Does ownership determine efficiency in an economic system? what determines efficiency in an economic system?
a) ownership
b) competition
c)both
d) none

I am confused between option (C) and option (B) because I am not sure about whether ownership can be said to be determining efficiency although it is often argued that private ownership brings efficiency. so if you can lend me some clarity would be grateful.
competition is certainly a force that brings forth efficiency so no doubt about it.
It's not unusual - in many real-world subjects - for there to be questions for which the right answer at one level of learning, under one teaching framework, is the wrong answer in a different context. This is one such question. It will depend as much on what the teaching framework's objectives are, and what its position is within normative economics.

As such, for your homework question, the right answer is whatever you've been taught is the right answer. You know what you've been taught. We do not.

So, given that, let's deal with the reality, rather than the homework question.

Ownership can matter ... but it's complicated.

Ultimately, competition matters most when it drives under-performing competitors out of the market. If a competitor can afford losses and still stay in business, then competition won't be fully effective: although the customers will benefit from lower prices, the push to greater efficiency won't flow up the supply chain. In theory, state-owned enterprises may be more able to sustain losses than private-owned ones, so ownership should matter. However, there are cases where private-owned enterprises have sustained losses for many years; just as there have been state-owned enterprises that do not tolerate losses and have cultivated economic efficiency.

Furthermore, if there are externalities, then one way to get a more efficient market is to have a state-owned enterprise rebalancing the market to price in those externalities. Indeed, one could take that further, and say that when externalities are the dominant part of the costs or benefits, and the lowest-cost supply is easy to establish, then a state-owned monopoly can provide the highest level of efficiency: in this case, there is neither competition nor private ownership in the most efficient market.