My Doubts: Elastic modulus should be in $N/m^2$ but here it is in Newtons only is this correct or they just mean here $k=30 N/m$ ? What do you want to say about $\lambda = 30 N$ ?
Do you think my equations are correct to solve for correct answer
My Try:
Let $x$ be the elongation, so we have $$T\cos\theta=mg$$ $$T\sin\theta=m\omega^2 r \;\;\;,\;\;\;r=(l+x)\sin\theta$$ $$T=kx$$
Using these three equations I can solve for $\theta\;\;,\;\; T$.
Is this correct ?
I suspect that the author defines elastic modulus here as $$\frac{\text{tension in the string}}{\text{relative extension}}\\=\text{tension in string}\times\frac{\text{original length}}{\text{elongation}},$$ which has units Newtons.
This replaces your third equation; everything else looks fine.