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AP EAPCET 19-Aug-2021 Shift 1 Paper

Section: Mathematics

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Question : 14 of 160

Marks: +1, -0

The sum of the roots of the equation

e^{4 t}-10 e^{3 t}+29 e^{2 t}-22 e^{t}+4=0

is _______

$\log_{e}10$
$2\log_{e}2$
$\log_{2}29$
$2\log_{10}2$

Solution:

e^{4 t}-10 e^{3 t}+29 e^{2 t}-22 e^{t}+4=0

...(i)

Let

e^{t}=x

and roots of (1) be

t_{1}, t_{2}, t_{3}, t_{4}

\therefore x^{4}-10 x^{3}+29 x^{2}-22 x+4=0

Then, roots are

x_{1}, x_{2}, x_{3}, x_{4}

product of roots

x_{1} \cdot x_{2} \cdot x_{3} \cdot x_{4}=4

\Rightarrow e^{t_1}\cdot e^{t_2}\cdot e^{t_3} e^{t_4}=4

\Rightarrow e^{t_{1}+t_{2}+t_{3}+t_{4}}=4

t_{1}+t_{2}+t_{3}+t_{4}=\log_{e}\;4=\log_{e}\;2^{2}=2\log_{e}\;2

Sum of roots

=2\log_{e}\;2

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