NIMCET 2017 — Mathematics PYQ

Concept:
Nature of roots of a cubic polynomial:
- If the maximum and minimum are of opposite signs, the cubic has three real roots.
- If one of them is zero, two of the three roots are equal.
- If both of them are zero, all three roots are equal.
- If the maximum and minimum are of the same sign, the cubic has one real and two unreal imaginary roots.

Calculations:

Given equation is $(\mathbf{x}-\mathbf{a}{)}^3+(\mathbf{x}-\mathbf{b}{)}^3+(\mathbf{x}-\mathbf{c}{)}^3=0$
Consider, f( x) = ( x- a${)}^3$+ ( x- b${)}^3$+ ( x- c${)}^3$
Differentiating on both side, we get $\mathbf{f}(\mathbf{x})=3(\mathbf{x}-\mathbf{a}{)}^2+3(\mathbf{x}-\mathbf{b}{)}^2+3(\mathbf{x}-\mathbf{c}{)}^2$

f(x)>0

$\Rightarrow$f is increasing function

To find the maximum and minimum of function, put x= $\infty$ in f(x)
# Now, f( $\infty$) = $\infty$

f( - $\infty$) = $\infty$

Here, the maximum and minimum of the function are of the same sign.
Hence, The equation (x-a)${}^3+(x-b{)}^3+(x-c{)}^3=0$ has one real and two imaginary root.