User:Pingusam

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EQUATION

$3x + 2y - 3z = -5\,$

$4x + 3y - 4z = -8\,$

$-5x - 4y + 4z = 9\,$

$B = \begin{bmatrix} 3 & 2 & -3 & | & 1 & 0 & 0 \\ 4 & 3 & -4 & | & 0 & 1 & 0 \\ -5 & -4 & 4 & | & 0 & 0 & 1 \end{bmatrix}$

$B = \begin{bmatrix} -3 & -4 & 0 & | & 4 & 0 & 3 \\ 4 & 3 & -4 & | & 0 & 1 & 0 \\ -5 & -4 & 4 & | & 0 & 0 & 1 \end{bmatrix}$

$B = \begin{bmatrix} -3 & -4 & 0 & | & 4 & 0 & 3 \\ -1 & -1 & 0 & | & 0 & 1 & 1 \\ -5 & -4 & 4 & | & 0 & 0 & 1 \end{bmatrix}$

$B = \begin{bmatrix} -3 & -4 & 0 & | & 4 & 0 & 3 \\ 0 & 1 & 0 & | & -4 & 3 & 0 \\ -5 & -4 & 4 & | & 0 & 0 & 1 \end{bmatrix}$

$B = \begin{bmatrix} -3 & -4 & 0 & | & 4 & 0 & 3 \\ 0 & 1 & 0 & | & -4 & 3 & 0 \\ 0 & -8 & -12 & | & 20 & 0 & 12 \end{bmatrix}$

$B = \begin{bmatrix} -3 & -4 & 0 & | & 4 & 0 & 3 \\ 0 & 1 & 0 & | & -4 & 3 & 0 \\ 0 & 0 & -12 & | & -12 & 24 & 12 \end{bmatrix}$

$B = \begin{bmatrix} -3 & 0 & 0 & | & -12 & 12 & 3 \\ 0 & 1 & 0 & | & -4 & 3 & 0 \\ 0 & 0 & -12 & | & -12 & 24 & 12 \end{bmatrix}$

$B = \begin{bmatrix} 1 & 0 & 0 & | & 4 & -4 & -1 \\ 0 & 1 & 0 & | & -4 & 3 & 0 \\ 0 & 0 & 1 & | & 1 & -2 & -1 \end{bmatrix}$

$B^{-1} = \begin{bmatrix} 4 & -4 & -1 \\ -4 & 3 & 0 \\ 1 & -2 & -1 \end{bmatrix}$

$\begin{bmatrix} x \\ y \\ z\end{bmatrix} = \begin{bmatrix} 4 & -4 & -1 \\ -4 & 3 & 0 \\ 1 & -2 & -1 \end{bmatrix} \begin{bmatrix} -5 \\ -8 \\ 9 \end{bmatrix} = \begin{bmatrix} 3 \\ -4 \\ 2\end{bmatrix}$

$x = 3\,$

$y=-4\,$

$z=2\,$

ECHELON

$6x + y + 2z = 5\,$

$8x + 3y + 4z = 5\,$

$2x - y - z = 6\,$

$A = \begin{bmatrix} 6 & 1 & 2 & | & 5 \\ 8 & 3 & 4 & | & 5 \\ 2 & -1 & -1 & | & 6 \end{bmatrix}$

$A = \begin{bmatrix} 8 & 0 & 1 & | & 11 \\ 8 & 3 & 4 & | & 5 \\ 2 & -1 & -1 & | & 6 \end{bmatrix}$

$A = \begin{bmatrix} 8 & 0 & 1 & | & 11 \\ 8 & 3 & 4 & | & 5 \\ 14 & 0 & 1 & | & 23 \end{bmatrix}$

$A = \begin{bmatrix} 8 & 0 & 1 & | & 11 \\ 0 & 3 & 3 & | & -6 \\ 14 & 0 & 1 & | & 23 \end{bmatrix}$

