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Question

# The arbitrary constant on which the value of the determinant ∣∣ ∣ ∣∣1αα2cos(p−d)acospacos(p+d)asin(p−d)asinpasin(p+d)a∣∣ ∣ ∣∣ does not depend is

A
α
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B
p
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C
d
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D
a
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Solution

## The correct option is D pΔ=∣∣ ∣ ∣∣1αα2cos(p−d)acospacos(p+d)asin(p−d)asinpasin(p+d)a∣∣ ∣ ∣∣Applying C1→C1+C3Δ=∣∣ ∣ ∣∣1+α2αα2cos(p−d)a+cos(p+d)acospacos(p+d)asin(p−d)a+sin(p+d)asinpasin(p+d)a∣∣ ∣ ∣∣=∣∣ ∣ ∣∣1+α2αα22cospacosdacospacos(p+d)a2sinpacosdasinpasin(p+d)a∣∣ ∣ ∣∣Applying C1→C1−2cosdaC2Δ=∣∣ ∣ ∣∣1+α2−2αcosdaαα20cospacos(p+d)a0sinpasin(p+d)a∣∣ ∣ ∣∣=(1+α2−2αcosda)(sin(p+d)acospa−cos(p+d)asinpa)=(1+α2−2αcosda)sindaIt does not depend upon the p.Hence, option 'B' is correct.

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