Question:
Let $f:[0,1] \rightarrow \mathrm{R}$ be such that $f(\mathrm{xy})=f(\mathrm{x}) . f(\mathrm{y})$ for all $\mathrm{x}, \mathrm{y}, \varepsilon[0,1]$, and $f(0) \neq 0$. If $\mathrm{y}=\mathrm{y}(\mathrm{x})$ satisfies the
differential equation, $\frac{\mathrm{dy}}{\mathrm{dx}}=f(\mathrm{x})$ with
$y(0)=1$, then $y\left(\frac{1}{4}\right)+y\left(\frac{3}{4}\right)$ is equal to
4
3
5
2
Correct Option: , 2
Solution:
$f(x y)=f(x) . f(y)$
$f(0)=1$ as $f(0) \neq 0$
$\Rightarrow f(x)=1$
$\frac{\mathrm{dy}}{\mathrm{dx}}=f(\mathrm{x})=1$
$\Rightarrow y=x+c$
At, $x=0, y=1 \Rightarrow \mathrm{c}=1$
$y=x+1$
$\Rightarrow \mathrm{y}\left(\frac{1}{4}\right)+\mathrm{y}\left(\frac{3}{4}\right)=\frac{1}{4}+1+\frac{3}{4}+1=3$
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All Study Material
- JEE Main
- Exam Pattern
- Previous Year Papers
- PYQ Chapterwise
- Physics
- Kinematics 1D
- Kinemetics 2D
- Friction
- Work, Power, Energy
- Centre of Mass and Collision
- Rotational Dynamics
- Gravitation
- Calorimetry
- Elasticity
- Thermal Expansion
- Heat Transfer
- Kinetic Theory of Gases
- Thermodynamics
- Simple Harmonic Motion
- Wave on String
- Sound waves
- Fluid Mechanics
- Electrostatics
- Current Electricity
- Capacitor
- Magnetism and Matter
- Electromagnetic Induction
- Atomic Structure
- Dual Nature of Matter
- Nuclear Physics
- Radioactivity
- Semiconductors
- Communication System
- Error in Measurement & instruments
- Alternating Current
- Electromagnetic Waves
- Wave Optics
- X-Rays
- All Subjects
- Physics
- Motion in a Plane
- Law of Motion
- Work, Energy and Power
- Systems of Particles and Rotational Motion
- Gravitation
- Mechanical Properties of Solids
- Mechanical Properties of Fluids
- Thermal Properties of matter
- Thermodynamics
- Kinetic Theory
- Oscillations
- Waves
- Electric Charge and Fields
- Electrostatic Potential and Capacitance
- Current Electricity
- Thermoelectric Effects of Electric Current
- Heating Effects of Electric Current
- Moving Charges and Magnetism
- Magnetism and Matter
- Electromagnetic Induction
- Alternating Current
- Electromagnetic Wave
- Ray Optics and Optical Instruments
- Wave Optics
- Dual Nature of Radiation and Matter
- Atoms
- Nuclei
- Semiconductor Electronics: Materials, Devices and Simple Circuits.
- Chemical Effects of Electric Current,
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