Let xk be real numbers such that x_k \geq k^4 + k^2 + 1 \for 1 \leq k \leq 2018. Denote N = \sum^{2018 }_{k=1} k . Consider the following inequalities :- I. (\sum^{2018 }_{k=1}kx_k )^2\leqN (\sum^{2018 }_{k= 1}kx_k^2 ) II. (\sum^{2018 }_{k=1}kx_k )^2\leqN(\sum^{2018 }_{k=1}k^2x_k^2 ) Then

Correct Answer is : both I and II are true

KVPY SX SB Exam 2018 Question Paper

Question : 18 of 120

Marks: +1, -0

Let xk be real numbers such that

x_k ≥ k^4 + k^2 + 1 \for 1 ≤ k ≤ 2018

. Denote

N = ∑↖{2018 }↙{k=1} k

. Consider the following inequalities :-
I.

(∑↖{2018 }↙{k=1}kx_k )^2≤N (∑↖{2018 }↙{k= 1}kx_k^2 )

II.

(∑↖{2018 }↙{k=1}kx_k )^2≤N(∑↖{2018 }↙{k=1}k^2x_k^2 )

Then

Solution:

I. Variance for variable

x_k

with frequency

k = {∑kx^2_k}/ {∑k}-({∑kx_k }/{∑k})^2≥0

II. Variance for variable kxk with frequency one =

{∑(kx_k)^2}/ {∑1}-({∑kx_k }/{∑1})^2≥0

⇒

2018 ∑k^2 x_k^2≥( ∑ kx_k )^2

( ∑1 =2018 )

⇒

N∑k^2 x_k^2≥( ∑ kx_k )^2

( N ≥2018 )

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