$I f t h e r o o t s o f t h e e q u a t i o n x^{n} - p x^{n - 1} + q x^{n - 2} . . . . . . . . . . . - (\frac{10}{n})^{n} = 0 a r e r e a l (n i s o d d p o s i t i v e i n t e g e r), t h e n t h e m i n i m u m v a l u e o f p a n d q i s a a n d b . T h e v a l u e o f a + b w h e n n = 601 i s$ ___ (approximately)

Open in App

Solution

The given equation has n roots.
p is the sum of the roots.
q is the sum of the roots taken two at a time.
The number of terms in q is $^{n} C_{2}$ .
Use A.M $\geq$ G.M,
$\frac{p}{n} \geq {(p r o d u c t o f t h e r o o t s)}^{\frac{1}{n}}$
$p \geq 10$
$\frac{q}{^{n} C_{2}} \geq ({(p r o d u c t o f t h e r o o t s)}^{n - 1})^{\frac{2}{n (n - 1)}}$
$q \geq \frac{100 (n - 1)}{2 n}$
When n = 601,
$q \geq 50$

Suggest Corrections

Similar questions

f (x) = (2011 + x)^{n}

x

n

f (0) + f^{'} (0) + \frac{f " (0)}{2!} + . . . . + \frac{f^{(n - 1)} (0)}{(n - 1)!}

α

x^{5} + 1 = 0

α^{10 n + 2} + α^{5 n + 2} + α^{5 n}

a, b

x^{2} + p x + 1 = 0,

p^{2} = 1

k

k \sum n = 1 (a^{2^{n}} + b^{2^{n}})

If l, m, n are 3 positive roots of the equation (p, q $ϵ$ R), then the minimum value of l + 2m + 3n is

Find the smallest positive integer value of n for which $\frac{(1 + i)^{n}}{(1 - n)^{n - 2}}$ is a real number.

Join BYJU'S Learning Program