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We locate the zeros of the modular forms $E_k^2(\tau) + E_{2k}(\tau),E_k^3(\tau) + E_{3k} (\tau),$ and $E_k(\tau)E_l(\tau) +E_{k+l}(\tau),$ where$E_k(\tau)$ is the Eisenstein series for the full modular group$\text{SL}_2(\mathbb{Z})$. By utilizing work of F.K.C. Rankin andSwinnerton-Dyer, we prove that for sufficiently large $k,l$, all zeros in thestandard fundamental domain are located on the lower boundary $\mathcal{A} = \{e^{i\theta} : \pi/2 \leq \theta \leq 2\pi/3\}$.

Zeros of certain combinations of Eisenstein series of weight 2k, 3k, and k + l