30. Refer to Question 8. If the grower wants to maximise the amount of nitrogen added to the garden, how many bags of each brand should be added? What is the maximum amount of nitrogen added?

$\Delta -\left|\begin{array}{ccc}\hfill 3\hfill & \hfill -2\hfill & \hfill -k\hfill \\ \hfill 2\hfill & \hfill -4\hfill & \hfill -2\hfill \\ \hfill 1\hfill & \hfill 2\hfill & \hfill -1\hfill \end{array}\right|=\left|\begin{array}{ccc}\hfill 3\hfill & \hfill -2\hfill & \hfill -k\hfill \\ \hfill 0\hfill & \hfill -8\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 2\hfill & \hfill -1\hfill \end{array}\right|,\left[R_2-R_1-2R_3\right]$

= -8 (-3 + k)

For inconsistent $\Delta =0\Rightarrow k=3$  

${\Delta }_x=\left|\begin{array}{ccc}\hfill 10\hfill & \hfill -2\hfill & \hfill -k\hfill \\ \hfill 6\hfill & \hfill -4\hfill & \hfill -2\hfill \\ \hfill 5m\hfill & \hfill 2\hfill & \hfill -1\hfill \end{array}\right|=32-40m\ne 0\Rightarrow m\ne \frac{4}{5}$              

  $l={\displaystyle {\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}\left(\left[x\right]+\left[-sinx\right]\right)dx}$ . (i)

$I={\displaystyle {\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}\left(\left[-x\right]+\left[sinx\right]\right)dx}$ . (ii)

by using property $\left({\displaystyle \underset{a}{\overset{b}{\int }}f\left(x\right)dx=}{\displaystyle \underset{a}{\overset{b}{\int }}f\left(a+b-x\right)dx}\right)$

Adding (i) and (ii) we get 2l = ${\displaystyle \underset{-\frac{\pi }{2}}{\overset{\frac{\pi }{2}}{\int }}\left(-2\right)dx=-2\pi \Rightarrow l=-\pi }$

$l_{n,m}={\displaystyle \underset{0}{\overset{\frac{1}{2}}{\int }}\frac{x^n}{x^m-1}dx,m,n\in N,n>m}$

$l_{6+i,3}-l_{3+i,3}$

$A=\frac{1}{2^5}\left[\begin{array}{ccc}\hfill \frac{1}{5}\hfill & \hfill \frac{1}{5}\hfill & \hfill \frac{1}{5}\hfill \\ \hfill 0\hfill & \hfill \frac{1}{12}\hfill & \hfill \frac{1}{12}\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{28}\hfill \end{array}\right]$ $=\frac{B}{32}$

$\left|A\right|={\left(\frac{1}{32}\right)}^3\left|B\right|=\frac{1}{105.2^{19}}$

$\alpha +\beta +\gamma =2\pi$

$\Delta =\left|\begin{array}{ccc}\hfill 1\hfill & \hfill cos\gamma \hfill & \hfill cos\beta \hfill \\ \hfill cos\gamma \hfill & \hfill 1\hfill & \hfill cos\alpha \hfill \\ \hfill cos\beta \hfill & \hfill cos\alpha \hfill & \hfill 1\hfill \end{array}\right|$

$=1-co{s}^2\alpha -co{s}^2\beta -co{s}^2\gamma +2cos\alpha .cos\beta .cos\gamma$

$=-cos\gamma cos\left(\alpha -\beta \right)+cos\gamma cos\left(\alpha -\beta \right)=0$

Given 2x + y – z = 3         . (i)

x – y – z = α        . (ii)

3x + 3y + βz = 3                . (iii)

(i) x 2 – (ii) – (iii) – (1 + β) z = 3 - α

For infinite solution 1 + β