Determine constants in partial fractions

1. $$\frac { 9 x ^ { 2 } } { ( x - 1 ) ^ { 2 } ( 2 x + 1 ) } = \frac { A } { ( x - 1 ) } + \frac { B } { ( x - 1 ) ^ { 2 } } + \frac { C } { ( 2 x + 1 ) }$$ Find the values of the constants \(A , B\) and \(C\).

16

1. Given that $$\frac { 1 } { 16 - 9 x ^ { 2 } } \equiv \frac { A } { 4 - 3 x } + \frac { B } { 4 + 3 x }$$ find the values of \(A\) and \(B\) 16
2. An empty container, in the shape of a cuboid, has length 1.6 metres, width 1.25 metres and depth 0.5 metres, as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{6a03a035-ff32-4734-864b-a076aa9cbec0-29_469_812_404_616} The container has a small hole in the bottom. Water is poured into the container at a rate of 0.16 cubic metres per minute.
   At time \(t\) minutes after the container starts to be filled, the depth of water is \(d\) metres and water leaks out at a rate of \(0.36 d ^ { 2 }\) cubic metres per minute. At time \(t\) minutes after the container starts to be filled, the volume of water in the container is \(V\) cubic metres. 16 (b) (i) Show that $$\frac { \mathrm { d } V } { \mathrm {~d} t } = \frac { 16 - 9 V ^ { 2 } } { 100 }$$ \includegraphics[max width=\textwidth, alt={}, center]{6a03a035-ff32-4734-864b-a076aa9cbec0-30_2493_1721_214_150} \includegraphics[max width=\textwidth, alt={}, center]{6a03a035-ff32-4734-864b-a076aa9cbec0-31_2492_1721_217_150} Question number Additional page, if required.
   Write the question numbers in the left-hand margin. Question number Additional page, if required.
   Write the question numbers in the left-hand margin. Question number Additional page, if required.
   Write the question numbers in the left-hand margin. \includegraphics[max width=\textwidth, alt={}, center]{6a03a035-ff32-4734-864b-a076aa9cbec0-36_2498_1723_213_148}

$$\frac{1 + 11x - 6x^2}{(x - 3)(1 - 2x)} \equiv A + \frac{B}{(x - 3)} + \frac{C}{(1 - 2x)}$$ Find the values of the constants \(A\), \(B\) and \(C\). [4]