1
JEE Main 2020 (Online) 8th January Morning Slot
+4
-1
Out of Syllabus
Let the volume of a parallelopiped whose coterminous edges are given by

$$\overrightarrow u = \widehat i + \widehat j + \lambda \widehat k$$, $$\overrightarrow v = \widehat i + \widehat j + 3\widehat k$$ and

$$\overrightarrow w = 2\widehat i + \widehat j + \widehat k$$ be 1 cu. unit. If $$\theta$$ be the angle between the edges $$\overrightarrow u$$ and $$\overrightarrow w$$ , then cos$$\theta$$ can be :
A
$${7 \over {6\sqrt 3 }}$$
B
$${7 \over {6\sqrt 6 }}$$
C
$${5 \over 7}$$
D
$${5 \over {3\sqrt 3 }}$$
2
JEE Main 2020 (Online) 7th January Evening Slot
+4
-1
Let $$\overrightarrow a$$ , $$\overrightarrow b$$ and $$\overrightarrow c$$ be three unit vectors such that
$$\overrightarrow a + \vec b + \overrightarrow c = \overrightarrow 0$$. If $$\lambda = \overrightarrow a .\vec b + \vec b.\overrightarrow c + \overrightarrow c .\overrightarrow a$$ and
$$\overrightarrow d = \overrightarrow a \times \vec b + \vec b \times \overrightarrow c + \overrightarrow c \times \overrightarrow a$$, then the ordered pair, $$\left( {\lambda ,\overrightarrow d } \right)$$ is equal to :
A
$$\left( {{3 \over 2},3\overrightarrow a \times \overrightarrow c } \right)$$
B
$$\left( { - {3 \over 2},3\overrightarrow c \times \overrightarrow b } \right)$$
C
$$\left( { - {3 \over 2},3\overrightarrow a \times \overrightarrow b } \right)$$
D
$$\left( {{3 \over 2},3\overrightarrow b \times \overrightarrow c } \right)$$
3
JEE Main 2020 (Online) 7th January Morning Slot
+4
-1
A vector $$\overrightarrow a = \alpha \widehat i + 2\widehat j + \beta \widehat k\left( {\alpha ,\beta \in R} \right)$$ lies in the plane of the vectors, $$\overrightarrow b = \widehat i + \widehat j$$ and $$\overrightarrow c = \widehat i - \widehat j + 4\widehat k$$. If $$\overrightarrow a$$ bisects the angle between $$\overrightarrow b$$ and $$\overrightarrow c$$, then:
A
$$\overrightarrow a .\widehat i + 3 = 0$$
B
$$\overrightarrow a .\widehat k - 4 = 0$$
C
$$\overrightarrow a .\widehat i + 1 = 0$$
D
$$\overrightarrow a .\widehat k + 2 = 0$$
4
JEE Main 2019 (Online) 12th April Evening Slot
+4
-1
Out of Syllabus
Let $$\alpha$$ $$\in$$ R and the three vectors

$$\overrightarrow a = \alpha \widehat i + \widehat j + 3\widehat k$$, $$\overrightarrow b = 2\widehat i + \widehat j - \alpha \widehat k$$

and $$\overrightarrow c = \alpha \widehat i - 2\widehat j + 3\widehat k$$.

Then the set S = {$$\alpha$$ : $$\overrightarrow a$$ , $$\overrightarrow b$$ and $$\overrightarrow c$$ are coplanar} :
A
contains exactly two numbers only one of which is positive
B
is singleton
C
contains exactly two positive numbers
D
is empty
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