JEE Main 2024 · MathsMultiple choiceSingle correctHardMulti-step

JEE Main 5 April 2024, Shift 2, Maths Q19

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

Consider three vectors

\vec{a}, \vec{b}, \vec{c}

. Let

|\vec{a}| = 2

|\vec{b}| = 3

and

\vec{a} = \vec{b} \times \vec{c}

. If

\alpha \in \left[0, \frac{\pi}{3}\right]

is the angle between the vectors

\vec{b}

and

\vec{c}

, then the minimum value of

27|\vec{c}-\vec{a}|^2

is equal to :

(1)105
(2)124Official answer
(3)110
(4)121

Official answer

Option 2

NTA final key.

Chapter: Vector Algebra
Topic: Cross Product
Idea tested: Magnitude and Sine Formula

Same idea in other shifts

Asked 13× in all

4 Apr 2026, Shift 2 · Q15Let $\hat u$ and $\hat v$ be unit vectors inclined at an acute angle such that $|\hat u\times\hat v|=\frac{\sqrt3}{2}$ . If…HardSingle correct
6 Apr 2024, Shift 2 · Q19Let $\vec{a}=6\hat{i}+\hat{j}-\hat{k}$ and $\vec{b}=\hat{i}+\hat{j}$ . If $\vec{c}$ is a vector such that $|\vec{c}|\ge 6$ ,…HardSingle correct
30 Jan 2024, Shift 1 · Q18Let $\vec{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$ and $\vec{b}=b_1\hat{i}+b_2\hat{j}+b_3\hat{k}$ be two vectors such that $|\vec{a}|=1$ ,…HardSingle correct
11 Apr 2023, Shift 1 · Q11Let $\vec{a}$ be a non-zero vector parallel to the line of intersection of the two planes described by $\hat{i}+\hat{j},\ \hat{i}+\hat{k}$ …HardSingle correct
11 Apr 2023, Shift 2 · Q29Let $\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}$ and $\vec{b} = \hat{i} + \hat{j} - \hat{k}$ . If $\vec{c}$ is a vector such that…HardNumerical value
25 Jul 2022, Shift 1 · Q15Let $ABC$ be a triangle such that $\overrightarrow{BC}=\vec a$ , $\overrightarrow{CA}=\vec b$ , $\overrightarrow{AB}=\vec c$ ,…HardTwo statements
26 Jul 2022, Shift 1 · Q15Let $\overrightarrow{\mathrm{a}} = \alpha \hat{i} + \hat{j} - \hat{k}$ and…MediumSingle correct
29 Jul 2022, Shift 1 · Q14Let $\hat{a}$ and $\hat{b}$ be two unit vectors such that the angle between them is $\frac{\pi}{4}$ . If $\theta$ is the angle between the…HardSingle correct

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.