KC Sinha Mathematics Solution Class 12 Chapter 6 सारणिक (Determinants) Exercise 6.1 (Q26-Q30)

Ex-6.1

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Exercise 6.1

Question 26

दिखाएँ कि [Show that]
$\left|\begin{array}{ccc}(y+z)^{2} & x y & z x \\ x y & (x+z)^{2} & y z \\ x z & z y & (x+y)^{2}\end{array}\right|=2 x y z(x+y+z)^{3}$

Sol: L.H.S
 $\left|\begin{array}{ccc}(y+z)^{2} & x y & z x \\ x y & (x+z)^{2} & y z \\ x z & z y & (x+y)^{2}\end{array}\right|$

$R_{1} \rightarrow x R_{1} , R_{2} \rightarrow y R_{2}, R_{3} \rightarrow zR_{3}$

$=\left|\begin{array}{ccc}x(y+z)^{2} & x^{2} y & =x^{2} \\ x y^{2} & y(x+z)^{2} & y^{2} z \\ x z^{2} & z^{2}, y & z(x+y)^{2}\end{array}\right|$

$=\frac{1}{xy z}\left|\begin{array}{lll}x(y+z)^{2} & x^{2} y & z x^{2} \\ x y^{2} & y(x+z)^{2} & y^{2} z \\ x z^{2} & z^{2} y & z(x+y)^{2}\end{array}\right|$

taking out x from C_{1 ,} y from C₂ , z from C₃

$=\frac{1}{x y z} \times x y z\left|\begin{array}{ccc}(y+z)^{2} & x^{2} & x^{2} \\ y^{2} & (x+z)^{2} & y^{2} \\ z^{2} & z^{2} & (x+y)^{2}\end{array}\right|$

C₁→C₁-C₂ , C₂→C₂-C₃

$=\left|\begin{array}{ccc}(y+z)^{2}-x^{2} & 0 & x^{2} \\ y^{2}-(x+z)^{2} & (x+z)^{2}-y^{2} & y^{2} \\ 0 & z^{2}-(x+y)^{2} & (x+y)^{2}\end{array}\right|$

$=\left|\begin{array}{ccc}(y+z+x)(y+z-x) & 0&x^2 \\ (y+x+z)(y-x-z) & (x+z+y)(x+z-y) & y^{2} \\ 0 & (z+x+y)(z-x-y) & (x+y)^{2}\end{array}\right|$

taking out (x+y+z) from C₁ and C₂

$=(x+y+z)^{2}\left|\begin{array}{ccc}y+z-x & 0 & x^{2} \\ y-x-z & x+z-y & y^{2} \\ 0 & z-x-y & (x+y)^{2}\end{array}\right|$

R₃→R₃-(R₁+R₂)

$=(x+y+z)^{2}\left|\begin{array}{ccc}y+z-x & 0 & x^{2} \\ y-x-z & x+z-y & y^{2} \\ 2 x-2 y & -2 x & 2 x y\end{array}\right|$

taking out 2 from R₃

$=2(x+y+z)^{2}\left|\begin{array}{ccc}y+z-x & 0 & x^{2} \\ y-x-z & x+z y & y^{2} \\ x-y & -x & x y\end{array}\right|$

taking out 2 from R₃

$=2(x+y+z)^{2}\left|\begin{array}{ccc}y+z-x & 0 & x^{2} \\ y-x-z & x+z- y & y^{2} \\ x-y & -x & x y\end{array}\right|$

R₂→R₂+R₃

$=2(x+y+x)^{2}\left|\begin{array}{ccc}y+z-x & 0 & x^{2} \\ -z & z-y & y^{2}+xy \\ x-y & -x & x y\end{array}\right|$

Expanding along R₁

$=2(x+y+z)^{2}\left[(y+z-z)\left|\begin{array}{cc}z-y & y^{2}+x y \\ -x & z y\end{array}\right|+x^{2} \bigg| \begin{array}{cc}-z & z-y \\ x -y & -x\end{array}\bigg| \right]$

=2(x+y+z)²[(y+z-z)(xyz-xy²+xy²+x²y)+x²(zx-zx+xy+yz-y²)]

=2(x+y+z)²[(y+z-z)(xy²z+x²y²+xyz²+x²yz-x²yz-x³y+x³y+x²yz-x²y²)]

=2(x+y+z)²xyz(y+z+x)

=2xyz(x+y+z)³

R₁→aR₁, R₂→bR₂ , R₃→cR₃

$=\frac{1}{a b c}\left|\begin{array}{ccc}1 & a^{3} & a b c \\ 1 & b^{3} & a b c \\ 1 & c^{3} & a b c\end{array}\right|$

taking out abc from C₃

$=\frac{1}{a b c} \times a b c\left|\begin{array}{ccc}1 & a^3 & 1 \\ 1 & b^{3} & 1 \\ 1 & c^{3} & 1\end{array}\right|$

= 0

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Question 27