If the shortest distance between the lines $\frac{x+2}{2}=\frac{y+3}{3}=\frac{z-5}{4}$ and $\frac{x-3}{1}=\frac{y-2}{-3}=\frac{z+4}{2}$ is $\frac{38}{3\sqrt{5}}k$, and $\int_0^k[x^2]dx=\alpha-\sqrt{\alpha}$, where $[x]$ denotes the greatest integer function, then $6\alpha^3$ is equal to ________.

JEE Main 2024 · MathsNumerical answerNumerical valueHardMulti-step

JEE Main 4 April 2024, Shift 1, Maths Q29

Question 29 of 90 in this shift, Maths question 29 of 30, Section B.

If the shortest distance between the lines

\frac{x+2}{2}=\frac{y+3}{3}=\frac{z-5}{4}

and

\frac{x-3}{1}=\frac{y-2}{-3}=\frac{z+4}{2}

\frac{38}{3\sqrt{5}}k

, and

\int_0^k[x^2]dx=\alpha-\sqrt{\alpha}

, where

[x]

denotes the greatest integer function, then

6\alpha^3

is equal to ________.

Official answer

NTA final key.

Chapter: Three Dimensional Geometry
Topic: Skew Lines
Idea tested: Vector Shortest Distance Formula

Same idea in other shifts

Asked 9× in all

4 Apr 2026, Shift 2 · Q13The shortest distance between the lines…EasySingle correct
2 Apr 2025, Shift 2 · Q15The line $L_1$ is parallel to the vector $\vec{a} = -3\hat{i} + 2\hat{j} + 4\hat{k}$ and passes through the point $(7, 6, 2)$ and the line…EasySingle correct
4 Apr 2025, Shift 2 · Q14Let the values of p, for which the shortest distance between the lines $\frac{x+1}{3}=\frac{y}{4}=\frac{z}{5}$ and…MediumSingle correct
6 Apr 2024, Shift 1 · Q17The shortest distance between the lines $\frac{x-3}{2}=\frac{y+15}{-7}=\frac{z-9}{5}$ and $\frac{x+1}{2}=\frac{y-1}{1}=\frac{z-9}{-3}$ isMediumSingle correct
8 Apr 2024, Shift 1 · Q14If the shortest distance between the lines $L_1:\vec{r}=(2+\lambda)\hat{i}+(1-3\lambda)\hat{j}+(3+4\lambda)\hat{k}$ ,…MediumSingle correct
24 Jun 2022, Shift 1 · Q30If the shortest distance between the lines $\vec{r}=(-\hat{i}+3\hat{k})+\lambda(\hat{i}-a\hat{j})$ and…MediumNumerical value
25 Jun 2022, Shift 2 · Q30Let $l_1$ be the line in $xy$ -plane with $x$ and $y$ intercepts $\frac{1}{8}$ and $\frac{1}{4\sqrt{2}}$ respectively, and $l_2$ be the…HardNumerical value
24 Feb 2021, Shift 2 · Q88Let $\lambda$ be an integer. If the shortest distance between the lines $x-\lambda=2y-1=-2z$ and $x=y+2\lambda=z-\lambda$ is…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.