Question

Let $\overset{ \rightarrow }{x}$ and $\overset{ \rightarrow }{y}$ are $2$ non-zero and non-collinear vectors, then the largest value of $k$ such that the non-zero vectors $\left(k^{2} - 5 k + 6\right)\overset{ \rightarrow }{x}+\left(k - 3\right)\overset{ \rightarrow }{y}$ and $2\overset{ \rightarrow }{x}+5\overset{ \rightarrow }{y}$ are collinear is

• A. $3$
• B. $6$
• C. $\frac{12}{5}$
• D. $-1$

Answer 1

Answer: (C)

Let, $\overset{ \rightarrow }{a}=\left(k^{2} - 5 k + 6\right)\overset{ \rightarrow }{x}+\left(k - 3\right)\overset{ \rightarrow }{y}$
and $\overset{ \rightarrow }{b}=2\overset{ \rightarrow }{x}+5\overset{ \rightarrow }{y}$
$\because \overset{ \rightarrow }{a}$ and $\overset{ \rightarrow }{b}$ are collinear vectors
$\Rightarrow \frac{k^{2} - 5 k + 6}{2}=\frac{k - 3}{5}$
$\Rightarrow k=3,\frac{12}{5}$
But, for $k=3, \, \overset{ \rightarrow }{a}=0$ (not possible)