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Question: If D(x) = \(\left| \begin{array} { c c c } 1 + x + 2 x ^ { 2 } & x + 3 & 1 \\ x + 2 x ^ { 2 } & x & ...

If D(x) = ∣1+x+2x2x+31x+2x2x33x+6x23x+119∣\left| \begin{array} { c c c } 1 + x + 2 x ^ { 2 } & x + 3 & 1 \\ x + 2 x ^ { 2 } & x & 3 \\ 3 x + 6 x ^ { 2 } & 3 x + 11 & 9 \end{array} \right| then dx is

A

1765\frac { 176 } { 5 }

B

– 1763\frac { 176 } { 3 }

C

1863\frac { 186 } { 3 }

D

None of these

Answer

– 1763\frac { 176 } { 3 }

Explanation

Solution

R2 ® R2 – R1 R­3 = R3 – 3R1

D(x) = ∣1+x+2x2x+31−1−32−326∣\left| \begin{array} { c c c } 1 + x + 2 x ^ { 2 } & x + 3 & 1 \\ - 1 & - 3 & 2 \\ - 3 & 2 & 6 \end{array} \right|

R1 = R1 –

D(x) = ∣1+x+2x2+1/2x+3+3/20−1−32−326∣\left| \begin{array} { c c c } 1 + x + 2 x ^ { 2 } + 1 / 2 & x + 3 + 3 / 2 & 0 \\ - 1 & - 3 & 2 \\ - 3 & 2 & 6 \end{array} \right|

R2 = R2 –

D(x) = ∣32+x+2x2x+9200−3−230−326∣\left| \begin{array} { c c c } \frac { 3 } { 2 } + x + 2 x ^ { 2 } & x + \frac { 9 } { 2 } & 0 \\ 0 & - 3 - \frac { 2 } { 3 } & 0 \\ - 3 & 2 & 6 \end{array} \right|

D(x) =

D(x) = 6 (−113(32+x+2x2))\left( - \frac { 11 } { 3 } \left( \frac { 3 } { 2 } + \mathrm { x } + 2 \mathrm { x } ^ { 2 } \right) \right)

∫01Δ(x)\int _ { 0 } ^ { 1 } \Delta ( \mathrm { x } ) =

= −663(32+12+23)\frac { - 66 } { 3 } \left( \frac { 3 } { 2 } + \frac { 1 } { 2 } + \frac { 2 } { 3 } \right)

= – 22 [9+3+46]\left[ \frac { 9 + 3 + 4 } { 6 } \right] = −1763\frac { - 176 } { 3 }