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STD 12 - 10. vector algebra — Mathematics STD 12 Science — Question

CBSE BoardEnglish MediumSTD 12 ScienceMathematicsSTD 12 - 10. vector algebra1 Mark
MCQ
If $\vec u,\vec v,\vec w $ are non-coplanar vectors and $p,q$ are real numbers,then the equality $[3\vec u \,p\vec v\, p\vec w]^-[p\vec v\, \vec w\, q\vec u]^-[2\vec w\, q\vec v\, q\vec u]=0$ hold for: 
  • A
    exactly two values of  $ (p,q)$
  • B
    more than two but not all values of $ (p,q)$
  • C
    all values of $ (p,q)$
  • ✓
    exactly one value of $ (p,q)$
✓

Answer

Correct option: D.
exactly one value of $ (p,q)$
d
$[3 \bar{u} p \bar{v} p \bar{w}]-[p \bar{v} \bar{w} q \bar{u}]-[2 \bar{w} q \bar{v} q \bar{u}]=0$

$\Rightarrow 3 p^{2}[\bar{u} \bar{v} \bar{w}]-p q[\bar{u} \bar{v} \bar{w}]+2 q^{2}[\bar{u} \bar{v} \bar{w}]=0$

$\Rightarrow\left(3 p^{2}-p q+2 q^{2}\right)[\bar{u} \bar{v} \bar{w}]=0$

$[\bar{u} \bar{v} \bar{w}] \neq 0$

$\therefore 3 p^{2}-p q+2 q^{2}=0$

$\Rightarrow q^{2}-24 q^{2} \geq 0 \Rightarrow-23 q^{2} \geq 0$

$q^{2} \leq 0 \Rightarrow q=0$

And thus $p=0$

Thus $(p, q) \equiv(0,0)$ is only the possibility.

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