SSC CGL 2024 PYQ — Question 64

The correct answer is option 2 i.e. r²  .

x = r cos t

y = r sin t cos u

z = r sin t sin u

x² + y² + z²

x² = (r cos t)² = r² cos² t

y² = (r sin t cos u)² = r² sin² t cos² u

z² = (r sin t sin u)² = r² sin² t sin² u

⇒$x^2 + y^2 + z^2 = r^2 \cos^2 t + r^2 \sin^2 t \cos^2 u + r^2 \sin^2 t \sin^2 u$

⇒ $x^2 + y^2 + z^2 = r^2 \left( \cos^2 t + \sin^2 t \cos^2 u + \sin^2 t \sin^2 u \right)$

cos² u + sin² u = 1

⇒ $\sin^2 t (\cos^2 u + \sin^2 u) = \sin^2 t (1) = \sin^2 t$

cos² t + sin² t = 1

⇒ x² + y² + z² = r² (1) = r²