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Question

If x=rsinθcosϕ,y=rsinθsinϕ and z=rcosθ, then x2+y2+z2 is independent of 



A

0,ϕ

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B

r,θ

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C

c,ϕ

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D

r

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Solution

The correct option is A

0,ϕ


0,ϕ

We have: x=rsinθcosϕ,y=rsinθsinϕ and z

=rcosθ,

x2+y2+z2

=(rsinθcosϕ)2+(rsinθsinϕ)2+(rcosθ)2

=r2sin2θcosϕ2+r2sin2θsin2ϕ+r2cos2θ

=r2sin62θ(cos2ϕ+sin2ϕ)+r2cos2θ

=r2sin2θ×1+r2cos2θ

=r2sin2θ+r2cos2θ

=r2(sin2θ+cos2θ)

=r2×1

=r2

Thus,x2+y2+z2 is independent of θ and ϕ.


Mathematics
RD Sharma
Standard XI

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