Q74 / 125 · BITSAT 2012 · Mathematics

If $a$.b $=a$.c and $a \times b=a \times c$, then correct statement is

  • A
    $\mathrm{a} \|(\mathrm{b}-\mathrm{c})$
  • B
    $a \perp(b-c)$
  • Correct answer
    C
    $a=0$ or $b=c$
  • D
    None of these

Solution

$a . b .=a \cdot c \Rightarrow a \cdot(b-c)=0$ $\Rightarrow \mathrm{a}=0 \mathrm{or} \mathrm{b}-\mathrm{c}=0$ or $\mathrm{a} \perp(\mathrm{b}-\mathrm{c})$ $\Rightarrow \mathrm{a}=0$ or $\mathrm{b}=\mathrm{c}$ or $\mathrm{a} \perp(\mathrm{b}-\mathrm{c})$ Also a $x b=a x c \Rightarrow a \times(b-c)=0$ $\Rightarrow \mathrm{a}=0$ or $\mathrm{b}-\mathrm{c}=0$ or a $\|(\mathrm{b}-\mathrm{c})$ $\Rightarrow a=0$ or $b=c$ or $a \|(b-c)$ Observing to ( 1 ) and ( 2 ) we find that $\mathrm{a}=0$ or $\mathrm{b}=\mathrm{c}$

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

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