1
MHT CET 2024 4th May Morning Shift
MCQ (Single Correct Answer)
+2
-0

The number of distinct real values of $\lambda$, for which the vectors $-\lambda^2 \hat{i}+\hat{j}+\hat{k}, \hat{i}-\lambda^2 \hat{j}+\hat{k}$ and $\hat{i}+\hat{j}-\lambda^2 \hat{k}$ are coplanar, is

A
zero.
B
one.
C
two.
D
three.
2
MHT CET 2024 4th May Morning Shift
MCQ (Single Correct Answer)
+2
-0

Let the vectors $\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}$ and $\overline{\mathrm{d}}$ be such that $(\overline{\mathrm{a}} \times \overline{\mathrm{b}}) \times(\overline{\mathrm{c}} \times \overline{\mathrm{d}})=\overline{0}$. Let $\mathrm{P}_1$ and $\mathrm{P}_2$ be the planes determined by the pair of vectors $\bar{a}, \bar{b}$ and $\bar{c}, \bar{d}$ respectively, then the angle between $P_1$ and $P_2$ is

A
0
B
$\frac{\pi}{4}$
C
$\frac{\pi}{3}$
D
$\frac{\pi}{2}$
3
MHT CET 2024 4th May Morning Shift
MCQ (Single Correct Answer)
+2
-0

Let $\bar{a}, \bar{b}$ and $\overline{\mathrm{c}}$ be three vectors having magnitude 1,1 and 2 respectively. If $\overline{\mathrm{a}} \times(\overline{\mathrm{a}} \times \overline{\mathrm{c}})+\overline{\mathrm{b}}=\overline{0}$, then the acute angle between $\overline{\mathrm{a}}$ and $\overline{\mathrm{c}}$ is

A
$\frac{\pi}{6}$
B
$\frac{\pi}{4}$
C
$\frac{\pi}{3}$
D
$\frac{\pi}{2}$
4
MHT CET 2024 3rd May Evening Shift
MCQ (Single Correct Answer)
+2
-0

The vectors $\overline{\mathrm{a}}$ and $\overline{\mathrm{b}}$ are not perpendicular and $\overline{\mathrm{c}}$ and $\overline{\mathrm{d}}$ are two vectors satisfying $\overline{\mathrm{b}} \times \overline{\mathrm{c}}=\overline{\mathrm{b}} \times \overline{\mathrm{d}}$ and $\overline{\mathrm{a}} \cdot \overline{\mathrm{d}}=0$, then the vector $\overline{\mathrm{d}}$ is equal to

A
$\overline{\mathrm{b}}+\left(\frac{\overline{\mathrm{b}} \cdot \overline{\mathrm{c}}}{\overline{\mathrm{a}} \cdot \overline{\mathrm{b}}}\right) \overline{\mathrm{c}}$
B
$\overline{\mathrm{c}}-\left(\frac{\overline{\mathrm{a}} \cdot \overline{\mathrm{c}}}{\overline{\mathrm{a}} \cdot \overline{\mathrm{b}}}\right) \overline{\mathrm{b}}$
C
$\bar{b}-\left(\frac{\bar{b} \cdot \bar{c}}{\bar{a} \cdot \bar{b}}\right) \bar{c}$
D
$\overline{\mathrm{c}}+\left(\frac{\overline{\mathrm{a}} \cdot \overline{\mathrm{c}}}{\overline{\mathrm{a}} \cdot \overline{\mathrm{b}}}\right) \overline{\mathrm{b}}$
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