1
JEE Main 2023 (Online) 10th April Evening Shift
+4
-1 Let $$\vec{a}=2 \hat{i}+7 \hat{j}-\hat{k}, \vec{b}=3 \hat{i}+5 \hat{k}$$ and $$\vec{c}=\hat{i}-\hat{j}+2 \hat{k}$$. Let $$\vec{d}$$ be a vector which is perpendicular to both $$\vec{a}$$ and $$\vec{b}$$, and $$\vec{c} \cdot \vec{d}=12$$. Then $$(-\hat{i}+\hat{j}-\hat{k}) \cdot(\vec{c} \times \vec{d})$$ is equal to :

A
24
B
42
C
44
D
48
2
JEE Main 2023 (Online) 10th April Evening Shift
+4
-1 If the points $$\mathrm{P}$$ and $$\mathrm{Q}$$ are respectively the circumcenter and the orthocentre of a $$\triangle \mathrm{ABC}$$, then $$\overrightarrow{\mathrm{PA}}+\overrightarrow{\mathrm{PB}}+\overrightarrow{\mathrm{PC}}$$ is equal to :

A
$$\overrightarrow {QP}$$
B
$$\overrightarrow {PQ}$$
C
$$2\overrightarrow {PQ}$$
D
$$2\overrightarrow {QP}$$
3
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
$$\frac{8}{3}$$
4
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
$${x^2} - x - 2 = 0$$
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