KC Sinha Mathematics Solution Class 12 Chapter 6 सारणिक (Determinants) Exercise 6.1 (Q31-Q36)
(i) $\left|\begin{array}{ll}2 & 4 \\ 5 & 1\end{array}\right|=\left|\begin{array}{cc}2 x & 4 \\ 6 & x\end{array}\right|$
Sol :
$\left|\begin{array}{cc}2 & 4 \\ 5 & 1\end{array}\right|=\left|\begin{array}{cc}2 x & 4 \\ 6 & x\end{array}\right|$
⇒2-20=2x2-24
⇒-18=2x2-24
⇒-18+24=2x2
⇒6=2x2
⇒x2=3
⇒$x=\pm \sqrt{3}$
(ii) $\left|\begin{array}{ll}3 & x \\ x & 1\end{array}\right|=\left|\begin{array}{ll}3 & 2 \\ 4 & 1\end{array}\right|$
Sol :
(iii) $\left|\begin{array}{ll}2 & 3 \\ 4 & 5\end{array}\right|=\left|\begin{array}{cc}x & 3 \\ 2 x & 5\end{array}\right|$
Question 32
निम्नलिखित समिकरणो को हल करें (Solve the following equations)
(i) $\left|\begin{array}{lll}a & a & x \\ a & a & a \\ b & x & b\end{array}\right|=0$
Sol :
$\left|\begin{array}{ccc}a & a & x \\ a & a & a \\ b & x & b\end{array}\right|=0$
R1→R1-R2
$\left[\begin{array}{ccc}0 & 0 & x-a \\ a & a & a \\ b & x & b\end{array}\right]=0$
$(x-a)\left|\begin{array}{cc}a & a \\ b & x\end{array}\right|=0$
$(x-a)(a x-a b)=0$
⇒x-a=0
⇒x=a
⇒ax-ab=0
⇒ax=ab
⇒x=b
(ii)$\left|\begin{array}{lll}x & 2 & 3 \\ 4 & x & 1 \\ x & 2 & 5\end{array}\right|=0$
Sol :
Sol :
$\left|\begin{array}{ccc}a & a & x \\ a & a & a \\ b & x & b\end{array}\right|=0$
R1→R1-R2
$\left[\begin{array}{ccc}0 & 0 & x-a \\ a & a & a \\ b & x & b\end{array}\right]=0$
$(x-a)\left|\begin{array}{cc}a & a \\ b & x\end{array}\right|=0$
$(x-a)(a x-a b)=0$
⇒x-a=0
⇒x=a
⇒ax-ab=0
⇒ax=ab
⇒x=b
(ii)$\left|\begin{array}{lll}x & 2 & 3 \\ 4 & x & 1 \\ x & 2 & 5\end{array}\right|=0$
Sol :
Question 33
निम्नलिखित समिकरणो को हल करें (Solve the following equations)
$\left|\begin{array}{ccc}x+a+b+c & b & c \\ x+a+b+c & x+b & c \\ x+a+b+c & b & x+c\end{array}\right|=0$
taking out (x+a+b+c) from C1
$(x+a+b+c)\left|\begin{array}{ccc}1 & b & c \\ 1 & x+b & c \\ 1 & b & x+c\end{array}\right|=0$
R1→R1-R2 , R2→R2-R3
$(x+a+b+c)\left|\begin{array}{ccc}0 & -x & 0 \\ 0 & x & -x \\ 1 & b & x+c\end{array}\right|=0$
Expanding along C1
$(x+a+b+c) \cdot \left| \begin{array}{rr}-x & 0 \\ x & -x\end{array}\right|=0$
$(x+a+b+c)\left(x^{2}-0\right)=0$
$(x+a+b+c) \cdot x^{2}=0$
$(x+a+b+c)\left(x^{2}-0\right)=0$
$(x+a+b+c) \cdot x^{2}=0$
⇒x+a+b+c=0
⇒x=-(a+b+c)
⇒$x^{2}=0$
⇒x=0
$\left|\begin{array}{ccc}x+a & b & c \\ a & x+b & c \\ a & b & x+c\end{array}\right|=0$
Sol :
$\left|\begin{array}{ccc}x+a & b & c \\ a & x+b & c \\ a & b & x+c\end{array}\right|=0$
C1→C1+C2+C3
$\left|\begin{array}{ccc}x+a+b+c & b & c \\ x+a+b+c & x+b & c \\ x+a+b+c & b & x+c\end{array}\right|=0$
taking out (x+a+b+c) from C1
$(x+a+b+c)\left|\begin{array}{ccc}1 & b & c \\ 1 & x+b & c \\ 1 & b & x+c\end{array}\right|=0$
R1→R1-R2 , R2→R2-R3
$(x+a+b+c)\left|\begin{array}{ccc}0 & -x & 0 \\ 0 & x & -x \\ 1 & b & x+c\end{array}\right|=0$
Expanding along C1
$(x+a+b+c) \cdot \left| \begin{array}{rr}-x & 0 \\ x & -x\end{array}\right|=0$
$(x+a+b+c)\left(x^{2}-0\right)=0$
$(x+a+b+c) \cdot x^{2}=0$
$(x+a+b+c)\left(x^{2}-0\right)=0$
$(x+a+b+c) \cdot x^{2}=0$
⇒x+a+b+c=0
⇒x=-(a+b+c)
⇒$x^{2}=0$
⇒x=0
Question 34
$\left|\begin{array}{ccc}15-2 x & 11-3 x & 7-x \\ 11 & 17 & 14 \\ 10 & 16 & 13\end{array}\right|=0$
Sol :
Question 35
$\left|\begin{array}{ccc}3 x-8 & 3 & 3 \\ 3 & 3 x-8 & 3 \\ 3 & 3 & 3 x-8\end{array}\right|=0$
Sol :
Question 36
$\left|\begin{array}{lll}x & 1 & 1 \\ 1 & x & 1 \\ 1 & 1 & x\end{array}\right|=0$
Sol :
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