MCQ
Equation $\frac{3}{{x - {a^3}}} + \frac{5}{{x - {a^5}}} + \frac{7}{{x - {a^7}}} = 0,a > 1$ has
- ✓Two real and positive roots
- BTwo real and negative roots
- CNo real roots
- Done positive and other negative roots
${\rm{ Now }}\,\,\,a > 1 \Rightarrow {a^3} < {a^5} < {a^7}$
${\rm{ Let }}f(x) = 3\left( {x - {a^5}} \right)\left( {x - {a^7}} \right) + 5\left( {x - {a^5}} \right)\left( {x - {a^7}} \right) + $
${7\left(x-a^{3}\right)\left(x-a^{5}\right) /\left(x-a^{3}\right)\left(x-a^{3}\right)\left(x-a^{7}\right)} $
$f\left( {{a^{3 - }}} \right) < 0,f\left( {{a^{3 + }}} \right) > 0,f\left( {{a^{5 - }}} \right) < 0$
$f\left( {{a^{5 + }}} \right) > 0,f\left( {{a^{7 - }}} \right) < 0\,\,and\,\,f\left( {{a^{7 + }}} \right) > 0$
$\therefore$ Equation has one root lying in $\left(a^{3}, a^{5}\right)$ and other in $\left(a^{5}, a^{7}\right)$
Now $a>1 \therefore$ Roots are real and positive
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