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If X = R Sin θ Cos ϕ, Y = R Sin θ Sin ϕ and Z = R Cos θ, Then X2 + Y2 + Z2 is Independent of - Mathematics

MCQ

If x = r sin θ cos ϕ, y = r sin θ sin ϕ and r cos θ, then x2 + y2 + z2 is independent of

Options

  • θ, ϕ

  • r, θ

  • r, ϕ

  • r

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Solution

θ, ϕ
We have:
x = r sin θ cos ϕ  ,  y = r sin θ sin ϕ and z = r cos θ,
∴ x2 + y2 + z2

\[= \left( r \sin\theta \cos\phi \right)^2 + \left( r \sin\theta \sin\phi \right)^2 + \left( r \cos\theta \right)^2 \]

\[ = r^2 \sin^2 \theta \cos^2 \phi + r^2 \sin^2 \theta \sin^2 \phi + r^2 \cos^2 \theta \]

\[ = r^2 \sin^2 \theta \left( \cos^2 \phi + \sin^2 \phi \right) + r^2 \cos^2 \theta \]

\[ = r^2 \sin^2 \theta \times 1 + r^2 \cos^2 \theta\]

\[ = r^2 \sin^2 \theta + r^2 \cos^2 \theta\]

\[ = r^2 \left( \sin^2 \theta + \cos^2 \theta \right)\]

\[ = r^2 \times 1\]

\[ = r^2 \]

\[\text{ Thus, }x^2 + y^2 + z^2\text{ is independent of }\theta\text{ and }\phi .\]

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APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 5 Trigonometric Functions
Q 7 | Page 41
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