1
MHT CET 2024 10th May Evening Shift
MCQ (Single Correct Answer)
+2
-0

Let two non-collinear unit vectors $\hat{\mathrm{a}}$ and $\hat{\mathrm{b}}$ form an acute angle. A point P moves, so that at any time $t$ the position vector $\overline{O P}$, where $O$ is the origin, is given by $\hat{a} \cos t+\hat{b} \sin t$. 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 $M=(1+\hat{\mathrm{a}} \cdot \hat{\mathrm{b}})^{\frac{1}{2}}$
B
$\hat{\mathrm{u}}=\frac{\hat{\mathrm{a}}-\hat{\mathrm{b}}}{|\hat{\mathrm{a}}-\hat{\mathrm{b}}|}$ and $M=(1+\hat{a} \cdot \hat{b})^{\frac{1}{2}}$
C
$\hat{\mathrm{u}}=\frac{\hat{\mathrm{a}}+\hat{\mathrm{b}}}{|\hat{\mathrm{a}}+\hat{\mathrm{b}}|}$ and $\mathrm{M}=(1+2 \hat{\mathrm{a}} \cdot \hat{\mathrm{b}})^{\frac{1}{2}}$
D
$\hat{\mathrm{u}}=\frac{\hat{\mathrm{a}}-\hat{\mathrm{b}}}{|\hat{\mathrm{a}}-\hat{\mathrm{b}}|}$ and $\mathrm{M}=(1-2 \hat{\mathrm{a}} \cdot \hat{\mathrm{b}})^{\frac{1}{2}}$
2
MHT CET 2024 10th May Evening Shift
MCQ (Single Correct Answer)
+2
-0

If $\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}$ are mutually perpendicular vectors having magnitudes $1,2,3$ respectively, then the value of $\left[\begin{array}{lll}\bar{a}+\bar{b}+\bar{c} & \bar{b}-\bar{a} & \bar{c}\end{array}\right]$ is

A
0
B
6
C
12
D
18
3
MHT CET 2024 10th May Evening Shift
MCQ (Single Correct Answer)
+2
-0

The vector of magnitude 6 units and perpendicular to vectors $2 \hat{i}+\hat{j}-3 \hat{k}$ and $\hat{\mathrm{i}}-2 \hat{\mathrm{j}}+\hat{\mathrm{k}}$ is

A
$2 \sqrt{3}(-\hat{i}+\hat{j}+\hat{k})$
B
$2 \sqrt{3}(\hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}})$
C
$2 \sqrt{3}(\hat{i}+\hat{j}+\hat{k})$
D
$2 \sqrt{3}(-\hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}})$
4
MHT CET 2024 10th May Evening Shift
MCQ (Single Correct Answer)
+2
-0

If $\overline{\mathrm{a}}=\hat{\mathrm{j}}-\hat{\mathrm{k}}$ and $\overline{\mathrm{c}}=\hat{\mathrm{i}}-\hat{\mathrm{j}}-\hat{\mathrm{k}}$, then the vector $\overline{\mathrm{b}}$ satisfying $\overline{\mathrm{a}} \times \overline{\mathrm{b}}+\overline{\mathrm{c}}=\overline{0}$ and $\overline{\mathrm{a}} \cdot \overline{\mathrm{b}}=3$ is

A
$-\hat{i}+2 \hat{j}-2 \hat{k}$
B
$-\hat{i}+\hat{j}-\hat{k}$
C
$-\hat{i}-\hat{j}+\hat{k}$
D
$\hat{i}+\hat{j}+\hat{k}$
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