1
JEE Main 2023 (Online) 24th January Evening Shift
Numerical
+4
-1 If the shortest between the lines $${{x + \sqrt 6 } \over 2} = {{y - \sqrt 6 } \over 3} = {{z - \sqrt 6 } \over 4}$$ and $${{x - \lambda } \over 3} = {{y - 2\sqrt 6 } \over 4} = {{z + 2\sqrt 6 } \over 5}$$ is 6, then the square of sum of all possible values of $$\lambda$$ is

2
JEE Main 2023 (Online) 24th January Morning Shift
Numerical
+4
-1 The shortest distance between the lines $${{x - 2} \over 3} = {{y + 1} \over 2} = {{z - 6} \over 2}$$ and $${{x - 6} \over 3} = {{1 - y} \over 2} = {{z + 8} \over 0}$$ is equal to ________

3
JEE Main 2022 (Online) 29th July Morning Shift
Numerical
+4
-1 Let a line with direction ratios $$a,-4 a,-7$$ be perpendicular to the lines with direction ratios $$3,-1,2 b$$ and $$b, a,-2$$. If the point of intersection of the line $$\frac{x+1}{a^{2}+b^{2}}=\frac{y-2}{a^{2}-b^{2}}=\frac{z}{1}$$ and the plane $$x-y+z=0$$ is $$(\alpha, \beta, \gamma)$$, then $$\alpha+\beta+\gamma$$ is equal to _________.

4
JEE Main 2022 (Online) 28th July Morning Shift
Numerical
+4
-1 Let $$\mathrm{P}(-2,-1,1)$$ and $$\mathrm{Q}\left(\frac{56}{17}, \frac{43}{17}, \frac{111}{17}\right)$$ be the vertices of the rhombus PRQS. If the direction ratios of the diagonal RS are $$\alpha,-1, \beta$$, where both $$\alpha$$ and $$\beta$$ are integers of minimum absolute values, then $$\alpha^{2}+\beta^{2}$$ is equal to ____________.

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