KVPY Sample Paper KVPY Stream-SX Model Paper-20

If the function/given by \[f(x)={{x}^{3}}-3\,(a-2){{x}^{2}}\]\[+3ax+7,\] for some \[a\in R\] is increasing in \[(0,1]\] and decreasing in \[[1,5),\] then a root of the equation,\[\frac{f(4)-14}{{{(x-1)}^{2}}}=0(x\ne 1)is:\]

A) \[-7\]

B) 5

C) 7

D) 6

Correct Answer: C

Solution :

\[f'(x)=3{{x}^{2}}-6(a-2)x+3a\]

\[f'(x)\ge 0\forall x\in (0,1]\]

\[f'(x)\le 0\forall x\in [1,5)\]

\[\Rightarrow \] \[f'(x)=0\]at \[x=1\]\[\Rightarrow \]\[a=5\]

\[f(x)-14={{(x-1)}^{2}}(x-7)\]

\[\frac{f(x)-14}{{{(x-1)}^{2}}}=x-7\]

Hence, root of equation \[\frac{f(x)-14}{{{(x-1)}^{2}}}=0\,\]is 7.