Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • School
  • Need Solution for R.D.Sharma Maths Class 12 Chapter determinants Exercise 5.2 Question 40 Maths Textbook Solution.

Need Solution for R.D.Sharma Maths Class 12 Chapter determinants  Exercise 5.2 Question 40  Maths Textbook Solution.

Answers (1)

Answer: a b c\left(a^{2}+b^{2}+c^{2}\right)^{3}

Hint Use determinant formula

Given: \left|\begin{array}{ccc} -a\left(b^{2}+c^{2}-a^{2}\right) & 2 b^{3} & 2 c^{3} \\ 2 a^{3} & -b\left(c^{2}+a^{2}-b^{2}\right) & 2 c^{3} \\ 2 a^{3} & 2 b^{3} & -c\left(a^{2}+b^{2}-c^{2}\right) \end{array}\right|=a b c\left(a^{2}+b^{2}+c^{2}\right)^{3}

Solution:

\begin{aligned} &\text { L.H.S }\left|\begin{array}{ccc} -a\left(b^{2}+c^{2}-a^{2}\right) & 2 b^{3} & 2 c^{3} \\ 2 a^{3} & -b\left(c^{2}+a^{2}-b^{2}\right) & 2 c^{3} \\ 2 a^{3} & 2 b^{3} & -c\left(a^{2}+b^{2}-c^{2}\right) \end{array}\right|\\ &\text { Common a from } \mathrm{C}_{1}, b \text { from } \mathrm{C}_{2} \text { and c from } \mathrm{C}_{3}\\ &=a b c\left|\begin{array}{ccc} -\left(b^{2}+c^{2}-a^{2}\right) & 2 b^{2} & 2 c^{2} \\ 2 a^{2} & -\left(c^{2}+a^{2}-b^{2}\right) & 2 c^{2} \\ 2 a^{2} & 2 b^{2} & -\left(a^{2}+b^{2}-c^{2}\right) \end{array}\right| \end{aligned}

\begin{aligned} &\text { Apply } \mathrm{C}_{1} \rightarrow C_{1}+C_{2}+C_{3} \\ &=a b c\left|\begin{array}{ccc} -b^{2}-c^{2}+a^{2}+2 b^{2}+2 c^{2} & 2 b^{2} & 2 c^{2} \\ 2 a^{2}+2 c^{2}-c^{2}-a^{2}+b^{2} & -c^{2}+a^{2}-b^{2} & 2 c^{2} \\ 2 a^{2}+2 b^{2}-a^{2}-b^{2}+c^{2} & 2 b^{2} & -a^{2}+b^{2}-c^{2} \end{array}\right| \\ &=a b c\left|\begin{array}{ccc} a^{2}+b^{2}+c^{2} & 2 b^{2} & 2 c^{2} \\ a^{2}+b^{2}+c^{2} & -c^{2}+a^{2}-b^{2} & 2 c^{2} \\ a^{2}+b^{2}+c^{2} & 2 b^{2} & -a^{2}+b^{2}-c^{2} \end{array}\right| \\ &=a b c\left(a^{2}+b^{2}+c^{2}\right)\left|\begin{array}{ccc} 1 & 2 b^{2} & 2 c^{2} \\ 1 & -c^{2}+a^{2}-b^{2} & 2 c^{2} \\ 1 & 2 b^{2} & -a^{2}+b^{2}-c^{2} \end{array}\right| \end{aligned}

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

Expanding w.r.t C_{1}

\begin{aligned} &=a b c\left(a^{2}+b^{2}+c^{2}\right)\left(a^{2}+b^{2}+c^{2}\right)^{2} \\ &=a b c\left(a^{2}+b^{2}+c^{2}\right)^{3} \\ &=R \cdot H \cdot S \end{aligned}

Posted by

infoexpert21

View full answer

Crack CUET with india's "Best Teachers"

  • HD Video Lectures
  • Unlimited Mock Tests
  • Faculty Support
cuet_ads

Similar Questions

Trending Articles/News

anam-khan

CBSE practical exams 2026 will be cancelled over non-compliance with guidelines, says board

Latest Question