1. Tardigrade
  2. Question
  3. Mathematics
  4. Let for i = 1, 2, 3, pi(x) be a polynomial of degree 2 in x, pi'(x) and pi''(x) be the first and second order derivatives of pi(x) respectively. Let, A(x) = [p1(x)&p1'(x)&p1''(x) p2(x)&p2'(x)&p2''(x) p3(x)&p3'(x)&p3''(x)] and B(x) = [A(x)]T A(x). Then determinant of B(x) :

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Q. Let for $i = 1, 2, 3, p_{i}\left(x\right)$ be a polynomial of degree 2 in x, $p_{i}'\left(x\right)$ and $p_{i}''\left(x\right)$ be the first and second order derivatives of $p_{i}\left(x\right)$ respectively. Let,
$A\left(x\right) = \begin{bmatrix}p_{1}\left(x\right)&p_{1}'\left(x\right)&p_{1}''\left(x\right)\\ p_{2}\left(x\right)&p_{2}'\left(x\right)&p_{2}''\left(x\right)\\ p_{3}\left(x\right)&p_{3}'\left(x\right)&p_{3}''\left(x\right)\end{bmatrix}$
and $B\left(x\right) = \left[A\left(x\right)\right]^{T} A\left(x\right).$ Then determinant of $B\left(x\right)$ :

JEE MainJEE Main 2014Continuity and Differentiability

Solution:

Let $P_{i} = a_{i} x^{2} + b_{i}X + C_{i} \,a_{i} \ne 0$
$b_{i},\,c_{i} \in R$
$A\left(X\right) = \begin{bmatrix}a_{1}x^{2}+b_{1}x+c_{1}&2a_{1}x+b_{1}&2a_{1}\\ a_{2}x^{2}+b_{2}x+c_{2}&2a_{2}x+b_{2}&2a_{2}\\ a_{3}x^{2}+b_{3}x+c_{3}&2a_{3}x+b_{3}&2a_{3}\end{bmatrix}$
use $\left(i\right) C_{2} \to C_{2} - x \,C_{3}$
then use $\left(ii\right) C_{1} \to C_{1} - x \,C_{2}-\frac{x^{2}}{2} C_{3}$
$A\left(X\right) = \begin{bmatrix}c_{1}&b_{1}&2a_{1}\\ c_{2}&b_{2}&2a_{2}\\ c_{3}&b_{3}&2a_{3}\end{bmatrix} \Rightarrow \left|A\right| =$ constant
So $\left|B\right| = \left|A^{T}\right| \left|A\right| = \left|A\right|^{2} = $ constant independent from n