#### Provide Solution For  R.D. Sharma Maths Class 12 Chapter determinants Exercise 5.2  Question 2 Sub Question 10 Maths Textbook Solution.

Answer:$\left|\begin{array}{llll} 1^{2} & 2^{2} & 3^{2} & 4^{2} \\ 2^{2} & 3^{2} & 4^{2} & 5^{2} \\ 3^{2} & 4^{2} & 5^{2} & 6^{2} \\ 4^{2} & 5^{2} & 6^{2} & 7^{2} \end{array}\right|=0$

Hint: We will try to do any two column or row equal

Given:$\left|\begin{array}{cccc} 1^{2} & 2^{2} & 3^{2} & 4^{2} \\ 2^{2} & 3^{2} & 4^{2} & 5^{2} \\ 3^{2} & 4^{2} & 5^{2} & 6^{2} \\ 4^{2} & 5^{2} & 6^{2} & 7^{2} \end{array}\right|$

Solution:$\left|\begin{array}{cccc} 1^{2} & 2^{2} & 3^{2} & 4^{2} \\ 2^{2} & 3^{2} & 4^{2} & 5^{2} \\ 3^{2} & 4^{2} & 5^{2} & 6^{2} \\ 4^{2} & 5^{2} & 6^{2} & 7^{2} \end{array}\right|$

\begin{aligned} &\text { On applying } C_{3} \rightarrow C_{3}-C_{2} \text { and } C_{4} \rightarrow C_{4}-C_{1}\\ &=\left|\begin{array}{cccc} 1^{2} & 2^{2} & 3^{2}-2^{2} & 4^{2}-1^{2} \\ 2^{2} & 3^{2} & 4^{2}-3^{2} & 5^{2}-2^{2} \\ 3^{2} & 4^{2} & 5^{2}-4^{2} & 6^{2}-3^{2} \\ 4^{2} & 5^{2} & 6^{2}-5^{2} & 7^{2}-4^{2} \end{array}\right|\\ &=\left|\begin{array}{cccc} 1^{2} & 2^{2} & 9-4 & 16-1 \\ 2^{2} & 3^{2} & 16-9 & 25-4 \\ 3^{2} & 4^{2} & 25-16 & 36-9 \\ 4^{2} & 5^{2} & 36-25 & 49-16 \end{array}\right|\\ &=\left|\begin{array}{cccc} 1^{2} & 2^{2} & 5 & 15 \\ 2^{2} & 3^{2} & 7 & 21 \\ 3^{2} & 4^{2} & 9 & 27 \\ 4^{2} & 5^{2} & 11 & 33 \end{array}\right|\\ &\text { On taking common } 3 \text { from } \mathrm{C}_{4}\\ &=3\left|\begin{array}{cccc} 1^{2} & 2^{2} & 5 & 5 \\ 2^{2} & 3^{2} & 7 & 7 \\ 3^{2} & 4^{2} & 9 & 9 \\ 4^{2} & 5^{2} & 11 & 11 \end{array}\right| \end{aligned}

If any two rows or columns of a determinant are identical.

The value of the determinant is zero

\begin{aligned} &=3 \times 0 \quad\left(\because C_{3}=C_{4}\right) \\ &=0 \end{aligned}

Hence$\left|\begin{array}{cccc} 1^{2} & 2^{2} & 3^{2} & 4^{2} \\ 2^{2} & 3^{2} & 4^{2} & 5^{2} \\ 3^{2} & 4^{2} & 5^{2} & 6^{2} \\ 4^{2} & 5^{2} & 6^{2} & 7^{2} \end{array}\right|=0$