6983manish wrote:sanju09 wrote:Value of the expression (n^2 - n + 1)/ (n - 1) cannot lie between
(A) -1, -3
(B) 1, -3
(C) -1, 2
(D) -1, 3
(E) 1, 3
I am not aware the exact mathematics approach for this kind of problem, but I try to find out the solution using plug-in methodology
(n^2-n+1) / ( n-1) = (n+1)(n^2-n+1) / ( n+1)(n-1)
= (n^3 +1) / ( n^2-1)
For n=-2 , value of expression comes -2.3
For n=-1 , value of expression comes 0
For n= 0 , value of expression comes -1
For n= 1 , value of expression comes "not defined"
For n= 1.5 , value of expression comes 3.5
For n= 2 , value of expression comes 3
For n= 3 , value of expression comes 3.5
First two options are out, as we have values under that range.
We cannot have the expression value when n=1.
Last 3 options seem to be in answer choices, but which one to select.
Please guide with accurate approach.
Assaulters please revise Quadratic Equations, Functions, and Inequalities.
If (n^2 - n + 1)/ (n - 1) = m, then question is simply asking for what's not within the range of m, not that of n here. Remember, in functions of the form y = f (x), we first need to derive it to the form x = f (y) in order to answer its range. In our case, we therefore need to find n in terms of m and then answer it for m. Here we go...
If (n^2 - n + 1)/ (n - 1) = m
then, (n^2 - n + 1) = m (n - 1)
or, n^2 - (m + 1) n + (m + 1) = 0
This is a quadratic in n, which may be solved by using the famous quadratic formula invented by an Indian genius
Aacharya Shridhar who proved that the roots of the quadratic of the form a x^2 + b x + c = 0 can be given by
x = [-b ± √(b^2 - 4 a c)]/ (2 a)
In our case, a = 1, b = -(m + 1), and c = m + 1, using we have
n = [(m + 1) ± √[(m + 1) ^2 - 4 × 1 × (m + 1)]/ (2 × 1)
Revise the concept of Discriminant too, for real roots (m + 1) ^2 - 4 × 1 × (m + 1) can never be negative, in other words
(m + 1) ^2 - 4 × 1 × (m + 1) ≥ 0
or, (m + 1) (m - 3) ≥ 0
We can see that m can take -1 and 3, and anything more than -1 and less than 3 won't make m defined.
Hence m or (n^2 - n + 1)/ (n - 1) cannot lie between [spoiler]
-1 and 3[/spoiler].
This is why [spoiler]
D[/spoiler] is the correct choice.
