A simple pendulum has a time period ‘T’ in vacuum. Its time period when it is completely immersed in a liquid of density one-eighth of the density of the material of the bob is

If <math><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math><xml> </xml>is a function such that <math><mrow><mi>f</mi><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow><mo>)</mo></mrow><mo>=</mo><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>+</mo><mi>f</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow></mrow></math><xml> </xml>and <math><mrow><mi>f</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow><mo>=</mo><mn>7</mn></mrow></math><xml> </xml>,then <math><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mrow><mi>f</mi><mrow><mo>(</mo><mi>r</mi><mo>)</mo></mrow></mrow></mstyle><mo>=</mo></mrow></math><xml> </xml>

If <math><mrow><mi>f</mi><mo>:</mo><mi>R</mi><mo>→</mo><mi>R</mi></mrow></math><xml> </xml>is defined by <math><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>2</mn></mrow></math><xml> </xml>, and <math><mrow><mi>f</mi><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>3</mn><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow><mo>=</mo><mi>a</mi><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mi>b</mi><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>c</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>d</mi><mi>x</mi><mo>+</mo><mi>e</mi></mrow></math><xml> </xml>then <math><mrow><mi>a</mi><mo>+</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo>+</mo><mi>d</mi><mo>+</mo><mi>e</mi><mo>=</mo></mrow></math><xml> </xml>

&nbsp;Let&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">f</mi><mo>(</mo><mi mathvariant="normal">x</mi><mo>)</mo><mo>=</mo><msup><mi>ax</mi><mn>7</mn></msup><mo>+</mo><msup><mi>bx</mi><mn>3</mn></msup><mo>+</mo><mi>cx</mi><mo>-</mo><mn>5</mn></math>, where&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mi>b</mi></math>&nbsp;and&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math>&nbsp;are constants. If&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mo>-</mo><mn>7</mn><mo>)</mo><mo>=</mo><mn>7</mn></math>, find&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mn>7</mn><mo>)</mo></math>.

If&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">f</mi><mfenced><mrow><mi mathvariant="normal">x</mi><mo>+</mo><mi mathvariant="normal">y</mi></mrow></mfenced><mo>=</mo><mi mathvariant="normal">f</mi><mo>(</mo><mi mathvariant="normal">x</mi><mo>)</mo><mo>+</mo><mi mathvariant="normal">f</mi><mo>(</mo><mi mathvariant="normal">y</mi><mo>)</mo><mo>,</mo><mo>&nbsp;</mo><mo>∀</mo><mi mathvariant="normal">x</mi><mo>,</mo><mi mathvariant="normal">y</mi><mo>∈</mo><mi mathvariant="normal">R</mi><mo>&nbsp;</mo><mi>and</mi><mo>&nbsp;</mo><mi mathvariant="normal">f</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>=</mo><mn>7</mn><mo>,</mo><mo>&nbsp;</mo><mi>then</mi><mo>&nbsp;</mo><munderover><mo>∑</mo><mrow><mi mathvariant="normal">r</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">n</mi></munderover><mi mathvariant="normal">f</mi><mo>(</mo><mi mathvariant="normal">r</mi><mo>)</mo><mo>=</mo></math>

A simple pendulum has a time period ‘T’ in vacuum. Its time period when it is completely immersed in a liquid of density one-eighth of the density of the material of the bob is

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