JEE Main 2014 — Mathematics Algebra

Let $p_1 x=a_1 x^2+b_1 x+c_1$p1​x=a1​x2+b1​x+c1​ $$p_2 x=a_2 x^2+b_2 x+c_2$$p2​x=a2​x2+b2​x+c2​ and $p_3 x=a_3 x^2+b_3 x+c_3$p3​x=a3​x2+b3​x+c3​ where $a_1, a_2, a_3, b_1, b_2, b_3, c_1, c_2, c_3$a1​,a2​,a3​,b1​,b2​,b3​,c1​,c2​,c3​ are real numbers. $$\begin{gathered} \therefore \mathrm{A}(x)=\left[\begin{array}{ccc} a_1 x^2+b_1 x+c_1 & 2 a_1 x+b_1 & 2 a_1 \\ a_2 x^2+b_2 x+c_2 & 2 a_2 x+b_2 & 2 a_2 \\ a_3 x^2+b_3 x+c_3 & 2 a_3 x+b_3 & 2 a_3 \end{array}\right] \\ =\left[\begin{array}{llll} a_1 x^2+b_1 x+c_1 & a_2 x^2+b_2 x+c_2 & a_3 x^2+b_3 x+c_3 \\ 2 a_1 x+b_1 & 2 a_2 x+b_2 & 2 a_3 x+b_2 \\ 2 a_1 & 2 a_2 & 2 a_3 \end{array}\right] \\ \times\left[\begin{array}{ccc} a_1 x^2+b_1 x+c_1 & 2 a_1 x+b_1 & 2 a_1 \\ a_2 x^2+b_2 x+c_2 & 2 a_2 x+b_2 & 2 a_2 \\ a_3 x^2+b_3 x+c_3 & 2 a_3 x+b_3 & 2 a_3 \end{array}\right] \end{gathered}$$∴A(x)=​a1​x2+b1​x+c1​a2​x2+b2​x+c2​a3​x2+b3​x+c3​​2a1​x+b1​2a2​x+b2​2a3​x+b3​​2a1​2a2​2a3​​​=​a1​x2+b1​x+c1​2a1​x+b1​2a1​​a2​x2+b2​x+c2​2a2​x+b2​2a2​​a3​x2+b3​x+c3​2a3​x+b2​2a3​​​×​a1​x2+b1​x+c1​a2​x2+b2​x+c2​a3​x2+b3​x+c3​​2a1​x+b1​2a2​x+b2​2a3​x+b3​​2a1​2a2​2a3​​​​ It is clear from the above multiplication, the degree of determinant of $\mathrm{B}(x)$B(x) can not be less than 4 .