Search any question & find its solution
Question:
Answered & Verified by Expert
A relation $\mathrm{R}$ is defined on the set $\mathrm{N}$ of natural numbers as $\mathrm{xRy} \Rightarrow \mathrm{x}^{2}-4 \mathrm{xy}+3 \mathrm{y}^{2}=0$, Then which one of the following is
correct?
Options:
correct?
Solution:
1058 Upvotes
Verified Answer
The correct answer is:
$\mathrm{R}$ is reflexive, but neither symmetric nor transitive
Given, $x R Y \Rightarrow x^{2}-4 x y+3 y^{2}=0$
$\Rightarrow x^{2}-x y-3 x y+3 y^{2}=0$
$\Rightarrow x(x-y)-3 y(x-y)=0$
$\Rightarrow(x-y)(x-3 y)=0$
Reflexive property:
$x R x \Rightarrow(x-x)(-3 x)=0$
So, $\mathrm{R}$ is reflexive ...(1)
Symmetric property:
Let us check using an example $(1,2)$ and $(2,1)$ for $(1,2) \Rightarrow(1-2)(1-6)=(-1)(-5)=10$
For $(2,1) \Rightarrow(2-1)(2-3)=(1)(-1)=-1$
So, $\mathrm{R}$ is not symmetric ...(2)
Transitive property :
$\operatorname{For}(9 \mathrm{x}, 3 \mathrm{x}) \Rightarrow(9 \mathrm{x}-3 \mathrm{x})(9 \mathrm{x}-9 \mathrm{x})=0$
for $(3 x, x) \Rightarrow(3 x-3 x)(3 x-9 x)=0$
For $(9 x, x) \Rightarrow(9 x-x)(9 x-3 x)$
So, $(9 x, 3 x) \in R,(3 x, x) \in R$ but $(9 x, x) \notin R$
So, $\mathrm{R}$ is not transitive...(3)
(3) From (1), (2), (3), R is reflexive, but not symmetr ic and transitive.
$\Rightarrow x^{2}-x y-3 x y+3 y^{2}=0$
$\Rightarrow x(x-y)-3 y(x-y)=0$
$\Rightarrow(x-y)(x-3 y)=0$
Reflexive property:
$x R x \Rightarrow(x-x)(-3 x)=0$
So, $\mathrm{R}$ is reflexive ...(1)
Symmetric property:
Let us check using an example $(1,2)$ and $(2,1)$ for $(1,2) \Rightarrow(1-2)(1-6)=(-1)(-5)=10$
For $(2,1) \Rightarrow(2-1)(2-3)=(1)(-1)=-1$
So, $\mathrm{R}$ is not symmetric ...(2)
Transitive property :
$\operatorname{For}(9 \mathrm{x}, 3 \mathrm{x}) \Rightarrow(9 \mathrm{x}-3 \mathrm{x})(9 \mathrm{x}-9 \mathrm{x})=0$
for $(3 x, x) \Rightarrow(3 x-3 x)(3 x-9 x)=0$
For $(9 x, x) \Rightarrow(9 x-x)(9 x-3 x)$
So, $(9 x, 3 x) \in R,(3 x, x) \in R$ but $(9 x, x) \notin R$
So, $\mathrm{R}$ is not transitive...(3)
(3) From (1), (2), (3), R is reflexive, but not symmetr ic and transitive.
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.