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Physics-

General

Easy

Question

A ring of mass m and radius R has three particles attached to the ring as shown in the figure. The centre of the ring has a speed v₀. The kinetic energy of the system is: (Slipping is absent)

$6 {m v}_{0}^{2}$
$12 {m v}_{0}^{2}$
$4 {m v}_{0}^{2}$
$8 {m v}_{0}^{2}$

The correct answer is: $6 m v subscript 0 end subscript superscript 2 end superscript$

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A plank with a uniform sphere placed on it rests on a smooth horizontal plane. Plank is pulled to right by a constant force F. If sphere does not slip over the plank. Which of the following is incorrect?

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Portion AB of the wedge shown in figure is rough and BC is smooth. A solid cylinder rolls without slipping from A to B. The ratio of translational kinetic energy to rotational kinetic energy, when the cylinder reaches point C is :

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A thin rod of length L is placed at angle $theta$ to vertical on a frictionless horizontal floor and released. If the center of mass has acceleration = A, and the rod an angular acceleration = $alpha$ at initial moment, then

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If $f$ is a real-valued differentiable function satisfying $vertical line f left parenthesis x right parenthesis minus f left parenthesis y right parenthesis vertical line less or equal than left parenthesis x minus y right parenthesis to the power of 2 end exponent comma x comma y element of R$ and f(0)=0, then f(1) equals

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If f(x) = $open curly brackets table row cell fraction numerator x to the power of 2 end exponent minus 1 over denominator x to the power of 2 end exponent plus 1 end fraction end cell cell comma 0 less than x less or equal than 2 end cell row cell fraction numerator 1 over denominator 4 end fraction open parentheses x to the power of 3 end exponent minus x to the power of 2 end exponent close parentheses end cell cell comma 2 less than x less or equal than 3 end cell row cell fraction numerator 9 over denominator 4 end fraction left parenthesis vertical line x minus 4 vertical line plus vertical line 2 minus x vertical line right parenthesis comma end cell cell 3 less than x less than 4 end cell end table comma close$ then:

If f(x) = $open curly brackets table row cell fraction numerator x to the power of 2 end exponent minus 1 over denominator x to the power of 2 end exponent plus 1 end fraction end cell cell comma 0 less than x less or equal than 2 end cell row cell fraction numerator 1 over denominator 4 end fraction open parentheses x to the power of 3 end exponent minus x to the power of 2 end exponent close parentheses end cell cell comma 2 less than x less or equal than 3 end cell row cell fraction numerator 9 over denominator 4 end fraction left parenthesis vertical line x minus 4 vertical line plus vertical line 2 minus x vertical line right parenthesis comma end cell cell 3 less than x less than 4 end cell end table comma close$ then:

where represents the integral part function, then:

where represents the integral part function, then:

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Which of the following function(s) has/have removable discontinuity at

parallel

Consider the function f(x)= $open square brackets table row cell x left curly bracket x right curly bracket plus 1 end cell cell 0 less or equal than x less than 1 end cell row cell 2 minus left curly bracket x right curly bracket end cell cell 1 less or equal than x less or equal than 2 end cell end table close$ where {x} denotes the fractional part function. Which one of the following statements is NOT correct?

Consider the function f(x)= $open square brackets table row cell x left curly bracket x right curly bracket plus 1 end cell cell 0 less or equal than x less than 1 end cell row cell 2 minus left curly bracket x right curly bracket end cell cell 1 less or equal than x less or equal than 2 end cell end table close$ where {x} denotes the fractional part function. Which one of the following statements is NOT correct?

Statement-I : If $a subscript 1 end subscript comma a subscript 2 end subscript comma a subscript 3 end subscript comma horizontal ellipsis horizontal ellipsis a subscript n end subscript greater than 0$ , then $l i m subscript x rightwards arrow infinity end subscript open curly brackets fraction numerator a subscript 1 end subscript superscript 1 divided by x end superscript plus a subscript 2 end subscript superscript 1 divided by x end superscript plus a subscript 3 end subscript superscript 1 divided by x end superscript plus horizontal ellipsis.. plus a subscript n end subscript superscript 1 divided by x end superscript over denominator n end fraction close curly brackets to the power of n x end exponent equals product subscript i equals 1 end subscript superscript n end superscript a subscript n end subscript$

Statement-I : If $a subscript 1 end subscript comma a subscript 2 end subscript comma a subscript 3 end subscript comma horizontal ellipsis horizontal ellipsis a subscript n end subscript greater than 0$ , then $l i m subscript x rightwards arrow infinity end subscript open curly brackets fraction numerator a subscript 1 end subscript superscript 1 divided by x end superscript plus a subscript 2 end subscript superscript 1 divided by x end superscript plus a subscript 3 end subscript superscript 1 divided by x end superscript plus horizontal ellipsis.. plus a subscript n end subscript superscript 1 divided by x end superscript over denominator n end fraction close curly brackets to the power of n x end exponent equals product subscript i equals 1 end subscript superscript n end superscript a subscript n end subscript$

$L i m subscript x rightwards arrow 0 end subscript invisible function application fraction numerator left parenthesis 1 minus c o s invisible function application 2 x right parenthesis left parenthesis 3 plus c o s invisible function application x right parenthesis over denominator x t a n invisible function application 4 x end fraction$ is equal to

$L i m subscript x rightwards arrow 0 end subscript invisible function application fraction numerator left parenthesis 1 minus c o s invisible function application 2 x right parenthesis left parenthesis 3 plus c o s invisible function application x right parenthesis over denominator x t a n invisible function application 4 x end fraction$ is equal to

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