Q42 / 160 · TS EAMCET 2023 (14 May Shift 1) · Mathematics

Let $x=\left\{\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]: a, b, c, d \in \mathbb{R}\right\}$. If $f: X \rightarrow \mathbb{R}$ is defined by $\mathrm{f}(\mathrm{A})=\operatorname{det}(\mathrm{A}) \forall \mathrm{A} \in \mathrm{X}$, then $\mathrm{f}$ is

  • A
    one-one but not onto
  • Correct answer
    B
    onto but not one-one
  • C
    one-one and onto
  • D
    neither one-one nor onto

Solution

$x=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]=\mathrm{A}$ $\operatorname{det} \mathrm{A}=\mathrm{f}(\mathrm{A})=a d-b c$ Let $a=1, d=1, b=0, c=1$ $$ f(A)=1 $$ Also if $a=2, d=1, b=1, c=1$ $$ f(A)=1 $$ $\therefore f$ is not one-one but we know that corresponding to any number in rcal number set, we can define an determinant. $\therefore f$ is onto

Hence, the correct answer is option: 'B'.

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