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4-2.Quadratic Equations and Inequations
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यदि $a, b, c, d,-5$ तथा 5 के बीच की वास्तविक संख्याएँ इस प्रकार हैं कि $|a|=\sqrt{4-\sqrt{5-a}}, \quad|b|=\sqrt{4+\sqrt{5-b}}, \quad|c|=\sqrt{4-\sqrt{5+c}},|d|=\sqrt{4+\sqrt{5+a}}$ तब गुणांक $abcd$ क्या होगा ?
A
$11$
B
$-11$
C
$121$
D
$-121$
(KVPY-2017)
Solution
(a)
Given,
$|a|=\sqrt{4-\sqrt{5-a}}$
$|b|=\sqrt{4+\sqrt{5-b}}$
$|c|=\sqrt{4-\sqrt{5+c}}$
$|d|=\sqrt{4+\sqrt{5+d}}$
On squaring, we get
$a^2=4-\sqrt{5-a}$
$=a^2-4=-\sqrt{5-a}$
Again squaring, we get
$a^4-8 a^2+16=5-a$
$a^4-8 a^2+a+11=0$
Similarly, squaring other given equation and solving we can say that $a, b,-c,-d$ are roots of equation
$x^4-8 x^2+x+11=0$
$\therefore$ The product of roots
i.e. $a b c d=11$
Standard 11
Mathematics
