QJEEM21Ma6d70: JEE Main 2021 Mathematics - Class 12 JEEM21Ma6d70 Solution: Let a be an integer such that all the real roots of the…

Let, $f(x) = 2{x^5} + 5{x^4} + 10{x^3} + 10{x^2} + 10x + 10$

$\Rightarrow f'(x) = 10({x^4} + 2{x^3} + 3{x^2} + 2x + 1)$

$= 10\left( {{x^2} + {1 \over {{x^2}}} + 2\left( {x + {1 \over x}} \right) + 3} \right)$

$= 10\left( {{{\left( {x + {1 \over x}} \right)}^2} + 2\left( {x + {1 \over x}} \right) + 1} \right)$

= 10{\left( {\left( {x + {1 \over x}} \right) + 1} \right)^2} > 0;\forall x \in R

$\therefore$ f(x) is strictly increasing function. Since, it is an odd degree polynomial it will have exactly one real root.

Now, by observation.

f( - 1) = 3 > 0

$f( - 2) = - 64 + 80 - 80 + 40 - 20 + 10$

= - 34 < 0

$\Rightarrow f(x)$ has at least one root in $( - 2, - 1) \equiv (a,a + 1)$

$\Rightarrow a = - 2$

$\Rightarrow$ |a| = - 2