Quant Boosters  Sagar Gupta  Set 3

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

Q11) 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

Q12) If the three altitudes in a triangle are 8 cm, 15 cm and 17 cm.which of the following could be the inradius 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

Q13) 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

Q14) 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 301 = 29 sec

Q15) 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

Q16) 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
100x = 33 * t
100/100x = 39/33
x = 200/13 = option A

Q17) Find the number of whole number solutions for A + B + C = 71 where 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

Q18) 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

Q19) There were 2 boxes A and B. Box A contained 1000 balls, consecutively numbered from 1 to 1000 such that there was only one number printed on each ball. Box B was initially empty. All the balls on which the number printed was a multiple of 7 were transferred to the box B by Amit. After that all the balls on which the number printed was a multiple of 27 were transferred to the box B by Bineet. Finally, all the balls in the box B on which the number printed was a multiple of 37 were transferred back to the box A by Ishu. The final count of the balls in Box B was :

a : 1000
b : 0
a : 1000  7k
b : 7k
a : 1000  7k  27m
b : 7k + 27m
a : 1000  7k  27m + 37n
b : 7k + 27m  37n
142 + 32  5 + 1 = 170

Q20) In a 800m race around a stadium having the circumference of 200m the top runner meets the last runner on the 5th minute of the race .If the top runner runs at twice the speed of the last runner,what is the time taken by the top runner to finish the race?
a) 20 min
b)15 min
c)10 min
d) 5 min