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rref in matlab : rref is a short form of row reduce Echelon form of a matrix. It is a matlab command to obtain row reduce Echelon form of a…

Question : Solve following system of linear equations x+2y+3z=1, 2x+y+3z=2, 5x+5y+9z=4 Solution : We have given system of linear equations \[x+2y+3z=1, 2x+y+3z=2, 5x+5y+9z=4 \]Writing these equations in Matrix form AX…

Question : Prove that following system of linear equation has unique solution x+2y+3z=1, 2x+y+3z=2, 5x+5y+9z=4 Solution : Given system of linear equations x+2y+3z=1, 2x+y+3z=2, 5x+5y+9z=4 Matrix form AX =B \[\left(\begin{matrix}1&2&3\\2&1&3\\5&5&9\\\end{matrix}\right)\left(\begin{matrix}x\\y\\z\\\end{matrix}\right)=\left(\begin{matrix}1\\2\\4\\\end{matrix}\right)\]…

Question : Find pgt math solution of the following problem Solution : Given system of linear equations x+2y+3z=1, 2x+y+3z=2, 5x+5y+9z=4 Matrix form AX =B \[\left(\begin{matrix}1&2&3\\2&1&3\\5&5&9\\\end{matrix}\right)\left(\begin{matrix}x\\y\\z\\\end{matrix}\right)=\left(\begin{matrix}1\\2\\4\\\end{matrix}\right)\] Augmented matrix \[\left[A:B\right]=\left(\begin{matrix}1&2&3&1\\2&1&3&2\\5&5&9&4\\\end{matrix}\right)\] Row reduce…

Question : Solve following up pgt math previous year question solution Solution : Let\[ A=\left|\begin{matrix}1&\log_x{y}&\log_x{z}\\\log_y{x}&1&\log_y{z}\\\log_z{x}&\log_z{y}&1\\\end{matrix}\right|\] \[A=1\left(1-\log_z{y}\log_y{z}\right)-\log_x{y}\left(\log_y{x}-\log_z{x}\log_y{z}\right)+\log_x{z}\left(\log_y{x}\log_z{y}-\log_z{x}\right)\] \[=1-\log_z{y}\log_y{z}-\log_x{y}\log_y{x}+\log_x{y}\log_z{x}\log_y{z}+\log_x{z}\log_y{x}\log_z{y}-\log_x{z}\log_z{x}\] By the property of logarithm \[\log_a{b}=\frac{\log_e{b}}{\log_e{a}}\] Therefore A=1-1-1+1+1-1 => A=0 Hence correct option…

Question : Find determinant of the following matrix \[A=\left[\begin{matrix}1&\log_x{y}&\log_x{z}\\\log_y{x}&1&\log_y{z}\\\log_z{x}&\log_z{y}&1\\\end{matrix}\right]\] Solution : Given matrix is \[A=\left[\begin{matrix}1&\log_x{y}&\log_x{z}\\\log_y{x}&1&\log_y{z}\\\log_z{x}&\log_z{y}&1\\\end{matrix}\right]\] Using formula of determinant of 3 by 3 matrix, It's determinant is given as \[determinant(A)=\left|\begin{matrix}1&\log_x{y}&\log_x{z}\\\log_y{x}&1&\log_y{z}\\\log_z{x}&\log_z{y}&1\\\end{matrix}\right|\]…

Logarithm product rule : The logarithm of product of two numbers a and b is the sum of logarithm of a and logarithm of b. This rule is called Logarithm…

Question : How to solve following problem of previous years solved question of pgt math Solution : \[x\vec{a}\ \ +y\vec{b}\ \ =0 \] (1) ⇒ \[ \vec{a}=-\frac{y}{x}\vec{b}\] Since \(\vec{a}\) and…

Question : how to solve following problem of pgt math past question paper solution Solution : \[\left(-i\right)^\frac{1}{2}=\left(\cos{\left(\frac{\pi}{2}\right)}-isin\left(\frac{\pi}{2}\right)\right)^\frac{1}{2}\] By Demoiver’s theorem \[\left(\cos{\left(\frac{\pi}{2}\right)}-isin\left(\frac{\pi}{2}\right)\right)^\frac{1}{2}\] \[=\cos{\left(\frac{\left(\frac{\pi}{2}+2\pi k\right)}{2}\right)}-isin\left(\frac{\left(\frac{\pi}{2}+2\pi k\right)}{2}\right)\ \ ,\ \ \ k=0,1\] When…

Question : Find pgt mathematics question solution Solution ; \[\left|z-\frac{4}{z}\right|=2\] \[\left|\left|z\right|-\left|\frac{4}{z}\right|\right|\le\left|z-\frac{4}{z}\right|\] \[\left|\left|z\right|-\left|\frac{4}{z}\right|\right|\le2\] Let \[\left|z\right|=r\] \[\left|r-\frac{4}{r}\right|\le2\] \[-2\le r-\frac{4}{r}\le2\] First \[-2\le r-\frac{4}{r}r^2+2r-4\geq0\] roots are \[r=-1\pm\sqrt5\] \[r\le-1-\sqrt5\geq0 or r\geq-1+\sqrt5\] Since r>0 \[r\geq-1+\sqrt5 \]…