If - x2-x 3x-1 -x-3- 2x-1 2-x2 x3-3- x-3 x2-4 3x -a0-a1x-a7x7 then the value of a0 is

Explanation for the correct optionDeterminant:It is given that |[ x^2+x 3x 1 x+3; 2x+1 2+x^2 x^3 3; x 3 x^2+4 3x ]|=a_0+a_1x+......+a_7x^7.Put x=0 in b ...

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Byju's Answer

Standard XIII

Mathematics

Absolute Value Function

If |[ x2+x 3x...

Question

If $|\begin{matrix} x^{2} + x & 3 x - 1 & - x + 3 \\ 2 x + 1 & 2 + x^{2} & x^{3} - 3 \\ x - 3 & x^{2} + 4 & 3 x \end{matrix}| = a_{0} + a_{1} x + . . . . . . + a_{7} x^{7}$ then the value of $a_{0}$ is

$35$

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$24$

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$23$

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$22$

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$21$

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Solution

The correct option is E
$21$

Explanation for the correct option
Determinant:
It is given that $|\begin{matrix} x^{2} + x & 3 x - 1 & - x + 3 \\ 2 x + 1 & 2 + x^{2} & x^{3} - 3 \\ x - 3 & x^{2} + 4 & 3 x \end{matrix}| = a_{0} + a_{1} x + . . . . . . + a_{7} x^{7}$ .
Put $x = 0$ in both sides of the equation.
$\Rightarrow |\begin{matrix} {(0)}^{2} + 0 & 3 (0) - 1 & - (0) + 3 \\ 2 (0) + 1 & 2 + {(0)}^{2} & {(0)}^{3} - 3 \\ (0) - 3 & {(0)}^{2} + 4 & 3 (0) \end{matrix}| = a_{0} + a_{1} (0) + . . . . . . + a_{7} {(0)}^{7} \Rightarrow |\begin{matrix} 0 & - 1 & 3 \\ 1 & 2 & - 3 \\ - 3 & 4 & 0 \end{matrix}| = a_{0} \Rightarrow a_{0} = 0 [(2 \times 0) - (- 3 \times 4)] - (- 1) [(1 \times 0) - \{(- 3) \times (- 3)\}] + 3 [(1 \times 4) - (- 3 \times 2)] \Rightarrow a_{0} = 0 - 9 + 3 (4 + 6) \Rightarrow a_{0} = 30 - 9 \Rightarrow a_{0} = 21$
Therefore, the value of $a_{0}$ is $21$ .
Hence, option(E) is the correct option

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If x2+x3x-1-x+32x+12+x2x3-3x-3x2+43x=a0+a1x+......+a7x7 then the value of a0 is

If $|\begin{matrix} x^{2} + x & 3 x - 1 & - x + 3 \\ 2 x + 1 & 2 + x^{2} & x^{3} - 3 \\ x - 3 & x^{2} + 4 & 3 x \end{matrix}| = a_{0} + a_{1} x + . . . . . . + a_{7} x^{7}$ then the value of $a_{0}$ is