Quant Boosters  Hemant Malhotra  Set 2

let base is n
61 * 61 = 4541
(1 + 6n) (1 + 6n) = 1 + 4n + 5n^2 + 4n^3
1 + 12n + 36n^2 = 4n^3 + 5n^2 + 4n + 1
so n = 8

Q28) What is the remainder when 27^11 + 9^11 + 3^11 + 1 is divided by 3^10  1

3^10 = x
so 27 * 3^30 + 3^2 * 3^20 + 3 * 3^11 + 1
so 27 * x^3 + 9x^2 + 3x + 1 put x = 1
so 27 + 9 + 3 + 1 = 40

Q29) What is the last digit of the LCM of 7^5678  1 and 7^5678 + 1 ?

HCF = 2
LCM * HCF = first number * second number
LCM * 2 = ( 7^5678  1) * (7^5678 + 1)
unit digit of 7^5678 = 9
so unit digit of 7^5678  1 will be 8
unit digit of 7^5678 + 1 will be 0
so unit digit of LCM will be 02nd method to solve this
LCM * 2 = (7^5678  1) * (7^5678 + 1)
= 49^5678  1
(491)(1 + 49 + 49^2 + ... 49^5677)/2 = 24 (1 + 49 + 49^2 + ... 49^5677)
49^k mod 10 = 1 when k is even and is 1 when k is odd
now check inside portion is zero

Q30) Seven Hockey teams A, B, C, D, E, F and G participated in a tournament. Each team played with every other team twice and D won all of its games and F lost all of its games. B and G scored equal points and are ahead of C. C, A and E also scored equal number of points. Each team gets 3 points, 1 point and no point for a win, draw and loss respectively. What is the highest possible number of points that can be scored by team A?

Total number of matches = 7C2 * 2 = 42
If there were no matches that resulted in a draw then total points = 42 * 3 = 126
For every match resulting in a draw the total points get reduced by 1.
Because in a WinLoss situation Winner + Loser = 3 + 0 = 3 points but in a draw it is 1 + 1 = 2 points.
Let the total points of A,C,E be y and that of B and G be x
3y + 2x + 36 = Total points [ also y < x ]
Total points can be 126 or 125 or 124 and so on
But our objective is to maximize y we have to take value of total as high as possible.
3y + 2x = 126 then max = 16
but if we take 3y + 2x = 125 then max(y) = 17 so OA=17

@hemant_malhotra Kindly explain the solution, I couldn't really comprehend it.

@Aayushiv This is a way of solving sets without venn diagrams.
Please go through the below notes which explains the method in great detail. Once you are done, revisit the problem and if the doubt persist, let us know. Will help you (I guarantee, it will be clear once you are done with the 2 notes given below!) :)https://www.mbatious.com/topic/376/settheorymaximaminimaconceptshemantmalhotra
https://www.mbatious.com/topic/379/settheorymaximaminimaconceptsgauravsharma

@hemant_malhotra shouldn't A point at the top so the distance become 2+3+2 = 7
as in room A at top face not at bottom face