<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>sin</mi><mn>2</mn></msup><mo>⁡</mo><mi>θ</mi><mo>=</mo><mfrac><mrow><mo>(</mo><mi>x</mi><mo>+</mo><mi>y</mi><msup><mo>)</mo><mn>2</mn></msup></mrow><mrow><mn>4</mn><mi>x</mi><mi>y</mi></mrow></mfrac></math>, where&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>R</mi></math>, gives real&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math>&nbsp;if and only if

A uniform metre stick is placed vertically on a horizontal frictionless surface and released. As the stick is in motion, the centre of mass moves

A sphere can roll on a surface inclined at an angle&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">θ</mi></math>&nbsp;if the friction coefficient is more than&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>7</mn></mfrac><mi mathvariant="normal">g</mi><mo>&nbsp;</mo><mi>tan</mi><mo>&nbsp;</mo><mi mathvariant="normal">θ</mi></math>&nbsp;. If a sphere is released from rest on the incline

A thin circular plate of mass M and radius R has its density varying as&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">ρ</mi><mo>(</mo><mi mathvariant="normal">r</mi><mo>)</mo><mo>=</mo><msub><mi mathvariant="normal">ρ</mi><mn>0</mn></msub><mi mathvariant="normal">r</mi></math>&nbsp;with&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">ρ</mi><mn>0</mn></msub></math>&nbsp;as constant and r is the distance from its center. The moment of Inertia of the circular plate about an axis perpendicular to the plate and passing through its edge is&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">I</mi><mo>=</mo><mo>&nbsp;</mo><mi mathvariant="normal">a</mi><mo>&nbsp;</mo><msup><mi>MR</mi><mn>2</mn></msup></math>. The value of the coefficient a is:

$\sin^{2} θ = \frac{(x + y)^{2}}{4 x y}$ , where $x, y \in R$ , gives real $θ$ if and only if

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sin2⁡θ=(x+y)24xy, where x,y∈R, gives real θ if and only if

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$\sin^{2} θ = \frac{(x + y)^{2}}{4 x y}$ , where $x, y \in R$ , gives real $θ$ if and only if