IPMAT Indore 2023 - QA (SA) Q14 Explanation

Explanation:

(8)

𝐺2𝑥     𝑥𝐺3

𝐺1𝑥     𝑥𝐺4

Let us assume there are x players in each of the four groups.

Let us consider a player from group-1(𝐺1)

He will play (x -1) matches within 𝐺1.

He will play 2x matches with players in 𝐺2.

He will play 3x matches with players in 𝐺3.

He will play 4x matches with players in 𝐺4.

Total matches played by this player = [x – 1 + 2x + 3x + 4x] = 10x - 1

According to the question, every player played matches in the similar manner.

Therefore, the total matches played by all the players = [ 10x – 1] × 4x

∴ Total unique matches played between any two players

= 1/2 [10x - 1] × 4x

= [10x - 1] × 2x

= 20𝑥2 − 2𝑥

According to the question, 20𝑥2 −2𝑥 > 1000

or 10𝑥^2 − 𝑥 > 500

or 10𝑥^2 − 𝑥 − 500 > 0

Solving the above inequality we get, x > 7

∴ 𝑥𝑚𝑖𝑛𝑖𝑚𝑢𝑚 = 8 Ans.