solve the following mathematical problem:

$if a=x{y}^{p-1}, b=x{y}^{q-1}, c=x{y}^{r-1} and p+q+r=3, then prove that a^{q-r}\times b^{r-p}\times c^{p-q}=1$

solve the following mathematical problem:

A family has 480kg of rice for X number of weeks. If they need to use the same amount for 4 more weeks, they need to cut down their weekly assumption of rice by 4kg. What is the value of X?

solve the following mathematical problem:

if $\cos \left(A+B\right)=\frac{1}{\sqrt{2}} and \cos \left(A-B\right)=\frac{\sqrt{3}}{2}, 0^{\circ }<\left(A+B\right)\le {90}^{\circ } and A>B, find the value of A and B.$

solve the following mathematical problem:

A shopkeeper sells two shirts at the same price. He makes 10% profit on one and loses 10% the other. How much in percentage does he gain or lose?

solve the following mathematical problem:

Price of 3 tables and 5 chairs is Tk. 2,000. Again, price of 5 tables and 7 chairs is Tk 3,200. What is the price of 1 table and 1 chair?

solve the following mathematical problem:

solve: $\frac{1}{2x}+\frac{6}{y}=3, \frac{5}{x}+\frac{3}{y}=11$