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Question

If f(x)=∣∣ ∣ ∣∣33x3x2+2a23x3x2+2a23x3+4a2x3x2+2a23x3+6a2x3x4+8a2x2+2a4∣∣ ∣ ∣∣ for some non-zero real constant a, then

A
f(x)=0
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B
f(x)=a2x2
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C
f(x)=x6
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D
y=f(x) is a straight line parallel to xaxis
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Solution

The correct option is D y=f(x) is a straight line parallel to xaxis
f(x)=∣ ∣ ∣33x3x2+2a23x3x2+2a23x3+4a2x3x2+2a23x3+6a2x3x4+8a2x2+2a4∣ ∣ ∣

Applying C3C3xC2 and C2C2xC1

f(x)=∣ ∣ ∣302a23x2a22a2x3x2+2a24a2x2a2x2+2a4∣ ∣ ∣

=4a4∣ ∣3013x1x3x2+2a22xx2+a2∣ ∣

Applying C1C13C3, we get
f(x)=4a4∣ ∣00101xa22xx2+a2∣ ∣=4a6

Hence, y=f(x) is a straight line parallel to xaxis.

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