1
JEE Main 2022 (Online) 28th June Evening Shift
+4
-1

Let $$\overrightarrow a = \alpha \widehat i + 2\widehat j - \widehat k$$ and $$\overrightarrow b = - 2\widehat i + \alpha \widehat j + \widehat k$$, where $$\alpha \in R$$. If the area of the parallelogram whose adjacent sides are represented by the vectors $$\overrightarrow a$$ and $$\overrightarrow b$$ is $$\sqrt {15({\alpha ^2} + 4)}$$, then the value of $$2{\left| {\overrightarrow a } \right|^2} + \left( {\overrightarrow a \,.\,\overrightarrow b } \right){\left| {\overrightarrow b } \right|^2}$$ is equal to :

A
10
B
7
C
9
D
14
2
JEE Main 2022 (Online) 28th June Evening Shift
+4
-1

Let $$\overrightarrow a$$ be a vector which is perpendicular to the vector $$3\widehat i + {1 \over 2}\widehat j + 2\widehat k$$. If $$\overrightarrow a \times \left( {2\widehat i + \widehat k} \right) = 2\widehat i - 13\widehat j - 4\widehat k$$, then the projection of the vector $$\overrightarrow a$$ on the vector $$2\widehat i + 2\widehat j + \widehat k$$ is :

A
$${1 \over 3}$$
B
1
C
$${5 \over 3}$$
D
$${7 \over 3}$$
3
JEE Main 2022 (Online) 27th June Evening Shift
+4
-1

Let $$\overrightarrow a$$ and $$\overrightarrow b$$ be the vectors along the diagonals of a parallelogram having area $$2\sqrt 2$$. Let the angle between $$\overrightarrow a$$ and $$\overrightarrow b$$ be acute, $$|\overrightarrow a | = 1$$, and $$|\overrightarrow a \,.\,\overrightarrow b | = |\overrightarrow a \times \overrightarrow b |$$. If $$\overrightarrow c = 2\sqrt 2 \left( {\overrightarrow a \times \overrightarrow b } \right) - 2\overrightarrow b$$, then an angle between $$\overrightarrow b$$ and $$\overrightarrow c$$ is

A
$${\pi \over 4}$$
B
$$-$$ $${\pi \over 4}$$
C
$${{5\pi } \over 6}$$
D
$${{3\pi } \over 4}$$
4
JEE Main 2022 (Online) 27th June Morning Shift
+4
-1

Let $$\overrightarrow a = \widehat i + \widehat j - \widehat k$$ and $$\overrightarrow c = 2\widehat i - 3\widehat j + 2\widehat k$$. Then the number of vectors $$\overrightarrow b$$ such that $$\overrightarrow b \times \overrightarrow c = \overrightarrow a$$ and $$|\overrightarrow b | \in$$ {1, 2, ........, 10} is :

A
0
B
1
C
2
D
3
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