CSIR NET Maths previous year question papers

Let $f : \mathbb{C} \to \mathbb{C}$ be a polynomial map. For R > 0, let $\gamma_R : [0, 1] \to \mathbb{C}$ be the map t ↦ $Re^{2\pi{}it}$. Suppose that there exists $c \in \mathbb{R}$ such that $\int_{0}^{1}$ |(f ∘ $\gamma_R)(t) \gamma'_R(t)$| dt → c as $R \to \infty$. Which of the following statements is FALSE?

Let f be an entire function such that $f(\mathbb{C}) \subset$ {x + iy | y = x + 1}. Which of the following statements is true?