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Question

Let A(x)=∣ ∣ ∣33x3x2+2a23x3x2+2a23x3+6a2x3x2+2a23x3+6a2x3x4+12a2x2+2a4∣ ∣ ∣,
where aR is a non-zero constant. Then which of the following option(s) is/are INCORRECT ?

A
There exists xR such that A(x)=0
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B
A(x) is independent of x
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C
11A(x)dx=16a6
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D
Area bounded by the curve y=A(x),x=2,x=3 and the x-axis is 20a6
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Solution

The correct options are
A There exists xR such that A(x)=0
C 11A(x)dx=16a6
A(x)=∣ ∣ ∣33x3x2+2a23x3x2+2a23x3+6a2x3x2+2a23x3+6a2x3x4+12a2x2+2a4∣ ∣ ∣

R2R2xR1, R3R3x2R1
A(x)=2a22a2∣ ∣ ∣33x3x2+2a2012x13x5x2+a2∣ ∣ ∣

A(x)=4a4∣ ∣ ∣33x3x2+2a2012x13x5x2+a2∣ ∣ ∣

Expanding along C1, we get
A(x)=4a4[3(a2x2)0+(3x22a2)]
A(x)=4a4×a2=4a6

A(x)0 as aR{0}

11A(x)dx=8a6

y=A(x)=4a6 is a straight line parallel to x-axis with y-intercept 4a6
So, area bounded is a rectangle with area equal to 4a6×(3(2))=20a6

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