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4-2.Quadratic Equations and Inequations
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Let $p, q$ be integers and let $\alpha, \beta$ be the roots of the equation, $x^2-x-1=0$, where $\alpha \neq \beta$. For $n=0,1,2, \ldots$, let $a_n=$ $p \alpha^n+q \beta^n$.
$FACT$ : If $a$ and $b$ are rational numbers and $a+b \sqrt{5}=0$, then $a=0=b$.
($1$) $a_{12}=$
$[A]$ $a_{11}-a_{10}$ $[B]$ $a_{11}+a_{10}$ $[C]$ $2 a_{11}+a_{10}$ $[D]$ $a_{11}+2 a_{10}$
($2$) If $a_4=28$, then $p+2 q=$
$[A] 21$ $[B] 14$ $[C] 7$ $[D] 12$
answer the quetion ($1$) and ($2$)
A
$A,C$
B
$A,B$
C
$A,D$
D
$B,D$
(IIT-2017)
Solution
($1$) $\text { As } a_{n+1}-a_n-a_{n-1}=0$
$\text { So } a_{12}=a_{11}+a_{10}$
($2$) $a_4=2 a_0+3 a_1$
$a_4=(q-p)(3 \beta)+5 p+2 q=28$
$\Rightarrow p=q=4$
Standard 11
Mathematics
