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10. vector algebra — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths10. vector algebra1 Mark
MCQ
If $a,\,b,\,c$ are non-coplanar vectors and $\lambda $ is a real number then $[\lambda (a + b)\,\,\,\,{\lambda ^2}b\,\,\,\,\,\lambda c] = \left[ {a\,\,b + c\,\,b} \right]$ for
  • A
    Exactly three values of $\lambda $
  • B
    Exactly two values of $\lambda $
  • C
    Exactly one value of $\lambda $
  • ✓
    No value of $\lambda $
✓

Answer

Correct option: D.
No value of $\lambda $
d
(d) $[\lambda (a + b)\,\,{\lambda ^2}b\,\,\lambda c] = [a\,\;b\,\, + c\,\,b]$

==> $\lambda (a + b).({\lambda ^2}b \times \lambda c)$$ = a.((b + c) \times b)$

==>$\lambda (a + b).{\lambda ^3}(b \times c)$$ = a.(b \times b + c \times b)$

==> ${\lambda ^4}[a.(b \times c) + b.(b \times c)] = a.(c \times b)$

==> ${\lambda ^4}[a\;b\;c] = - [a\;b\;c]$ ==> $[a\;b\;c]({\lambda ^4} + 1) = 0$

Since $a, b, c$  are non-coplanar, so $[a\;b\;c] \ne 0$

$\therefore $ ${\lambda ^4} = - 1$. Hence no real value of $\lambda $.

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