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  • Need Solution for R. D. Sharma Maths Class 12 Chapter determinants Exercise 5.2 Question 33 Maths Textbook Solution.

Need Solution for R. D.  Sharma Maths Class 12 Chapter determinants  Exercise 5.2 Question 33  Maths Textbook Solution.

Answers (1)

Answer: 4 a^{2} b^{2} c^{2}

Hint: Use determinant formula

Given: \left|\begin{array}{ccc} b^{2}+c^{2} & a b & a c \\ b a & c^{2}+a^{2} & b c \\ c a & c b & a^{2}+b^{2} \end{array}\right|=4 a^{2} b^{2} c^{2}

Solution:

\begin{aligned} &\text { L.H.S }\left|\begin{array}{ccc} b^{2}+c^{2} & a b & a c \\ b a & c^{2}+a^{2} & b c \\ c a & c b & a^{2}+b^{2} \end{array}\right| \\ &\text { Use } R_{1} \rightarrow a R_{1}, R_{2} \rightarrow b R_{2}, R_{3} \rightarrow c R_{3} \\ &=\frac{1}{a b c}\left|\begin{array}{ccc} a\left(b^{2}+c^{2}\right) & a^{2} b & a^{2} c \\ b^{2} a & b\left(c^{2}+a^{2}\right) & b^{2} c \\ c^{2} a & c^{2} b & c\left(a^{2}+b^{2}\right) \end{array}\right| \end{aligned}

\begin{aligned} &\text { Take common a from } \mathrm{R}_{1}, b \text { from } \mathrm{R}_{2} \text { and } \mathrm{c} \text { from } \mathrm{R}_{3}\\ &=\frac{a b c}{a b c}\left|\begin{array}{ccc} \left(b^{2}+c^{2}\right) & a^{2} & a^{2} \\ b^{2} & \left(c^{2}+a^{2}\right) & b^{2} \\ c^{2} & c^{2} & \left(a^{2}+b^{2}\right) \end{array}\right| \end{aligned}

\begin{aligned} &\text { Use } R_{1} \rightarrow R_{1}-R_{2}-R_{3} \\ &=\left|\begin{array}{ccc} 0 & -2 c^{2} & -2 b^{2} \\ b^{2} & \left(c^{2}+a^{2}\right) & b^{2} \\ c^{2} & c^{2} & \left(a^{2}+b^{2}\right) \end{array}\right| \end{aligned}

\begin{aligned} &\text { Expanding w.r.t } C_{1}\\ &=-b^{2}\left|\begin{array}{cc} -2 c^{2} & -2 b^{2} \\ c^{2} & a^{2}+b^{2} \end{array}\right|+c^{2}\left|\begin{array}{cc} -2 c^{2} & -2 b^{2} \\ c^{2}+a^{2} & b^{2} \end{array}\right|\\ &=-b^{2}\left(-2 a^{2} c^{2}-2 b^{2} c^{2}+2 b^{2} c^{2}\right)+c^{2}\left(-2 c^{2} b^{2}+2 b^{2} c^{2}+2 a^{2} b^{2}\right)\\ &=2 a^{2} b^{2} c^{2}+2 a^{2} b^{2} c^{2}=4 a^{2} b^{2} c^{2}\\ &=R . H . S \end{aligned}

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