KVPY Sample Paper KVPY Stream-SX Model Paper-11

Let f : \[[0,1]\to \text{R}\]be such that\[f(xy)=f(x)f(y)\] for all \[x,y\in [0,1]\] and\[f(0)\ne 0.\] If \[y=y(x)\] satisfies the differential equation, \[\frac{dy}{dx}=f(x)\] with \[y(0)=1,\]then \[y\left( \frac{1}{4} \right)+y\left( \frac{3}{4} \right)\] is equal to:

A) 4

B) 3

C) 5

D) 2

Correct Answer: B

Solution :

\[f(xy)=f(x).f(y)\]

\[f(0)=1\]as \[f(0)\ \ne 0\]

\[\Rightarrow \] \[f(x)=1\]

\[\frac{dy}{dx}=f(x)=1\]

\[\Rightarrow \] \[y=x+c\]

At, \[x=0,\text{ }y=1\]

\[\Rightarrow \] \[c=1\]

\[y=x+1\]

\[\Rightarrow \] \[y\left( \frac{1}{4} \right)+y\left( \frac{3}{4} \right)=\frac{1}{4}+1+\frac{3}{4}+1=3\]