=8$ , then $|\vec{a}\cdot\vec{b}|$ is equal to :EasySingle correct
2 Sep 2020, Shift 2 · Q73Let the position vectors of points 'A' and 'B' be $\hat{i}+\hat{j}+\hat{k}$ and $2\hat{i}+\hat{j}+3\hat{k}$ , respectively. A point 'P'…MediumNumerical value

Question text from the official JEE Main paper published by NTA; answer from the final answer key. Chapter, topic and idea tags, difficulty and skill are Lumi’s. Lumi analyses 108 of the 174 JEE Main shifts held from 2017 to 2026.