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We prove a type of the Lefschetz hyperplane section theorem on Q-Fano 3-foldswith Picard number one and $1/2(1,1,1)$-singularities by using somedegeneration method. As a byproduct, we obtain a new example of a Calabi-Yau3-fold $X$ with Picard number one whose invariants are $$(H_X^3, c_2 (X) \cdotH_X, {e} (X)) = (8, 44, -88),$$ where $H_X$, $e(X)$ and $c_2(X)$ are an amplegenerator of $\Pic(X)$, the topological Euler characteristic number and thesecond Chern class of $X$ respectively.

A type of the Lefschetz hyperplane section theorem on \Q-Fano 3-folds with Picard number one and $1/2(1,1,1)$-singularities