A) \[c<0\]
B) \[0<c<\frac{4}{3}\]
C) \[-\frac{4}{3}<c<0\]
D) \[c>0\]
Correct Answer: C
Solution :
For an obtuse angle \[(cx\,\,\widehat{\mathbf{i}}-6\widehat{\mathbf{j}}+3\widehat{\mathbf{k}})\cdot (x\widehat{\mathbf{i}}+2\widehat{\mathbf{j}}+2cx\,\,\widehat{\mathbf{k}})<0\] \[\Rightarrow \] \[c{{x}^{2}}-12+7cx<0\] \[\Rightarrow \] \[c{{x}^{2}}-6cx-12<0\] We know that, if \[a{{x}^{2}}+bx+c>\]or\[<0,\,\,\forall x\] Then, \[{{b}^{2}}-4ac<0\] \[\therefore \] \[{{(6c)}^{2}}-4c(-12)<0\] \[\Rightarrow \] \[3{{c}^{2}}+4c<0\] \[\Rightarrow \] \[3{{c}^{2}}\left( c+\frac{4}{3} \right)<0\] \[\Rightarrow \] \[-\frac{4}{3}<c<0\]
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