CAT 2019 – Slot 1 – Quantitative Ability – Let f(x) = min{2x2,52−5x}, where x is any positive real number

Q. 1: Let f(x) = min{2×2,52−5x}, where x is any positive real number. Then the maximum possible value of f(x) is 

for maximum possible value , 2×2= 52−5x

2×2+ 5x – 52 = 0

(x -4)*(x+6.5) = 0

So x = 4 ( as x is positive real number )

Maximum possible value of f(x) = 2×2= 52−5x = 32

 

Q. 2: The distance from A to B is 60 km. Partha and Narayan start from A at the same time and move towards B. Partha takes four hours more than Narayan to reach B. Moreover, Partha reaches the mid-point of A and B two hours before Narayan reaches B. The speed of Partha, in km per hour, is
1.  4
2   6
3.  5
4.  3

let time taken by Narayan to reach B = t hour

So time taken by Partha to reach B = (t+4) hours

Time taken by Partha to reach mid point of A and B = (t+ 4)/2

Given t – (t+ 4)/2 = 2

2t – t – 4 = 4

t = 8 hours

Speed of Partha = 60/(t+4) = 60/12 = 5 km/hour

Q. 3:Humans and robots can both perform a job but at different efficiencies. Fifteen humans and five robots working together take thirty days to finish the job, whereas five humans and fifteen robots working together take sixty days to finish it. How many days will fifteen humans working together (without any robot) take to finish it?
1. 36
2. 32
3. 45
4. 40

Given, 15/H+5/R=1/30  ——-1)

And 5/H+15/R=1/60  ——–2)

By solving eq 1) & eq 2) we get

1/(H )=1/(40×12)

Thus H = 40×12

So time taken by 15 humans = 40×12/15=32

Q. 4:When they work alone, B needs 25% more time to finish a job than A does. They two finish the job in 13 days in the following manner: A works alone till half the job is done, then A and B work together for four days, and finally B works alone to complete the remaining 5% of the job. In how many days can B alone finish the entire job?
1. 20
2. 16
3. 22
4. 18

ratio of time taken by A and B to complete the work alone = 1: 1.25 = 4x : 5x

As A complete half of the work alone  and B complete 5% of the work alone, so they will complete remaining 45% of work in 4 days .

So time taken by them to complete the whole work together = 4 * 100/45 = 80/9

Or (4x*5x)/(4x+5x) = 80/9

x = 4

Thus time taken by B to complete the work alone = 5x = 20 days

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