Mark the tick against the correct answer in the following:
Let $Q$ be the set of all rational numbers, and $*$ be the binary operation, defined by $a * b=a+2 b$, then
A. ${ }^{*}$ is commutative but not associative
B. $*$ is associative but not commutative
C. $*$ is neither commutative nor associative
D. $*$ is both commutative and associative
According to the question,
$\mathrm{Q}$ is set of all rarional numbers
$R=\{(a, b): a * b=a+2 b\}$
Formula
$*$ is commutative if $a * b=b * a$
$*$ is associative if $\left(a^{*} b\right) * c=a^{*}\left(b^{*} c\right)$
Check for commutative
Consider, $a * b=a+2 b$
Both equations will not always be true .
Therefore , * is not commutative ……. (1)
Check for associative
Consider, $(a * b) * c=(a+2 b) * c=a+2 b+2 c$
And,$a^{*}\left(b^{*} c\right)=a^{*}(b+2 c)=a+2(b+2 c)=a+2 b+4 c$
Both the equation are not the same and therefore will not always be true.
Therefore , * is not associative ……. (2)
Now , according to the equations (1) , (2)
Correct option will be (C)
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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,