Question

Assertion (A) : The set of all real numbers ‘$a$' such that $a^2 + 2a, 2a + 3, a^2 + 3a + 8$ are sides of a triangle is $(5, \infty)$.
Reason (R) : In a triangle, sum of two sides is greater than the third side and sides are positive.

• A. Both (A) & (R) are individually true & (R) is correct explanation of (A)
• B. Both (A) & (R) are individually true but (R) is not the correct (proper) explanation of (A)
• C. (A) is true but (R) is false
• D. (A) is false but (R) is true

Answer 1

Answer: (A)

$a^{2}+2 a > 0, a \in(-\infty,-2) \cup(0, \infty)$
Also, $2 a+3 > 0, a \in(-3 / 2, \infty)$
$a^{2}+3 a + 8 > 0 \forall a \in R$
Now, $2 a^{2}+5 a+8 > 2 a+3$
$ \Rightarrow 2 a^{2} + 3 a + 5 > 0 \forall a \in R$
Also, $a^{2} + 5 a + 11 > a^{2} + 2 a$
$ \Rightarrow 3 a + 11 > 0$
$ \Rightarrow a>-11 / 3$
$a^{2}+4 a + 3 > a^{2}+3 a + 8$
$ \Rightarrow a - 5 > 0$
$ \Rightarrow a > 5$
So, $a \in(5, \infty)$
(By combining all the equations)