On the vector bundles over rationally connected varieties

Let X be a rationally connected smooth projective variety defined over C and E →X a vector bundle such that for every morphism γ :CP1 → X , the pullback γ E is trivial. We prove that E is trivial. Using this we show that if γ∗ E is isomorphic to L(γ) ⊕r for all γ of the above type, where L(γ) → CP1 is some line bundle, then there is a line bundle ζ over X such that E=ζ⊕r