Question

The value of $c$, so that for all real $x$, the vectors $cx\, i -6 j +3 \hat{ k },\,\, x i +2 j +2 c x\hat{k}$ make an obtuse angle, are

• A. $ c < 0 $
• B. $ 0 < c < \frac{4}{3} $
• C. $ -\frac{4}{3} < c < 0 $
• D. $ c>0 $

Answer 1

Answer: (C)

For an obtuse angle
$(c x \hat{ i }-6 \hat{ j }+3 \widehat{ k }) \cdot(x \hat{ i }+2 \hat{ j }+2 c x \widehat{ k })<0$
$\Rightarrow c x^{2}-12+6 c x<0$
$\Rightarrow c x^{2} +6 c x- 12 < 0$
We know that, if
$a x^{2}+b x+c>$ or $<0, \forall x$
Then, $b^{2}-4 a c<0$
$\therefore (6 c)^{2}-4 c(-12)<0$
$\Rightarrow 3 c^{2}+4 c<0$
$\Rightarrow 3 c^{2}\left(c+\frac{4}{3}\right)<0$
$\Rightarrow -\frac{4}{3} < c < 0$