$A = \begin{bmatrix} -6 & 0 & 0 & | & -12 \\ 0 & 3 & 3 & | & -6 \\ 14 & 0 & 1 & | & 23 \end{bmatrix}$

$A = \begin{bmatrix} 1 & 0 & 0 & | & 2 \\ 0 & 3 & 3 & | & -6 \\ 14 & 0 & 1 & | & 23 \end{bmatrix}$

$x + 0y + 0z = 2\,$

$0x + 3y + 3z = -6\,$

$14x + 0y z = 23\,$

$x = 2\,$ as $[1]\,$

$3y + 3z = -6\,$ as $[2]\,$

$14x + z = 23\,$ as $[3]\,$

Substituting $[1]\,$ in to $[3]\,$

$14(2) + z = 23\,$

$28 + z = 23\,$

$z = -5\,$ as $[4]\,$

Substituting $[4]\,$ in to $[2]\,$

$3y + 3(-5) = -6\,$

$3y -15 = -6\,$

$3y = 9\,$

$y = 3\,$

MAGIC SQUARE

$a + g = 13\,$

$c + e = 13\,$

$b + f = 26\,$

$d + h = 24\,$

$a + b = 11\,$

$c + d = 15\,$

$e + f = 25\,$

$g + h = 25\,$

$e + g = 21\,$

$a + c + f+h = 34\,$

$M= \begin{bmatrix} 1 & 0 & 1 & 0 & 0 & 1 & 0 & 1\\ 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1\\ 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0\\ \end{bmatrix}$

$A= \begin{bmatrix} 34\\ 11\\ 15\\ 25\\ 25\\ 26\\ 24\\ 13 \end{bmatrix}$

$\begin{bmatrix} 1 & 0 & 1 & 0 & 0 & 1 & 0 & 1\\ 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1\\ 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0\\ \end{bmatrix} \begin{bmatrix} 34\\ 11\\ 15\\ 25\\ 25\\ 26\\ 24\\ 13 \end{bmatrix} = \begin{bmatrix} a\\ b\\ c\\ d\\ e\\ f\\ g\\ h \end{bmatrix}$

$\begin{bmatrix} a\\ b\\ c\\ d\\ e\\ f\\ g\\ h \end{bmatrix} = \begin{bmatrix} 1\\ 10\\ 4\\ 11\\ 9\\ 16\\ 12\\ 13 \end{bmatrix}$

MATHS B QUESTION 1 A

$y = 2(x-4)^3 - 7\,$

$y = 2((x-4)(x-4)(x-4)) - 7\,$

$y = 2((x^2-8x+16)(x-4)) - 7\,$

$y = 2(x^3 - 4x^2 - 8x^2 + 32x + 16x -64)-7\,$

$y = 2x^3 - 8x^2 - 16x^2 + 64x + 32x - 128 - 7\,$

$y = 2x^3 -24x^2 + 96x -135\,$

$a = 2\,$

$b = -24\,$

$c = 96\,$

$d = -135\,$

MATHS C LAST QUESTION

$A = \begin{bmatrix} 360\div1400 & 40\div1700 & 400\div2000 \\ 240\div1400 & 160\div1700 & 520\div2000 \\ 50\div1400 & 650\div1700 & 430\div2000 \end{bmatrix}$

$= \begin{bmatrix} 9\div35 & 40\div85 & 1\div5 \\ 6\div35 & 8\div85 & 13\div50 \\ 1\div28 & 1\div34 & 43\div200 \end{bmatrix}$

$D= \begin{bmatrix} 800 \\ 900 \\ 1000 \end{bmatrix}$

$P.Inp.= \begin{bmatrix} 750 \\ 850 \\ 650 \end{bmatrix}$

$Labour= \begin{bmatrix} 750\times .8\\ 850\times.6 \\ 650\times.7 \end{bmatrix}$

