1
MHT CET 2023 10th May Morning Shift
+2
-0

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

A
$$\frac{2 \sqrt{2}}{3}$$
B
$$\frac{-\sqrt{2}}{3}$$
C
$$\frac{\sqrt{2}}{3}$$
D
$$\sqrt{\frac{2}{3}}$$
2
MHT CET 2023 9th May Evening Shift
+2
-0

$$\overline{\mathrm{a}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}, \overline{\mathrm{b}}=\hat{\mathrm{j}}-\hat{\mathrm{k}}$$, then vector $$\overline{\mathrm{r}}$$ satisfying $$\overline{\mathrm{a}} \times \overline{\mathrm{r}}=\overline{\mathrm{b}}$$ and $$\overline{\mathrm{a}} \cdot \overline{\mathrm{r}}=3$$ is

A
$$\frac{5}{3} \hat{\mathrm{i}}+\frac{2}{3} \hat{\mathrm{j}}+\frac{2}{3} \hat{\mathrm{k}}$$
B
$$-\frac{5}{3} \hat{\mathrm{i}}+\frac{2}{3} \hat{\mathrm{j}}+\frac{2}{3} \hat{\mathrm{k}}$$
C
$$\frac{5}{3} \hat{\mathrm{i}}-\frac{2}{3} \hat{\mathrm{j}}+\frac{2}{3} \hat{\mathrm{k}}$$
D
$$-\frac{5}{3} \hat{\mathrm{i}}+\frac{2}{3} \hat{\mathrm{j}}+\frac{1}{3} \hat{\mathrm{k}}$$
3
MHT CET 2023 9th May Evening Shift
+2
-0

The magnitude of the projection of the vector $$2 \hat{i}+\hat{j}+\hat{k}$$ on the vector perpendicular to the plane containing the vectors $$\hat{i}+\hat{j}+\hat{k}$$ and $$\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}$$ is

A
$$\frac{2}{\sqrt{6}}$$
B
$$\frac{1}{\sqrt{6}}$$
C
$$\frac{5}{\sqrt{6}}$$
D
$$\frac{7}{\sqrt{6}}$$
4
MHT CET 2023 9th May Evening Shift
+2
-0

If $$\bar{a}, \bar{b}$$ and $$\bar{c}$$ are any three non-zero vectors, then $$(\bar{a}+2 \bar{b}+\bar{c}) \cdot[(\bar{a}-\bar{b}) \times(\bar{a}-\bar{b}-\bar{c})]=$$

A
$$\left[\begin{array}{lll}\bar{a} & \bar{b} & \bar{c}\end{array}\right]$$
B
$$2\left[\begin{array}{lll}\overline{\mathrm{a}} & \overline{\mathrm{b}} & \overline{\mathrm{c}}\end{array}\right]$$
C
$$3\left[\begin{array}{lll}\bar{a} & \bar{b} & \bar{c}\end{array}\right]$$
D
$$4\left[\begin{array}{lll}\overline{\mathrm{a}} & \overline{\mathrm{b}} & \overline{\mathrm{c}}\end{array}\right]$$
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