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Maths-

General

Easy

Question

$5 cos space theta plus 3 cos space open parentheses theta minus pi over 3 close parentheses plus 8 element of$

$[- 7, 1]$
$[1, 8]$
$[1, 15]$
$[2, 15]$

Hint:

In this question we have to find the range of the given equation. For this we will first simplify the given equation then we know that $negative square root of a squared plus b squared end root less or equal than a space sin x plus b cos x less or equal than square root of a squared plus b squared end root$ , using this we can find the range.

The correct answer is: $left square bracket 1 comma 15 right square bracket$

$5 cos space theta plus 3 cos space open parentheses theta minus pi over 3 close parentheses plus 8 equals 5 space cos theta plus 3 open parentheses cos theta. cos straight pi over 3 plus sin theta. sin straight pi over 3 close parentheses plus 8 equals 5 cos theta plus 3 open parentheses fraction numerator cos theta over denominator 2 end fraction plus fraction numerator square root of 3 sin theta over denominator 2 end fraction close parentheses plus 8 equals 5 cos theta plus 3 over 2 cos theta plus fraction numerator 3 square root of 3 over denominator 2 end fraction sin theta plus 8 equals 13 over 2 cos theta plus fraction numerator 3 square root of 3 over denominator 2 end fraction sin theta plus 8 minus square root of a squared plus b squared end root less or equal than a space sin x plus b cos x less or equal than square root of a squared plus b squared end root rightwards double arrow negative square root of open parentheses fraction numerator 3 square root of 3 over denominator 2 end fraction close parentheses squared plus open parentheses 13 over 2 close parentheses squared end root less or equal than 13 over 2 cos theta plus fraction numerator 3 square root of 3 over denominator 2 end fraction sin theta less or equal than square root of open parentheses fraction numerator 3 square root of 3 over denominator 2 end fraction close parentheses squared plus open parentheses 13 over 2 close parentheses squared end root rightwards double arrow negative 7 less or equal than 13 over 2 cos theta plus fraction numerator 3 square root of 3 over denominator 2 end fraction sin theta less or equal than 7 rightwards double arrow negative 7 plus 8 less or equal than 13 over 2 cos theta plus fraction numerator 3 square root of 3 over denominator 2 end fraction sin theta plus 8 less or equal than 7 plus 8 rightwards double arrow 1 less or equal than 13 over 2 cos theta plus fraction numerator 3 square root of 3 over denominator 2 end fraction sin theta less or equal than 15 f left parenthesis x right parenthesis element of open square brackets 1 comma 15 close square brackets$

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sin space open parentheses x plus pi over 4 close parentheses sin space open parentheses x minus pi over 4 close parentheses equals sin squared x minus sin squared straight pi over 4 equals fraction numerator 1 minus cos 2 x over denominator 2 end fraction minus open parentheses fraction numerator 1 over denominator square root of 2 end fraction close parentheses squared equals 1 half minus fraction numerator cos 2 x over denominator 2 end fraction minus 1 half equals negative 1 half cos 2 x F o r space m i n i m u m space v a l u e space o f space f left parenthesis x right parenthesis comma space cos 2 x equals 1 f left parenthesis x right parenthesis equals negative 1 half cross times 1 equals fraction numerator negative 1 over denominator 2 end fraction F o r space m a x i m u m space v a l u e space o f space f left parenthesis x right parenthesis comma space cos 2 x equals negative 1 f left parenthesis x right parenthesis equals negative 1 half cross times negative 1 equals 1 half S o comma space sin space open parentheses x plus pi over 4 close parentheses sin space open parentheses x minus pi over 4 close parentheses element of open square brackets fraction numerator negative 1 over denominator 2 end fraction comma 1 half close square brackets

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A piston fitted in cylindrical pipe is pulled as shown in the figure. A tuning fork is sounded at open end and loudest sound is heard at open length 13cm, 41 cm and 69 cm, the frequency of tuning fork if velocity of sound is $350 m s to the power of negative 1 end exponent$ is

In a closed organ pipe in which length of air-column can be increased or decreased, the first resonance occurs at

lambda divided by 4

and second resonance occurs at

3 lambda divided by 4.

Thus, at first resonance

fraction numerator lambda over denominator 4 end fraction equals 13 blank horizontal ellipsis open parentheses i close parentheses

And a second resonance

fraction numerator 3 lambda over denominator 4 end fraction equals 41 blank horizontal ellipsis open parentheses i i close parentheses

Subtracting Eq.(i) from Eq.(ii), we have

fraction numerator 3 lambda over denominator 4 end fraction minus fraction numerator lambda over denominator 4 end fraction equals 41 minus 13

⟹ fraction numerator lambda over denominator 2 end fraction equals 28

therefore blank lambda equals 56 blank c m

Hence, frequency of tuning fork

v equals fraction numerator v over denominator lambda end fraction equals fraction numerator 350 over denominator 56 cross times 10 to the power of negative 2 end exponent end fraction equals 365 blank H z

A piston fitted in cylindrical pipe is pulled as shown in the figure. A tuning fork is sounded at open end and loudest sound is heard at open length 13cm, 41 cm and 69 cm, the frequency of tuning fork if velocity of sound is $350 m s to the power of negative 1 end exponent$ is

physics-General

In a closed organ pipe in which length of air-column can be increased or decreased, the first resonance occurs at

lambda divided by 4

and second resonance occurs at

3 lambda divided by 4.

Thus, at first resonance

fraction numerator lambda over denominator 4 end fraction equals 13 blank horizontal ellipsis open parentheses i close parentheses

And a second resonance

fraction numerator 3 lambda over denominator 4 end fraction equals 41 blank horizontal ellipsis open parentheses i i close parentheses

Subtracting Eq.(i) from Eq.(ii), we have

fraction numerator 3 lambda over denominator 4 end fraction minus fraction numerator lambda over denominator 4 end fraction equals 41 minus 13

⟹ fraction numerator lambda over denominator 2 end fraction equals 28

therefore blank lambda equals 56 blank c m

Hence, frequency of tuning fork

v equals fraction numerator v over denominator lambda end fraction equals fraction numerator 350 over denominator 56 cross times 10 to the power of negative 2 end exponent end fraction equals 365 blank H z

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As is clear from figure, at

t equals 0 comma x equals 0

y equals 0

. Therefore, option (a)or (d)may be correct.
In case of (d);

y equals A sin invisible function application left parenthesis k x minus omega t right parenthesis

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physics-General

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In case of (d);

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x equals 0

t equals 0

is positive, in figure. Therefore, particle velocity is in negative y-direction.

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