Find the value of  $x^6+\frac{1}{x^6},$ if $x+\frac{1}{x}=3$ (Algebraic Formula)

Given that, total notes = 12
Let, the number of Tk. 10 notes = x 

The number of Tk. 5 notes = 12-x
According to the question 

$$\begin{gathered} \left(10\times x\right)+5\left(12-x\right)<95 \\ \Rightarrow 10x+60-5x<95 \\ \Rightarrow 5x<95-60 \\ \Rightarrow x<\frac{35}{5} \\ \therefore x=7 \end{gathered}$$

So, the number of Tk. 10 notes will be less than 7, i.e 6.

 Repeat: Pubali Bank Ltd. (TAT) 2017 

(A + B)'s 1 day's work $=\left(\frac{1}{15}+\frac{1}{10}\right)=\frac{1}{6}$ Part
. (A + B) 's 2 day's work $=\frac{2}{6}=\frac{1}{3}$ Part
Remaining work= $\left(1-\frac{1}{3}\right)=\frac{2}{3}$ Part
Now, part of work is done by A in 1 day

$\frac{2}{3}$ part of work is done by A in=$\frac{15\times 2}{3\times 1}$ = 10 days
Time taken to complete the work = 10+2 = 12 days

Repeat: Dhaka Bank Ltd. (TACO) 2018 

Repeat: Rupali Bank Ltd. (Officer - Cash) 2018

$$\begin{gathered} \frac{3}{x+2}+\frac{x-1}{x-5}=2 \\ \Rightarrow \frac{3\left(x-5\right)+\left(x-1\right)\left(x+2\right)}{\left(x+2\right)\left(x-5\right)}=2 \\ \Rightarrow \frac{3x-15+x^2+2x-x-2}{x^2-5x+2x-10}=2 \\ \Rightarrow \frac{x^2+4x-17}{x^2-3x-10}=2 \\ \Rightarrow 2x^2-6x-20=x^2+4x-17 \\ \Rightarrow x^2-10x-3=0 \\ Now u\sin g a{x}^2+bx+c=0 \\ Here, a=1;b=-10;c=-3 \\ We know, x=\frac{-b\pm \sqrt{b^2-4ac}}{2a} \\ x=\frac{-\left(-10\right)\pm \sqrt{{\left(-10\right)}^2-4\times 1\times \left(-3\right)}}{2\times 1} \\ \Rightarrow x=\frac{10\pm \sqrt{100+12}}{2} \\ \Rightarrow x=\frac{10\pm \sqrt{112}}{2} \\ \Rightarrow x=\frac{10\pm \sqrt{16\times 7}}{2} \\ \Rightarrow x=\frac{10\pm 4\sqrt{7}}{2} \\ \Rightarrow x=\frac{2\left(5\pm 2\sqrt{7}\right)}{2} \\ \therefore x=5\pm 2\sqrt{7} \end{gathered}$$