Ashita

From the question we know (1,-2)i.e x=1 &y=-2

Putting the values of x and y in the equation 

2x-3y=8

2*1-3*(-2)=8                  (2*1=2 and 3*(-2)=-6)

2-(-6)=8                          (-*-=+)

2+6=8

8=8

It proves that both sides are equal.   

Therefore ,(1,-2)is a solution of equation 2x-3y=8

Let the average speed of the passenger train be 'x' km/hr and the average speed of the express train be (x + 11) km/hr
Distance between Mysore and Bangalore = 132 km
It is given that the time taken by the express train to cover the distance of 132 km is 1 hour less than the passenger train to cover the same distance.
So, time taken by passenger train = 132/x hr
The time taken by the express train = {(132)/x+11} hr
Now, according to the question
{132/(x+11)} = 132/x + 1
After taking L.C.M. of 132/x +1 and then solving it we get (132+x)/x.
Now,
{132/(x+11)} = (132+x)/x
By cross multiplying, we get
132x = x²+132x+11x+1452
x² + 11x - 1452 = 0
x² + 44x - 33x - 1452 = 0
x(x+44) - 33(x+44) = 0
(x+33) (x+44) = 0
x - 44 or x = 33
As the speed cannot be in negative therefore, x = 33 or the speed of the passenger train = 33 km/hr and the speed is 33 + 11 = 44 km/hr.

The longest side of the triangles i.e. the hypotenuse.
So, by Pythagoras Theorem,
(Diagonal)² = (Smaller Side)² + (Longer Side)²
(x + 60)² = (x)² + (x + 30)²
x² + 120x + 3600 = x² + x² + 60x + 900
x² + 60x - 120x + 900 - 3600 = 0
x² - 60x - 2700 = 0
x² - 90x + 30x - 2700 = 0
x(x - 90) + 30(x - 90) = 0
(x - 90) (x + 30) = 0
x = 90 because x = -30 as length cannot be possible.
So the length of the shorter side is 90 meters and the length of the longer side is 90 + 30 = 120 meters.

Let the time taken by the smaller diameter tap = x

larger = x-10

total time taken = 75 /8

portion filled in one hour by smaller diameter tap = 1/x

and by larger diamter tap = 1/x-10

1/x + 1/x-10 = 8/75

x-10+x/x(x-10) = 8/75

2x+10/x²-10x = 8/75

8(x²-10x) = 75 ×2 (x-5)

8/2 (x²-10x) = 75 (x-5)

4x²-40x = 75x-375

4x² -40x-75x +375 = 0

4x²-115x + 375 = 0

4x²-100x-15x +375 = 0

4x(x-25)-15(x-25)=0

(x-25)(4x-15)

x= 25

x= 15/4

If x= 25 

the x-10 = 25-10 = 15

if x = 15/4

x-10 = 15/4 - 10 = 15-40/4 = -25/4

since time cannot be negative therefore x = 25.