Q88 / 160 · WBJEE 2015 · Mathematics

If $(2+i)$ and $(\sqrt{5}-2 i)$ are the roots of the equation $\left(x^{2}+a x+b\right)\left(x^{2}+c x+d\right)=0$ where $a, b, c$ and $d$ are real constants, then product of all the roots of the equation is

  • A
    40
  • B
    $9 \sqrt{5}$
  • Correct answer
    C
    45
  • D
    35

Solution

If one root of a quadratic equation is of the form $a+i b,$ then other root will be $a-i b$ So, all the roots are $2 \pm 1, \sqrt{5} \pm 2 i$ $\therefore$ Product of all the roots $=(2+i)(2-i)(\sqrt{5}+2 i)(\sqrt{5}-2 i)$ $=(4+1)(5+4)=5 \times 9=45$

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

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