True or False?
The answer is True. A monetary expansion engineered by cutting taxes and printing money to cover the short-fall of revenue from expenditure will generally lead to some increase in perceived income and consumption when the price level is fixed. The increase in disposable income resulting from the tax cut represents a flow of additions to monetary wealth. This higher wealth will result in an increase in consumption, though the magnitude of the effect may not be large in relation to the tax cut. The IS curve will shift to the right. And the LM curve will end up shifting to the right by the same amount---any effect on it in excess of the shift in the IS curve will be neutralized by a loss of official foreign exchange reserves as the authorities maintain the official exchange rate.
You have learned that monetary policy will be ineffective under fixed exchange rates because equilibrium is determined by the intersection of IS and ZZ and fiscal policy does not work under flexible exchange rates because equilibrium is determined in that regime by the intersection of LM and ZZ. Fiscal policy works under fixed exchange rates because it operates on the IS curve and monetary policy is effective under flexible exchange rates because it operates on the LM curve.
The reason why monetary policy works in the situation outlined in this question is that it is takes the form of fiscal policy. It works not because it causes people to buy assets abroad and leads to a devaluation of the domestic currency---the usual mechanism through which monetary policy operates---but because it increases permanent income a little bit and causes people to spend more on consumption.