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AP EAPCET 23-Aug-2021 Shift 2 Paper

Section: Mathematics

Question : 7 of 160

Marks: +1, -0

\begin{vmatrix} x^{3}+2x^{2}+3x-2 & x^{2}+2x+4 \\ x^{3}-x^{2}-2x-1 & 3x^{3}-2x^{2}+4x-2 \end{vmatrix}

=a x^{6}+b x^{5}+c x^{4}+d x^{3}+e x^{2}+f x+g

a+b+c+d+e+f=

Solution:

Given,

\begin{vmatrix} x^{3}+2x^{2}+3x-2 & x^{2}+2x+4 \\ x^{3}-x^{2}-2x-1 & 3x^{3}-2x^{2}+4x-2 \end{vmatrix}

=a x^{6}+b x^{5}+c x^{4}+d x^{3}+e x^{2}+f x+g

On expanding the determinant,

(x^{3}+2x^{2}+3x-2)(3x^{3}-2x^{2}+4x-2)

-(x^{2}+2x+4)(x^{3}-x^{2}-2x-1)

\Rightarrow 3x^{6}-2x^{5}+4x^{4}-2x^{3}+6x^{5}

-4x^{4}+8x^{3}-4x^{2}+9x^{4}-6x^{3}

+12x^{2}-6x-6x^{3}+4x^{2}-8x+4

-\{x^{5}-x^{4}-2x^{3}-x^{2}+2x^{4}

-2x^{3}-4x^{2}-2x+4x^{3}-4x^{2}-8x-4\}

\Rightarrow 3x^{6}+4x^{5}+9x^{4}-6x^{3}+12x^{2}

-14x+4-x^{5}-x^{4}+9x^{2}+10x+4

3x^{6}+3x^{5}+8x^{4}-6x^{3}+21x^{2}-4x+8

=a x^{6}+b x^{5}+c x^{4}+d x^{3}+e x^{2}+f x+g

\Rightarrow a=3, b=3, c=8

d=-6, e=21, f=-4,9=8

\therefore a+b+c+d+e+f=25

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