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4-2.Quadratic Equations and Inequations
hard
यदि समीकररण $x^2-7 x-1=0$ के मूल $a$ तथा $b$ हैं, तो $\frac{a^{21}+b^{21}+a^{17}+b^{17}}{a^{19}+b^{19}}$ का मान बराबर _______________ है।
A
$50$
B
$51$
C
$52$
D
$53$
(JEE MAIN-2023)
Solution
$x^2-7 x-1=0 < _b^a$
By newton's theorem
$S _{ n +2}-7 S _{ n +1}- S _{ n }=0$
$S _{21}-7 S _{20}- S _{19}=0$
$S _{20}-7 S _{19}- S _{18}=0$
$S _{19}-7 S _{18}- S _{17}=0$
$\frac{ S _{21}+ S _{17}}{ S _{19}}=\frac{ S _{21}+\left( S _{19}-7 S _{18}\right)}{ S _{19}}$
$=\frac{ S _{21}+ S _{19}-7\left( S _{20}-7 S _{19}\right)}{ S _{19}}$
$=\frac{50 S _{19}+\left( S _{21}-7 S _{20}\right)}{ S _{19}}$
$=51 \cdot \frac{ S _{19}}{ S _{19}}=51$
Standard 11
Mathematics
