KVPY Sample Paper KVPY Stream-SX Model Paper-1

The total number of distinct \[x\in [0,1]\] for which \[\int\limits_{0}^{x}{\frac{{{t}^{2}}}{1+{{t}^{4}}}dt=2x-1}\]

A) 1

B) 2

C) 3

D) 4

Correct Answer: A

Solution :

[a]

Let \[f(x)=\int\limits_{0}^{x}{\frac{{{t}^{2}}}{1+{{t}^{4}}}dt-2x+1}\]

\[\Rightarrow \]\[f'(x)=\frac{{{x}^{2}}}{1+{{x}^{4}}}-2\] as \[\frac{1+{{x}^{4}}}{{{x}^{2}}}\ge 2\]

\[\therefore \]\[\frac{{{x}^{2}}}{1+{{x}^{4}}}\le \frac{1}{2}\]

\[\Rightarrow \]\[f'(x)\le -\frac{3}{2}\]

\[\Rightarrow \]\[f\,(x)\]is continuous and decreasing.

\[\therefore \,\,f(0)=1\]and \[f(1)=\int\limits_{0}^{1}{\frac{{{t}^{2}}}{1+{{t}^{4}}}dt-2\le -\frac{3}{2}}\]

By intermediate value theorem \[f\,(x)=0\]possesses exactly one solution in [0, 1].