- Home
- Standard 11
- Mathematics
4-2.Quadratic Equations and Inequations
hard
જો $a, b, c, d$ અને $p$ ભિન્ન વાસ્તવિક સંખ્યાઑ છે કે જેથી $(a^2 + b^2 + c^2)\,p^2 -2p\, (ab + bc + cd) + (b^2 + c^2 + d^2) \le 0$ થાય તો ...
A
$a, b, c, d$ સમાંતર શ્રેણીમાં છે
B
$ab =cd$
C
$ac = bd$
D
$a, b, c, d$ સમગુણોત્તર છે
(AIEEE-2012)
Solution
The given relation can be written as
$\left(a^{2} p^{2}-2 a b p+b^{2}\right)+\left(b^{2} p^{2}+c^{2}-2 b p c\right)+$
$\left(c^{2} p^{2}+d^{2}-2 p c d\right) \leq 0$
or $\quad(a p-b)^{2}+(b p-c)^{2}+(c p-d)^{2} \leq 0……….(1)$
since $a, b, c, d$ and $p$ are all real, the inequality $(1)$ is possible only when each of factor is zero
i.e., $a p-b=0, bp-c=0$ and $cp-d=0$
or $\quad p=\frac{b}{a}=\frac{c}{b}=\frac{d}{c}$
or $\quad a, b, c, d$ are in $G.P.$
Standard 11
Mathematics
