Question

If \[\left( {m - 1} \right)\left( {1 - k} \right) = 0\] , which of the following can be true?
I. \[m = 1\]
II. \[k = 1\]
III. \[m = k\]

A.None
B. \[II\] only
C. \[III\] only
D. \[I\] and \[II\] only
E. \[I,II\] , and \[III\] only

Answer 1

Hint: Here, in the question, we have been given an equation in the form of two variables \[m,k\] . And some values of the variables are given. We have to check which of the given values of \[m,k\] are correct. We will put each of the values given one by one and then check if that value satisfies the given equation or not. This will give us the desired result.

Complete step-by-step answer:
Given equation: \[\left( {m - 1} \right)\left( {1 - k} \right) = 0\;\,\, \ldots \left( 1 \right)\]
For \[m = 1\] , left hand side of equation \[\left( 1 \right)\] becomes,
 \[\left( {1 - 1} \right)\left( {1 - k} \right)\]
  \[ = 0\] , which is equal to the right hand side of the equation.
It means, \[m = 1\] satisfies the given equation. So it’s true.
For \[k = 1\] , left hand side of equation \[\left( 1 \right)\] becomes,
 \[\left( {m - 1} \right)\left( {1 - 1} \right)\]
  \[ = 0\] , which is equal to the right hand side of the equation.
It means, \[k = 1\] satisfies the given equation. So it’s true.
For \[m = k\] , left hand side of equation \[\left( 1 \right)\] becomes,
 \[\left( {k - 1} \right)\left( {1 - k} \right)\]
It is not the right hand side of the equation, which means \[m = k\] doesn’t satisfy the given equation. So it’s not true.
Therefore, only \[m = 1\] and \[k = 1\] are true. \[I\] and \[II\] only.
So, the correct answer is “Option D”.

Note: We have selected only two options i.e. \[m = 1\] and \[k = 1\] to be true out of the three options given. When we put \[m = k\] , we get \[\left( {k - 1} \right)\left( {1 - k} \right) = 0\] which further gives \[{k^2} - 2k + 1 = 0\] . Solving for the roots of this equation, we get \[{\left( {k - 1} \right)^2} = 0\] which further gives \[k = 1\] . It means \[m = k\] is true only and only when \[m = k = 1\] . For all other values except \[1\] , \[m = k\] is not true.