1
JEE Main 2023 (Online) 10th April Morning Shift
+4
-1

Let O be the origin and the position vector of the point P be $$- \widehat i - 2\widehat j + 3\widehat k$$. If the position vectors of the points A, B and C are $$- 2\widehat i + \widehat j - 3\widehat k,2\widehat i + 4\widehat j - 2\widehat k$$ and $$- 4\widehat i + 2\widehat j - \widehat k$$ respectively, then the projection of the vector $$\overrightarrow {OP}$$ on a vector perpendicular to the vectors $$\overrightarrow {AB}$$ and $$\overrightarrow {AC}$$ is :

A
$$\frac{7}{3}$$
B
3
C
$$\frac{10}{3}$$
D
$$\frac{8}{3}$$
2
JEE Main 2023 (Online) 10th April Morning Shift
+4
-1

An arc PQ of a circle subtends a right angle at its centre O. The mid point of the arc PQ is R. If $$\overrightarrow {OP} = \overrightarrow u ,\overrightarrow {OR} = \overrightarrow v$$, and $$\overrightarrow {OQ} = \alpha \overrightarrow u + \beta \overrightarrow v$$, then $$\alpha ,{\beta ^2}$$ are the roots of the equation :

A
$${x^2} + x - 2 = 0$$
B
$$3{x^2} + 2x - 1 = 0$$
C
$$3{x^2} - 2x - 1 = 0$$
D
$${x^2} - x - 2 = 0$$
3
JEE Main 2023 (Online) 8th April Evening Shift
+4
-1
Out of Syllabus

Let the vectors $$\vec{u}_{1}=\hat{i}+\hat{j}+a \hat{k}, \vec{u}_{2}=\hat{i}+b \hat{j}+\hat{k}$$ and $$\vec{u}_{3}=c \hat{i}+\hat{j}+\hat{k}$$ be coplanar. If the vectors $$\vec{v}_{1}=(a+b) \hat{i}+c \hat{j}+c \hat{k}, \vec{v}_{2}=a \hat{i}+(b+c) \hat{j}+a \hat{k}$$ and $$\vec{v}_{3}=b \hat{i}+b \hat{j}+(c+a) \hat{k}$$ are also coplanar, then $$6(\mathrm{a}+\mathrm{b}+\mathrm{c})$$ is equal to :

A
12
B
6
C
0
D
4
4
JEE Main 2023 (Online) 8th April Evening Shift
+4
-1

The area of the quadrilateral $$\mathrm{ABCD}$$ with vertices $$\mathrm{A}(2,1,1), \mathrm{B}(1,2,5), \mathrm{C}(-2,-3,5)$$ and $$\mathrm{D}(1,-6,-7)$$ is equal to :

A
48
B
$$8 \sqrt{38}$$
C
54
D
$$9 \sqrt{38}$$
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