1
MHT CET 2024 16th May Morning Shift
MCQ (Single Correct Answer)
+2
-0

Let $\overline{\mathrm{a}}, \overline{\mathrm{b}}$ and $\overline{\mathrm{c}}$ be three non-zero vectors such that no two of them are collinear and $(\overline{\mathrm{a}} \times \overline{\mathrm{b}}) \times \overline{\mathrm{c}}=\frac{1}{3}|\overline{\mathrm{~b}}||\mathrm{c}| \overline{\mathrm{a}}$. If $\theta$ is the angle between vectors $\bar{b}$ and $\bar{c}$, then the value of $\sin \theta$ is

A
$\frac{2}{3}$
B
$\frac{-2 \sqrt{2}}{3}$
C
$\frac{2 \sqrt{2}}{3}$
D
$\frac{-\sqrt{2}}{3}$
2
MHT CET 2024 16th May Morning Shift
MCQ (Single Correct Answer)
+2
-0

If $\left[\begin{array}{lll}\overline{\mathrm{a}} \times \overline{\mathrm{b}} & \overline{\mathrm{b}} \times \overline{\mathrm{c}} & \overline{\mathrm{c}} \times \overline{\mathrm{a}}\end{array}\right]=\lambda\left[\begin{array}{lll}\overline{\mathrm{a}} & \overline{\mathrm{b}} & \overline{\mathrm{c}}\end{array}\right]^2$, then $\lambda$ is equal to

A
3
B
0
C
1
D
2
3
MHT CET 2024 16th May Morning Shift
MCQ (Single Correct Answer)
+2
-0

If the vectors $\overline{A B}=3 \hat{i}+4 \hat{k}$ and $\overline{A C}=5 \hat{i}-2 \hat{j}+4 \hat{k}$ are the sides of the triangle $A B C$, then the length of the median, through $A$, is

A
$\sqrt{45}$ units.
B
 $\sqrt{18}$ units.
C
$\sqrt{72}$ units.
D
$\sqrt{33}$ units
4
MHT CET 2024 16th May Morning Shift
MCQ (Single Correct Answer)
+2
-0

If $\bar{a}$ and $\bar{b}$ are two unit vectors such that $\bar{a}+2 \bar{b}$ and $5 \overline{\mathrm{a}}-4 \overline{\mathrm{~b}}$ are perpendicular to each other, then the angle between $\bar{a}$ and $\bar{b}$ is

A
$\frac{\pi}{4}$
B
$\frac{\pi}{3}$
C
$\cos ^{-1}\left(\frac{1}{3}\right)$
D
$\cos ^{-1}\left(\frac{3}{7}\right)$
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