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Question

# The parameter on which the value of the determinant ā£ā£ ā£ ā£ā£1aa2cos(pād)xcospxcos(p+d)xsin(pād)xsinpxsin(p+d)xā£ā£ ā£ ā£ā£ does not depend on

A
a
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B
p
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C
d
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D
x
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Solution

## The correct option is B pLetΔ=∣∣ ∣ ∣∣1aa2cos(p−d)xcospxcos(p+d)xsin(p−d)xsinpxsin(p+d)x∣∣ ∣ ∣∣ Applying C1→C1+C3 ​​⇒Δ=∣∣ ∣ ∣∣1+a2aa2cos(p−d)x+cos(p+d)xcospxcos(p+d)xsin(p−d)x+sin(p+d)xsinpxsin(p+d)x∣∣ ∣ ∣∣⇒Δ=∣∣ ∣ ∣∣1+a2aa22cospxcosdxcospxcos(p+d)x2sinpxcosdxsinpxsin(p+d)x∣∣ ∣ ∣∣ Applying C1→C1−2cosdxC2 ⇒Δ=∣∣ ∣ ∣∣1+a2−2acosdxaa20cospxcos(p+d)x0sinpxsin(p+d)x∣∣ ∣ ∣∣ ⇒Δ=(1+a2−2acosdx)[sin(p+d)xcospx−sinpxcos(p+d)x ⇒Δ=(1+a2−2acosdx)sindx Which is independent of p

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