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JEE Main 2021 (Online) 25th July Morning Shift
Numerical
+4
-1
Let $$\overrightarrow p = 2\widehat i + 3\widehat j + \widehat k$$ and $$\overrightarrow q = \widehat i + 2\widehat j + \widehat k$$ be two vectors. If a vector $$\overrightarrow r = (\alpha \widehat i + \beta \widehat j + \gamma \widehat k)$$ is perpendicular to each of the vectors ($$(\overrightarrow p + \overrightarrow q )$$ and $$(\overrightarrow p - \overrightarrow q )$$, and $$\left| {\overrightarrow r } \right| = \sqrt 3 $$, then $$\left| \alpha \right| + \left| \beta \right| + \left| \gamma \right|$$ is equal to _______________.
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2
JEE Main 2021 (Online) 20th July Evening Shift
Numerical
+4
-1
For p > 0, a vector $${\overrightarrow v _2} = 2\widehat i + (p + 1)\widehat j$$ is obtained by rotating the vector $${\overrightarrow v _1} = \sqrt 3 p\widehat i + \widehat j$$ by an angle $$\theta$$ about origin in counter clockwise direction. If $$\tan \theta = {{\left( {\alpha \sqrt 3 - 2} \right)} \over {\left( {4\sqrt 3 + 3} \right)}}$$, then the value of $$\alpha$$ is equal to _____________.
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3
JEE Main 2021 (Online) 20th July Morning Shift
Numerical
+4
-1
Let $$\overrightarrow a $$, $$\overrightarrow b $$, $$\overrightarrow c $$ be three mutually perpendicular vectors of the same magnitude and equally inclined at an angle $$\theta$$, with the vector $$\overrightarrow a $$ + $$\overrightarrow b $$ + $$\overrightarrow c $$. Then 36cos22$$\theta$$ is equal to ___________.
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4
JEE Main 2021 (Online) 20th July Morning Shift
Numerical
+4
-1
Let P be a plane passing through the points (1, 0, 1), (1, $$-$$2, 1) and (0, 1, $$-$$2). Let a vector $$\overrightarrow a = \alpha \widehat i + \beta \widehat j + \gamma \widehat k$$ be such that $$\overrightarrow a $$ is parallel to the plane P, perpendicular to $$(\widehat i + 2\widehat j + 3\widehat k)$$ and $$\overrightarrow a \,.\,(\widehat i + \widehat j + 2\widehat k) = 2$$, then $${(\alpha - \beta + \gamma )^2}$$ equals ____________.
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