For an optimal control problem with fixed initial state and constraints on admissible controls we propose new second-order necessary optimality conditions using second-order tangents to the set of controls. In the absence of constraints on controls, our conditions are stronger than previously known second-order conditions. Then, we investigate optimal control problems with fixed initial state and mixed state-control equality constraints. Reducing these problems to problems without mixed constraints, we provide first- and second-order necessary optimality conditions. Our results improve some existing results. In particular, we do not make the assumption that the optimal control is piecewise continuous.