1
MHT CET 2023 9th May Evening Shift
+2
-0

Two adjacent sides of a parallelogram are $$2 \hat{i}-4 \hat{j}+5 \hat{k}$$ and $$\hat{i}-2 \hat{j}-3 \hat{k}$$, then the unit vector parallel to its diagonal is

A
$$\frac{3}{7} \hat{\mathrm{i}}-\frac{6}{7} \hat{\mathrm{j}}+\frac{2}{7} \hat{\mathrm{k}}$$
B
$$\frac{2}{7} \hat{\mathrm{i}}-\frac{6}{7} \hat{\mathrm{j}}+\frac{3}{7} \hat{\mathrm{k}}$$
C
$$\frac{6}{7} \hat{\mathrm{i}}-\frac{2}{7} \hat{\mathrm{j}}+\frac{3}{7} \hat{\mathrm{k}}$$
D
$$\frac{1}{7} \hat{\mathrm{i}}+\frac{1}{7} \hat{\mathrm{j}}-\frac{3}{7} \hat{\mathrm{k}}$$
2
MHT CET 2023 9th May Evening Shift
+2
-0

If $$\mathrm{D}, \mathrm{E}$$ and $$\mathrm{F}$$ are the mid-points of the sides $$\mathrm{BC}$$, $$\mathrm{CA}$$ and $$\mathrm{AB}$$ of triangle $$\mathrm{ABC}$$ respectively, then $$\overline{\mathrm{AD}}+\frac{2}{3} \overline{\mathrm{BE}}+\frac{1}{3} \overline{\mathrm{CF}}=$$

A
$$\frac{1}{2} \overline{\mathrm{AB}}$$
B
$$\frac{1}{2} \overline{\mathrm{AC}}$$
C
$$\frac{1}{2} \overline{\mathrm{BC}}$$
D
$$\frac{2}{3} \overline{\mathrm{AC}}$$
3
MHT CET 2023 9th May Morning Shift
+2
-0

If two vertices of a triangle are $$\mathrm{A}(3,1,4)$$ and $$\mathrm{B}(-4,5,-3)$$ and the centroid of the triangle is $$G(-1,2,1)$$, then the third vertex $$C$$ of the triangle is

A
$$(2,0,2)$$
B
$$(-2,0,2)$$
C
$$(0,-2,2)$$
D
$$(2,-2,0)$$
4
MHT CET 2023 9th May Morning Shift
+2
-0

Let two non-collinear vectors $$\hat{a}$$ and $$\hat{b}$$ form an acute angle. A point $$\mathrm{P}$$ moves, so that at any time $$t$$ the position vector $$\overline{\mathrm{OP}}$$, where $$\mathrm{O}$$ is origin, is given by $$\hat{a} \sin t+\hat{b} \cos t$$, when $$P$$ is farthest from origin $$O$$, let $$M$$ be the length of $$\overline{\mathrm{OP}}$$ and $$\hat{\mathrm{u}}$$ be the unit vector along $$\overline{\mathrm{OP}}$$, then

A
$$\hat{\mathrm{u}}=\frac{\hat{\mathrm{a}}+\hat{\mathrm{b}}}{|\hat{\mathrm{a}}+\hat{\mathrm{b}}|}$$ and $$\mathrm{M}=(1+\hat{\mathrm{a}} \cdot \hat{\mathrm{b}})^{\frac{1}{2}}$$
B
$$\hat{\mathrm{u}}=\frac{\hat{\mathrm{a}}-\hat{\mathrm{b}}}{|\hat{\mathrm{a}}-\hat{\mathrm{b}}|}$$ and $$\mathrm{M}=(1+\hat{\mathrm{a}} \cdot \hat{\mathrm{b}})^{\frac{1}{2}}$$
C
$$\hat{u}=\frac{\hat{a}+\hat{b}}{|\hat{a}+\hat{b}|}$$ and $$M=(1+2 \hat{a} \cdot \hat{b})^{\frac{1}{2}}$$
D
$$\hat{u}=\frac{\hat{a}-\hat{b}}{|\hat{a}-\hat{b}|}$$ and $$M=(1-2 \hat{a} \cdot \hat{b})^{\frac{1}{2}}$$
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