# Quant Boosters - Sagar Gupta - Set 8

• Find the number of positive solutions for 2x + 3y more than 60 and 2y + 3x less than 60

Draw 3x+2y = 60 and 2x+3y = 60
find the common region as per question
a triangle will be formed with cordinates (12,12 ) ( 0,20 ) ( 0,30 )
area of this triangle = 60
use picks theorem : A = i + b/2 - 1
61 = i + b/2
calculate the number of boundary pts of triangle boundary , total 20 boundary pts will be there
9 on the vertical line between 0,20 and 0,30
3 on the line between 0,20 and 12,12
5 on the line between 0,30 and 12,12
3 vertices
total 20 boundary points
61 - 20/2 = 51 integral points

Remainder of 1776^1816! divided by 2000 is ?

1776^1816! mod 2000
2000 = 16 * 125
1776^1816! mod 16 = 0
Euler of 125 is 100.
1816! mod 100 = 0.
1776^1816! mod 125 => 1716^0 mod 125 =1
16a = 125b+1
Remainder = 1376

In how many ways, 18 identical chocolates can be distributed among 3 identical boxes?

a + b + c = 18 total 20C2 = 190 ways
when all equal : 1 way
when 2 equal : 2a + c = 18 -> a = 0 to a = 9 , except a = 6 :
total 9 ways 3c2 * 9 = 27
Req cases = (190 - 1 - 3 * 9) / 3! + 1 + 9 = 37

If the three altitudes in a triangle are 8 cm, 15 cm and 17 cm.which of the following could be the in-radius of the triangle?
a) 9 cm
b) 16 cm
c) 4 cm
d) none of these

let sides be a,b,c opp to angles A,B,C
4a = 15b/2 = 17c/2 = k
we know that : area = inradius * semi perimeter
semi perimeter =
( a + (8a/15) + (8a/17) ) /2 = 511a/510
inradius = to find
4a = r * 511a / 510
r = 510 * 4 / 511 = 2040 / 511 , which is less than 4
Hence none of these

Find the number of pairs of x and y if 3x + 7y less than or equal to 210 (x and y are whole numbers)

3x+7y less than equal to 210
y=0 -> x= 0 to 70
y=1 -> x= 0 to 67
y=2 -> x= 0 to 65
y=3 -> x=0 to 63
. .
. .
y=30 -> x=0
and so on
(71 + 68 + 66) + (64 + 61 + 59) + (57 + 54 + 52) + ... 1
205 + 184 + 163 +142 + 121 + 100 + 79 + 58 + 37 + 16 + 1
1106

A bird is sitting on the last coach at the very end of a 100 m long train, travelling at speed of 36 km/ hr. The bird starts flying in the direction of the train at a speed 54 km/hr but rests on the train for 1 s after flying for every 2 s. Find the time taken by the bird to reach the starting of the engine?

100 = 5t
t = 20 seconds had there been no resting times , the bird would have reached in 20 seconds
for every 2 seconds , bird takes rest for 1 sec : therefore total time now is 30 sec
But for the last 2 sec when bird reaches the engine , bird doesn't take rest.. so 30-1 = 29 sec

A rod of length 30 units is to be cut into P parts so that the magnitude of the product of the lengths of the parts thus obtained is greater than 1 unit. The maximum possible value of P is

30/p , 30/p , 30/p , 30/p , 30/p.........
(30/p)^p > 1
p can be max 29

A and B are running on a circular track of length 3km in the clockwise direction. A laps B in the middle of A's 7th lap. If they are racing on a 100m dash, find the head start that A has to give B so that they finish the race in a dead heat?
a) 15 (5/13) m
b) 15 (7/13)m
c) 17 (2/11)m
d) None of the above

19.5 = Sa * t
16.5 = Sb * t
39 / 33 = Sa / Sb
100 = 39 * t
100-x = 33 * t
100/100-x = 39/33
x=200/13 =option A

Find the number of whole number solutions for A + B + C = 71 (A > B > C )

total whole number solutions => 73C2
73 * 72/2 = 73 * 36
a = b = c -> 0
a = b
2a + b = 71 -> a = 0 to a = 35
(73 * 36 - 3 * 36)/ 6 = 420

A 10% solution is one in which 10 gram of solute is dissolved in 100gram of water. On heating the solution water evaporates, so 1 kg of the 10% solution is reduced to 0.4 kg. What is the concentration now?

10 grams sugar : 100 grams water : 110 grams solution
1000/11 grams sugar : 10000/11 grams water : 1 kg solution
1000/11 grams sugar : 3400 /11 grams water : 0.4 kg solution
10 grams sugar in 34 grams water
10 * 100/34 grams sugar in 100 grams water , which is 29.4

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