1
JEE Main 2023 (Online) 29th January Evening Shift
+4
-1

Let $$\overrightarrow a = 4\widehat i + 3\widehat j$$ and $$\overrightarrow b = 3\widehat i - 4\widehat j + 5\widehat k$$. If $$\overrightarrow c$$ is a vector such that $$\overrightarrow c .\left( {\overrightarrow a \times \overrightarrow b } \right) + 25 = 0,\overrightarrow c \,.(\widehat i + \widehat j + \widehat k) = 4$$, and projection of $$\overrightarrow c$$ on $$\overrightarrow a$$ is 1, then the projection of $$\overrightarrow c$$ on $$\overrightarrow b$$ equals :

A
$$\frac{3}{\sqrt2}$$
B
$$\frac{1}{\sqrt2}$$
C
$$\frac{1}{5}$$
D
$$\frac{5}{\sqrt2}$$
2
JEE Main 2023 (Online) 29th January Morning Shift
+4
-1

If the vectors $$\overrightarrow a = \lambda \widehat i + \mu \widehat j + 4\widehat k$$, $$\overrightarrow b = - 2\widehat i + 4\widehat j - 2\widehat k$$ and $$\overrightarrow c = 2\widehat i + 3\widehat j + \widehat k$$ are coplanar and the projection of $$\overrightarrow a$$ on the vector $$\overrightarrow b$$ is $$\sqrt {54}$$ units, then the sum of all possible values of $$\lambda + \mu$$ is equal to :

A
24
B
0
C
18
D
6
3
JEE Main 2023 (Online) 25th January Evening Shift
+4
-1
Out of Syllabus

Let $$\overrightarrow a = - \widehat i - \widehat j + \widehat k,\overrightarrow a \,.\,\overrightarrow b = 1$$ and $$\overrightarrow a \times \overrightarrow b = \widehat i - \widehat j$$. Then $$\overrightarrow a - 6\overrightarrow b$$ is equal to :

A
$$3\left( {\widehat i + \widehat j + \widehat k} \right)$$
B
$$3\left( {\widehat i - \widehat j - \widehat k} \right)$$
C
$$3\left( {\widehat i + \widehat j - \widehat k} \right)$$
D
$$3\left( {\widehat i - \widehat j + \widehat k} \right)$$
4
JEE Main 2023 (Online) 25th January Evening Shift
+4
-1
Out of Syllabus

If the four points, whose position vectors are $$3\widehat i - 4\widehat j + 2\widehat k,\widehat i + 2\widehat j - \widehat k, - 2\widehat i - \widehat j + 3\widehat k$$ and $$5\widehat i - 2\alpha \widehat j + 4\widehat k$$ are coplanar, then $$\alpha$$ is equal to :

A
$${{73} \over {17}}$$
B
$$- {{73} \over {17}}$$
C
$$- {{107} \over {17}}$$
D
$${{107} \over {17}}$$
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