$= \begin{bmatrix} 600\\ 510 \\ 455 \end{bmatrix}$

$X = (I_3 - A)^{-1} D\,$

$X = \begin{bmatrix} 1766.40\\ 1995.42 \\ 2326.17 \end{bmatrix}$

$New Labour = \begin{bmatrix} 1766.40\times(3\div7)\\ 1995.42\times.3 \\ 2326.17\times.2275 \end{bmatrix}$

$= \begin{bmatrix} 756.86\\ 598.63 \\ 529.20 \end{bmatrix}$

EQUATION

$-2x - 2y + 3z = -5\,$

$6x + 5y - 8z = -8\,$

$-3x - y + 3z = 9\,$

$Q = \begin{bmatrix} -2 & -2 & 3 \\ 6 & 5 & -8 \\ -3 & -1 & 3 \end{bmatrix}$

${det}Q = ( -2 \times {det}\begin{bmatrix} 5 & -8\\ -1 & 3 \\ \end{bmatrix}) - ( -2 \times {det}\begin{bmatrix} 6 & -8\\ -3 & 3 \\ \end{bmatrix}) + ( 3 \times {det}\begin{bmatrix} 6 & 5\\ -3 & -1 \\ \end{bmatrix})$

${det}Q = (-2 \times ((5 \times 3) - (-8 \times -1))) - (-2 \times ((6 \times 3) - (-8 \times -3))) + (3 \times ((6 \times -1) - (5 \times -3)))$

${det}Q = (-2 \times 7) - (-2 \times -6) + (3 \times 9)$

${det}Q = 1\,$

↔↔↔↔↔↔↔↔↔↔↔↔

$Q_x = \begin{bmatrix} -3 & -2 & 3 \\ 5 & 5 & -8 \\ 5 & -1 & 3 \end{bmatrix}$

${det}Q_x = ( -3 \times {det}\begin{bmatrix} 5 & -8\\ -1 & 3 \\ \end{bmatrix}) - ( -2 \times {det}\begin{bmatrix} 5 & -8\\ 5 & 3 \\ \end{bmatrix}) + ( 3 \times {det}\begin{bmatrix} 5 & 5\\ 5 & -1 \\ \end{bmatrix})$

${det}Q_x = (-2 \times ((5 \times 3) - (-8 \times -1))) - (-2 \times ((5 \times 3) - (-8 \times 5))) + (3 \times ((5 \times -1) - (5 \times 5)))$

${det}Q_x = (-3 \times 7) - (-2 \times 55) + (3 \times -30)$

${det}Q = -1\,$

↔↔↔↔↔↔↔↔↔↔↔↔

$Q_y = \begin{bmatrix} -2 & -3 & 3 \\ 6 & 5 & -8 \\ -3 & 5 & 3 \end{bmatrix}$

${det}Q_y = ( -3 \times {det}\begin{bmatrix} 5 & -8\\ -1 & 3 \\ \end{bmatrix}) - ( -3 \times {det}\begin{bmatrix} 6 & -8\\ -3 & 3 \\ \end{bmatrix}) + ( 3 \times {det}\begin{bmatrix} 6 & 5\\ -3 & 5 \\ \end{bmatrix})$

${det}Q_y = (-2 \times ((5 \times 3) - (-8 \times 5))) - (-3 \times ((6 \times 3) - (-8 \times -3))) + (3 \times ((6 \times 5) - (5 \times -3)))$

${det}Q_y = (-2 \times 55) - (-3 \times -6) + (3 \times 45)$

${det}Q_y = 7\,$

↔↔↔↔↔↔↔↔↔↔↔↔

$Q_z = \begin{bmatrix} -2 & -2 & -3 \\ 6 & 5 & 5 \\ -3 & -1 & 5 \end{bmatrix}$

${det}Q_z = ( -2 \times {det}\begin{bmatrix} 5 & 5\\ -1 & 5 \\ \end{bmatrix}) - ( -2 \times {det}\begin{bmatrix} 6 & 5\\ -3 & 5 \\ \end{bmatrix}) + ( -3 \times {det}\begin{bmatrix} 6 & 5\\ -3 & -1 \\ \end{bmatrix})$