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Physics-

General

Easy

Question

A billiard table whose length and width are as shown in the figure. A ball is placed at point A. At what angle ‘ $theta$ ’ the ball be projected so that after colliding with two walls, the ball will fall in the pocket B. Assume that all collisions are perfectly elastic (neglect friction)

$θ = {c o t}^{- 1} \frac{2 a - c}{2 b}$
$θ = {t a n}^{- 1} \frac{2 a - c}{2 b}$
$θ = {c o t}^{- 1} \frac{c - a}{2 b}$
$θ = {c o t}^{- 1} \frac{c - a}{b}$

The correct answer is: $theta equals c o t to the power of negative 1 end exponent invisible function application fraction numerator 2 a minus c over denominator 2 b end fraction$

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$I f tan space alpha comma tan space beta$ are the roots of the equation $x squared plus p x plus q equals 0 left parenthesis p not equal to 0 right parenthesis$ then $s i n squared invisible function application left parenthesis alpha plus beta right parenthesis plus p s i n invisible function application left parenthesis alpha plus beta right parenthesis c o s invisible function application left parenthesis alpha plus beta right parenthesis plus q c o s squared invisible function application left parenthesis alpha plus beta right parenthesis equals$

$I f tan space alpha comma tan space beta$ are the roots of the equation $x squared plus p x plus q equals 0 left parenthesis p not equal to 0 right parenthesis$ then $s i n squared invisible function application left parenthesis alpha plus beta right parenthesis plus p s i n invisible function application left parenthesis alpha plus beta right parenthesis c o s invisible function application left parenthesis alpha plus beta right parenthesis plus q c o s squared invisible function application left parenthesis alpha plus beta right parenthesis equals$

parallel

$cos space left parenthesis x minus y right parenthesis plus cos space left parenthesis y minus z right parenthesis plus cos space left parenthesis z minus x right parenthesis equals negative 3 over 2 not stretchy rightwards double arrow sum left parenthesis cos space x right parenthesis equals$

$cos space left parenthesis x minus y right parenthesis plus cos space left parenthesis y minus z right parenthesis plus cos space left parenthesis z minus x right parenthesis equals negative 3 over 2 not stretchy rightwards double arrow sum left parenthesis cos space x right parenthesis equals$

$text In a end text straight triangle A B C comma text if end text sin text end text A sin space B sin space C equals 1 third$ then the value of $c o t space A c o t space B plus c o t space B c o t space C plus c o t space C c o t space A$ is

$text In a end text straight triangle A B C comma text if end text sin text end text A sin space B sin space C equals 1 third$ then the value of $c o t space A c o t space B plus c o t space B c o t space C plus c o t space C c o t space A$ is

$cot invisible function application A plus cot invisible function application B plus cot invisible function application C plus fraction numerator cos invisible function application left parenthesis A plus B plus C right parenthesis over denominator sin invisible function application A sin invisible function application B sin invisible function application C end fraction equals$

$cot invisible function application A plus cot invisible function application B plus cot invisible function application C plus fraction numerator cos invisible function application left parenthesis A plus B plus C right parenthesis over denominator sin invisible function application A sin invisible function application B sin invisible function application C end fraction equals$

parallel

$text In a end text straight triangle A B C comma tan invisible function application A plus tan invisible function application B plus tan invisible function application C equals 6$ and $tan invisible function application A tan invisible function application B equals 2$ then the triangle $A B C$ is

$text In a end text straight triangle A B C comma tan invisible function application A plus tan invisible function application B plus tan invisible function application C equals 6$ and $tan invisible function application A tan invisible function application B equals 2$ then the triangle $A B C$ is

$text If end text tan invisible function application straight A equals 1 comma tan invisible function application straight B equals 2 comma tan invisible function application straight C equals 3 text then end text straight A plus straight B plus straight C equals$

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$text If end text alpha plus beta plus gamma equals pi over 2 text and end text cot invisible function application alpha comma cot invisible function application beta comma cot invisible function application gamma text are in end text straight A stack P with _ below with _ below text then end text cot invisible function application alpha times cot invisible function application gamma equals$

$text If end text alpha plus beta plus gamma equals pi over 2 text and end text cot invisible function application alpha comma cot invisible function application beta comma cot invisible function application gamma text are in end text straight A stack P with _ below with _ below text then end text cot invisible function application alpha times cot invisible function application gamma equals$

parallel

if $tan space alpha comma tan space beta$ are the roots of the equation, $x squared plus p x plus q equals 0 left parenthesis p not equal to 0 right parenthesis$ then

if $tan space alpha comma tan space beta$ are the roots of the equation, $x squared plus p x plus q equals 0 left parenthesis p not equal to 0 right parenthesis$ then

$cot space B equals 2 tan space left parenthesis A minus B right parenthesis not stretchy rightwards double arrow 2 tan space B plus cot space B equals$

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$text If end text sin text end text B equals 1 fifth times sin space left parenthesis 2 A plus B right parenthesis text then end text fraction numerator tan begin display style text end text end style left parenthesis A plus B right parenthesis over denominator tan begin display style text end text end style A end fraction equals$

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parallel

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