1
MHT CET 2023 12th May Morning Shift
MCQ (Single Correct Answer)
+2
-0

Two adjacent sides of a parallelogram $$\mathrm{ABCD}$$ are given by $$\overline{A B}=2 \hat{i}+10 \hat{j}+11 \hat{k}$$ and $$\overline{\mathrm{AD}}=-\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+2 \hat{\mathrm{k}}$$. The side $$\mathrm{AD}$$ is rotated by an acute angle $$\alpha$$ in the plane of parallelogram so that $$\mathrm{AD}$$ becomes $$\mathrm{AD}^{\prime}$$. If $$\mathrm{AD}^{\prime}$$ makes a right angle with side AB, then the cosine of the angle $$\alpha$$ is given by

A
$$\frac{8}{9}$$
B
$$\frac{\sqrt{17}}{9}$$
C
$$\frac{1}{9}$$
D
$$\frac{4 \sqrt{5}}{9}$$
2
MHT CET 2023 12th May Morning Shift
MCQ (Single Correct Answer)
+2
-0

If the area of the triangle with vertices $$(1,2,0)$$, $$(1,0,2)$$ and $$(0, x, 1)$$ is $$\sqrt{6}$$ square units, then the value of $x$ is

A
1
B
2
C
3
D
4
3
MHT CET 2023 12th May Morning Shift
MCQ (Single Correct Answer)
+2
-0

Let $$\overline{\mathrm{A}}$$ be a vector parallel to line of intersection of planes $$P_1$$ and $$P_2$$ through origin. $$P_1$$ is parallel to the vectors $$2 \hat{j}+3 \hat{k}$$ and $$4 \hat{j}-3 \hat{k}$$ and $$P_2$$ is parallel to $$\hat{j}-\hat{k}$$ and $$3 \hat{i}+3 \hat{j}$$, then the angle between $$\bar{A}$$ and $$2 \hat{i}+\hat{j}-2 \hat{k}$$ is

A
$$\frac{\pi}{3}$$
B
$$\frac{\pi}{2}$$
C
$$\frac{\pi}{6}$$
D
$$\frac{3 \pi}{4}$$
4
MHT CET 2023 12th May Morning Shift
MCQ (Single Correct Answer)
+2
-0

$$\overline{\mathrm{u}}, \overline{\mathrm{v}}, \overline{\mathrm{w}}$$ are three vectors such that $$|\overline{\mathrm{u}}|=1, |\bar{v}|=2,|\bar{w}|=3$$. If the projection of $$\bar{v}$$ along $$\bar{u}$$ is equal to projection of $$\bar{w}$$ along $$\bar{u}$$ and $$\bar{v}, \bar{w}$$ are perpendicular to each other, then $$|\bar{u}-\bar{v}+\bar{w}|=$$

A
4
B
$$\sqrt{7}$$
C
$$\sqrt{14}$$
D
2
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