Which of the following pair of graphs intersect ?
(i) y = `x^(2) - x ` and y = 1
(ii) y = `x^(2) - 2x + 3` and y = sin x
(iii) `y`= `x^(2) - x + 1` and y = x - 4

Explanation

(i) ` y = x^(2) - x and y = 1" intersect if "x^(2) - x = 1 or x^(2)-x-1 = 0`, which has real roots .
Hence, the graphs intersect.
(ii) ` y = x^(2) - 2x+3 and y = sin x` intersect if ` x^(2) - 2x+3 = sin x or (x-1)^(2) + 2=sin x`, which is not possible since L.H.S. has least value 2, while R.H.S. has maximum value 1.
`(iii) y = x^(2)-x + 1 and y = x - 4" intersect if " x^(2) - x+1 = x-4 or x^(2) -2x + 5 = 0`, which has non-real roots. Hence, the graph do not intersect.