CAT 2003 Slot 2 — QA Question 28
Answer the following question based on the information given below.
Consider three circular parks of equal size with centres at A1, A2 and A3 respectively. The parks touch each other at the edge as shown in the figure (not drawn to scale). There are three paths formed by the triangles A1A2A3, B1B2B3 and C1C2C3, as shown. Three sprinters A, B, and C begin running from points A1, B1 and C1 respectively. Each sprinter traverses her respective triangular path clockwise and returns to her starting point.
If x and y are integers then the equation 5x + 19y = 64 has:
Answer & solution
- A
no solution for x < 300 and y < 0
- B
no solution for x > 250 and y > –100
a solution for 250 < x < 300
- D
a solution for –59 < y < –56
5x + 19y = 64 where x, y ∈ I
This means that the values of x have an interval of 19 between each other and the values of y will have an interval of 5 between each other.
Now, there are 2 possible cases; y could either be positive or negative:
Case 1:
When y = 1, then x = 9
When y = 6, then x = −10
When y = 11, then x = −29 and so on
You will notice that the values of y are in intervals of 5 and that of x are in intervals of 19.
Generally speaking, when y is positive, we will get integral values of x when y’s unit’s digit is either 1 or 6.
Case 2:
When y = −4, then x = 28
When y = −9, then x = 47
Again, the values of y are in intervals of 5 and that of x are in intervals of 19.
That is, when y is negative, we will get integral values of x when y’s unit’s digit is either 4 or 9.
Now, let’s evaluate the options:
Option 1: “no solution for x < 300 and y < 0” is False.
âµ According to Case 2, we should get integral values of x when y is −4, −9 or −14 and so on.
Option 2: “no solution for x > 250 and y > –100” is False.
According to Case 2, we should get integral values of x when y is −99, −94, −74 or −69 etc.
Now, when y = −74, x = 294
∴ A solution exists.
Option 3: “a solution for 250 < x < 300” is True.
âµ y = −74, x = 294 is a possible solution
Option 4: “a solution for –59 < y < –56” is False.
âµ From Case 2, when y is negative, we will get integral values of x only when y’s unit’s digit is either 4 or 9.
Hence, option (c).