# Quant Boosters - Sagar Gupta - Set 10

• Find the maximum power of 12 in C(100,60)

100! / 60! * 40!
12 = 2^2 * 3
max power of 3 in 100! = 100/3+100/9+100/27+100/81 = 33+11+3+1 = 48
max power of 3 in 60! = 60/3 + 60/9 + 60/27 = 20 + 6 + 2 = 28
max power of 3 in 40! = 40/3 + 40/9 + 40/27 = 13 + 4 + 1 = 18
3^48 / 3^28 * 3^18 = ( 3^2 )
max power of 4 in 100! = 100/4 + 100/16 + 10/64 = 32
max power of 4 in 60! = 60/4 + 60/16 = 18
max power of 4! in 40! = 40/4 + 40/16 =12
2^32 / 2^18 * 2^12 = 2^2
so highest power is 1

A jet is flying 2400 miles from Hawaii to San Francisco. In still air, it flies at 600 mph. There is 40 mph tailwind in the same direction. Exactly how many hours after takeoff would it becomes neutral for the plane to either go to San Francisco or to return to Hawaii in the case of an emergency?
a. 1.25 hours
b. 1.5 hours
c. 1.75 hours
d. 2 hours

d / 640 = 2400 - d / 560
d = 1280 miles
1280 = 640 * t
t = 2 hours

Find the sum, S = 1 + 1/2 + 1/3 + 1/4 + 1/5 + ... + 1/2^32

Integrate it!
Integration 1/x = [ ln x ] , x goes from 0 to 2^32 : so answer simply is 32 * ln2 = 32 * 0.693 = 22

Two motorists set out at the same time to go from A to B, a distance of 100 miles. They both followed the same route and travelled at different, though uniform speeds of an integral number of miles per hour. The difference in their speeds was a prime number of miles per hour and after they had been driving for 2 hours, the distance of the slower car from A was 5 times that of the faster car from B. At what speed did the two motorists drive?

A-------100miles------------B
Sp and Sq and | Sp - Sq | = prime number
Sp * 2 = b
Sq *2 = 500 - 5b
10Sp + 2 Sq = 500
(42, 40) satisfies

If there are "n" students in a circular arrangement facing toward the center. It is known that their roll numbers are Integral value from 1 to "n" and they are sitting in the order of increasingroll numbers. Then what would be value of "n" if Roll number 13 is just opposite to Roll Number 29 ?

14 to 28 : 15 students
30 , 31 ,32 and 1 to 12 : 15 students
so total 32

If x/(2a+b) = y/(2b+c) = z/(2c+a) = 8 and x/2a = y/2b = z/2c = k where a+b+c not equal to 0 then k=?

x = 16a + 8b
y = 16b + 8c
z = 16c + 8a
x/2a = 8 + 4b/a
y/2b = 8 + 4c/b
z/2c = 8 + 4a/c
all are equal :
b/a = c/b = a/c : numbers are equal
8 + 4 = 12

How many liters of a 12 liter mixture containing milk and water in the ratio 2:3 be replaced with pure milk so that the resultant mixture contains milk and water in equal proportion?

milk = 4.8 + 3x/5
water = 7.2 - 3x/5
4.8 + 3x/5 = 7.2 - 3x/5
x=2

There are two vessels A and B, both containing vinegar solution of 40% concentration. I add some pure vinegar to A to bring the concentration to 50%. In the vessel B, I take out some quantity of the solution and replace it with an equal quantity of pure vinegar, to bring the concentration to 50%. What is the ratio of the amount of vinegar added in A and B, if the quantity of initial solutions in A and B are in the ratio of 1 : 2?

For A
vinegar : 40
water : 60
conc of vinegar = 50%

For B
vinegar =80 + 3x/5
water = 120-3x/5
80+3x/5 = 120 - 3x/5
x = 100/3 ml was added
20 : 100/3 = 3:5

In a polygon, the sum of all the exterior angles is equal to sum of all the interior angles. Each interior angle is equal to X degrees or its supplementary, where X is not equal to 90. How many of the angles must be X degrees?

Sum of all interior angles= Sum of all exterior angles = 360
x + x + 180 - x + 180 - x
total 2 angles have x degree

Two tangents pa and pb are drawn from an external point p length Length of pa=156 and length of ab =120 . Find radius of circle

Let O be the center and Angle APB = θ
Let AB cuts PA at X
AX =XB = 120/2 = 60
Sin θ = 60/156
Sin(90-θ) = 60/ r
Solve above : r =65

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