Question

Assertion :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: In a triangle, sum of two sides is greater than the third side & sides are positive.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion.
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion.
• C. Assertion is correct but Reason is incorrect.
• D. Both Assertion and Reason are incorrect.

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion.

$Letx=a^2+2a>0,a\in (-\infty ,-2)\cup (0,\infty )Thus(0,\infty )$
$Lety=2a+3>0,a\in (-3/2,\infty )$
$Letz=a^2+3a+8>0\forall a\in R$
Now we have to check for the sum of two sides is greater than the third is true or not
Now, $(z<x+y)a^2+3a+8<2a+3+a^2+2a\Rightarrow a>5\forall a\in R$
Also, $(x<y+z)a^2+2a<a^2+3a+8+2a+3\Rightarrow 6a+11>0\Rightarrow a>-\frac{11}{6}$
$(y<x+z)2a+3<a^2+3a+8+a^2+2a\Rightarrow x$ is Real
So, $\forall a\in (5,\infty )$