Verified Solution • 2025–26 Session moderate

QJEEM19Maa897: JEE Main 2019 Mathematics - Class 12 JEEM19Maa897 Solution: Let f : [0,1]   R be such that f(xy) = f(x).f(y), for all…

Let f : [0,1] →\to  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) satiesfies the differential equation, dydx{{dy} \over {dx}} = f(x) with y(0) = 1, then y (14)\left( {{1 \over 4}} \right) + y (34)\left( {{3 \over 4}} \right) is equal to :

Multiple Choice Options

  • A 3
  • B 4
  • C 2
  • D 5

Step-by-Step Verified Solution

If f(xy) = f(x) f(y) ∀\forall x, y ∈\in R and f(0) ≠\ne 0

put x = y = 0

⇒\Rightarrow   f(0) = [f(0)]2

⇒\Rightarrow   f(0) = 1

put y = 0 ⇒\Rightarrow  f(0) = f(x) f(0)

⇒\Rightarrow  f(x) = 1

given that dydx{{dy} \over {dx}} = f(x)

∴\therefore    dydx{{dy} \over {dx}} = 1 ⇒\Rightarrow  y = x + k

given that y(0) = 1

∴\therefore   k = 1

hence y = x + 1

y (14)\left( {{1 \over 4}} \right) + y (34)\left( {{3 \over 4}} \right) = (14+1)\left( {{1 \over 4} + 1} \right) + (34+1)\left( {{3 \over 4} + 1} \right) = 3

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