Basic economic principles?

First of all, the effect on the unemployment rate for a price change in one product would have to be so small that it would not be noticed at all, unless the product happens to be very widely used, so much so that it amounts to a noticably large percentage of the overall economy. I'm not sure why the price of a single product would even be part of a macroeconomics exam.

Second, if your answer considers the change in employment relating to this product only, regardless of how miniscule it is in the overall economy, the answer cannot be determined by the information given. If the price of the product decreases, the effect on employment will depend on what market (or other) forces caused the price change. Market forces causing a price change can be related to the supply of the product, or the demand for the product. Depending on which it is, you get a different answer.

For example, the price could decrease because new technology allows for the product to be produced more cheaper. Producers will be able to sell more of the product at a lower price, and still cover production costs. In this case, the supply will increase, which will increase the quantity demanded. In order to produce more, the producers will likely need to hire more workers, decreasing unemployment.

Or, on the other hand, the price could decrease because consumers suddenly decide to purchase fewer of this product; such as because of studies showing that the product is unsafe. In this case, the demand will decrease, which will decrease the quantity supplied. In order to meet the lower demand, producers will have to cut back on production, likely laying off workers, increasing unemployment.

Or the price could decrease because of a price ceiling, below the market price, imposed by the government. In this case, the quantity demanded will be higher than the quantity supplied, resulting in a shortage of that product. As long as the price ceiling is in effect, producers will receive a lower price, which will decrease profits. In order to cover costs (and make normal profits), they will cut back on production, which will increase unemployment. The shortage will get worse, and consumers will purchase what is available, and do without once the supplies run out. Of course, this could be an incentive for the development of an underground market in this product, which will allow consumers to purchase what they want at the higher market price, but this will happen "off the books", and the official unemployment rate will not factor in the additional employment this creates.

In short, the change in the unemployment rate will likely be so small as to be irrelevant, but even the small change will depend on the source of the price change. It cannot be determined by the information given.

You didn't explore the question enough. Sure there may be a larger demand for the product. But why do you equate larger demand with an automatic increase in supply? In actuallity as the increase in price isn't explained in this case AND one reason it happens could be a decrease in demand. Or it could be an increase in supply. From both of those points in time see bellow for what happens next.

What really happens is the price decreased so suppliers will likely be willing to produce less at the new lower price. Producing less means less employment and more unemployed.

The unemployment rate means that the question is in the area of macroeconomics. The price of a product is in the area of microeconomics. So the answer is concerning the market of demand and supply of one product. It will increase quantity demanded. But the supply will decrease. Some guys will be unemployed and be on welfare. This causes the unemployment rate to rise.

They are not directly related. If the price of something decreases, it may be because there is little demand for the product. People have decided they prefer something else instead. Why do you think that would affect employment?

A price decreases and enables more people to buy it (increased demand). To meet that demand the supply increases, creating more jobs and lowering unemployment.