1
MHT CET 2023 14th May Morning Shift
+2
-0

If $$\bar{a}, \bar{b}, \bar{c}$$ are three vectors such that $$|\bar{a}+\bar{b}+\bar{c}|=1, \overline{\mathrm{c}}=\lambda(\overline{\mathrm{a}} \times \overline{\mathrm{b}})$$ and $$|\overline{\mathrm{a}}|=\frac{1}{\sqrt{3}},|\overline{\mathrm{b}}|=\frac{1}{\sqrt{2}},|\overline{\mathrm{c}}|=\frac{1}{\sqrt{6}}$$, then the angle between $$\bar{a}$$ and $$\bar{b}$$ is

A
$$\frac{\pi}{6}$$
B
$$\frac{\pi}{4}$$
C
$$\frac{\pi}{3}$$
D
$$\frac{\pi}{2}$$
2
MHT CET 2023 14th May Morning Shift
+2
-0

Let $$\bar{a}, \bar{b}, \bar{c}$$ be three vectors such that $$|\bar{a}|=\sqrt{3}, |\bar{b}|=5, \bar{b} \cdot \bar{c}=10$$ and the angle between $$\bar{b}$$ and $$\bar{c}$$ is $$\frac{\pi}{3}$$. If $$\bar{a}$$ is perpendicular to the vector $$\bar{b} \times \bar{c}$$, then $$|\bar{a} \times(\bar{b} \times \bar{c})|$$ is equal to

A
$$10 \sqrt{3}$$
B
$$5 \sqrt{3}$$
C
60
D
30
3
MHT CET 2023 14th May Morning Shift
+2
-0

If $$\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}$$ are unit vectors and $$\theta$$ is angle between $$\overline{\mathrm{a}}$$ and $$\bar{c}$$ and $$\bar{a}+2 \bar{b}+2 \bar{c}=\overline{0}$$, then $$|\bar{a} \times \bar{c}|=$$

A
$$\frac{\sqrt{15}}{2}$$
B
$$\frac{\sqrt{15}}{4}$$
C
$$\sqrt{15}$$
D
$$\frac{\sqrt{15}}{3}$$
4
MHT CET 2023 14th May Morning Shift
+2
-0

If $$\bar{a}, \bar{b}, \bar{c}$$ are three vectors with magnitudes $$\sqrt{3}$$, 1, 2 respectively, such that $$\bar{a} \times(\bar{a} \times \bar{c})+3 \bar{b}=\overline{0}$$, if $$\theta$$ is the angle between $$\bar{a}$$ and $$\bar{c}$$, then $$\sec ^2 \theta$$ is

A
1
B
$$\frac{3}{2}$$
C
$$\frac{4}{3}$$
D
$$\frac{2}{\sqrt{3}}$$
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