1
JEE Main 2022 (Online) 24th June Morning Shift
Numerical
+4
-1 If the shortest distance between the lines

$$\overrightarrow r = \left( { - \widehat i + 3\widehat k} \right) + \lambda \left( {\widehat i - a\widehat j} \right)$$

and $$\overrightarrow r = \left( { - \widehat j + 2\widehat k} \right) + \mu \left( {\widehat i - \widehat j + \widehat k} \right)$$ is $$\sqrt {{2 \over 3}}$$, then the integral value of a is equal to ___________.

2
JEE Main 2021 (Online) 1st September Evening Shift
Numerical
+4
-1
Let $$\overrightarrow a = 2\widehat i - \widehat j + 2\widehat k$$ and $$\overrightarrow b = \widehat i + 2\widehat j - \widehat k$$. Let a vector $$\overrightarrow v$$ be in the plane containing $$\overrightarrow a$$ and $$\overrightarrow b$$. If $$\overrightarrow v$$ is perpendicular to the vector $$3\widehat i + 2\widehat j - \widehat k$$ and its projection on $$\overrightarrow a$$ is 19 units, then $${\left| {2\overrightarrow v } \right|^2}$$ is equal to _____________.
3
JEE Main 2021 (Online) 27th August Morning Shift
Numerical
+4
-1
Let $$\overrightarrow a = \widehat i + 5\widehat j + \alpha \widehat k$$, $$\overrightarrow b = \widehat i + 3\widehat j + \beta \widehat k$$ and $$\overrightarrow c = - \widehat i + 2\widehat j - 3\widehat k$$ be three vectors such that, $$\left| {\overrightarrow b \times \overrightarrow c } \right| = 5\sqrt 3$$ and $${\overrightarrow a }$$ is perpendicular to $${\overrightarrow b }$$. Then the greatest amongst the values of $${\left| {\overrightarrow a } \right|^2}$$ is _____________.
4
JEE Main 2021 (Online) 26th August Evening Shift
Numerical
+4
-1
If the projection of the vector $$\widehat i + 2\widehat j + \widehat k$$ on the sum of the two vectors $$2\widehat i + 4\widehat j - 5\widehat k$$ and $$- \lambda \widehat i + 2\widehat j + 3\widehat k$$ is 1, then $$\lambda$$ is equal to __________.
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