Question

A relation R is defined from C to R by xRy iff $\left| x \right| = y$ . Which of the following is correct?
(A) $\left( {2 + 3i} \right)R13$
(B) $3R\left( { - 3} \right)$
(C) $\left( {1 + i} \right)R2$
(D) $iR1$

Answer 1

The given question revolves around the concepts of relations and complex numbers. We are provided with a relation defined from C to R and some options related to the same. We are supposed to select the correct answer on the basis of the knowledge of the modulus of complex numbers. We calculate the modulus of the complex numbers given in all the options and then match with the given relation to find the correct answer to the problem.

Complete step by step solution:
The relation provided to us is defined from C to R by xRy iff $\left| x \right| = y$ .
Now, in option (A), we have, $\left( {2 + 3i} \right)R13$ .
So, we have to calculate the modulus of the complex number $\left( {2 + 3i} \right)$ .
We know that the modulus of complex number $Z = x + iy$ is $\left| Z \right| = \sqrt {{x^2} + {y^2}}$ where x and y are the real and imaginary parts of the complex number respectively.
So, we get the modulus of complex number $\left( {2 + 3i} \right)$ as,
$\left| {2 + 3i} \right| = \sqrt {{2^2} + {3^2}} = \sqrt {4 + 9} = \sqrt {13}$
Hence, $\left( {2 + 3i} \right)R\sqrt {13}$ .
So the option (A) is incorrect.

Now, in option (B), we have, $3R\left( { - 3} \right)$ .
We know that the modulus of a complex number is always positive.
So the option (B) is incorrect.

Now, in option (C), we have, $\left( {1 + i} \right)R2$ .
So, we have to calculate the modulus of the complex number $\left( {1 + i} \right)$ .
So, we get the modulus of complex number $\left( {1 + i} \right)$ as,
$\left| {1 + i} \right| = \sqrt {{1^2} + {1^2}} = \sqrt {1 + 1} = \sqrt 2$
Hence, $\left( {1 + i} \right)R\sqrt 2$.
So the option (C) is incorrect.

Now, in option (D), we have the complex number $i$ .
So, we have to calculate the modulus of the complex number $i$ .
So, we get the modulus of complex number $i$ as,
$\left| i \right| = \sqrt {{0^2} + {1^2}} = \sqrt 1 = 1$
Hence, $iR1$.
So the option (D) is correct.
So, the correct answer is “Option D”.

Note : The modulus of a complex number represents the distance of the point represented by the complex number on the Argand plane from the origin. We must know how to calculate the modulus of complex numbers in order to solve the given problem. We first understand the problem and then jump to the options to find which one of them is correct by judging them one by one.