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PHXI15:WAVES

355079 Assertion :
If two waves of same amplitude, produce a resultant wave of same amplitude, then the phase difference between them will be $120^{\circ}$ .
Reason :
The resultant amplitude of two waves is equal to sum of amplitude of two waves.

1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.

2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.

3 Assertion is correct but Reason is incorrect.

4 Assertion is incorrect but reason is correct.

Explanation:

$\sqrt{A_{1}^{2} + A_{2}^{2} + 2 A_{1} A_{2} \cos ϕ}$
$A_{1} = A_{2} \Rightarrow A_{n e t} = 2 A \cos \frac{ϕ}{2}$
$ϕ = 120^{\circ} \Rightarrow A_{n e t} = 2 A \cos 60^{\circ} = A$
So correct option is (3).

PHXI15:WAVES

355080 Three coherent waves of equal frequencies having amplitude $10 μ m, 4 μ m$ and $7 μ m$ respectively, arrive at a given point with succesive phase difference of $π / 2$ . The amplitude of the resulting wave in $μ m$ is given by:

1 5

2 6

3 3

4 4

PHXI15:WAVES

355081 The amplitude of a wave represented by displacement equation $y = \frac{1}{\sqrt{a}} \sin ω t \pm \frac{1}{\sqrt{b}} \cos ω t$ will be

1

\sqrt{\frac{a + b}{a b}}

2

\frac{a + b}{a b}

3

\frac{\sqrt{a} + \sqrt{b}}{a b}

4

\frac{\sqrt{a} \pm \sqrt{b}}{a b}

Explanation:

$y = \frac{1}{\sqrt{a}} \sin ω t \pm \frac{1}{\sqrt{b}} \sin (ω t + \frac{π}{2})$
Here phase difference $= \frac{π}{2}$
Resultant amplitude
$= \sqrt{{(\frac{1}{\sqrt{b}})}^{2} + {(\frac{1}{\sqrt{b}})}^{2}} = \sqrt{\frac{1}{a} + \frac{1}{b}} = \sqrt{\frac{a + b}{a b}}$

PHXI15:WAVES

355082 The ratio of intensities of two waves is 16:9. If they produce interference, then the ratio of maximum and minimum will be

1

4 : 3

2

49 : 1

3

64 : 27

4

81 : 49

Explanation:

Since, $\frac{I_{1}}{I_{2}} = \frac{16}{9} = \frac{a_{1}^{2}}{a_{2}^{2}}$
$\Rightarrow \frac{a_{1}}{a_{2}} = \sqrt{\frac{16}{9}} = \frac{4}{3}$
$\Rightarrow a_{1} = \frac{4}{3} a_{2}$
$∴ \frac{I_{max}}{I_{min}} = \frac{{(a_{1} + a_{2})}^{2}}{{(a_{1} - a_{2})}^{2}} = \frac{{(\frac{4}{3} a_{2} + a_{2})}^{2}}{(\frac{4}{3} a_{2} - a_{2})}$
$= 49 : 1$

PHXI15:WAVES

355079 Assertion :
If two waves of same amplitude, produce a resultant wave of same amplitude, then the phase difference between them will be $120^{\circ}$ .
Reason :
The resultant amplitude of two waves is equal to sum of amplitude of two waves.

1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.

2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.

3 Assertion is correct but Reason is incorrect.

4 Assertion is incorrect but reason is correct.

Explanation:

$\sqrt{A_{1}^{2} + A_{2}^{2} + 2 A_{1} A_{2} \cos ϕ}$
$A_{1} = A_{2} \Rightarrow A_{n e t} = 2 A \cos \frac{ϕ}{2}$
$ϕ = 120^{\circ} \Rightarrow A_{n e t} = 2 A \cos 60^{\circ} = A$
So correct option is (3).

PHXI15:WAVES

355080 Three coherent waves of equal frequencies having amplitude $10 μ m, 4 μ m$ and $7 μ m$ respectively, arrive at a given point with succesive phase difference of $π / 2$ . The amplitude of the resulting wave in $μ m$ is given by:

1 5

2 6

3 3

4 4

PHXI15:WAVES

355081 The amplitude of a wave represented by displacement equation $y = \frac{1}{\sqrt{a}} \sin ω t \pm \frac{1}{\sqrt{b}} \cos ω t$ will be

1

\sqrt{\frac{a + b}{a b}}

2

\frac{a + b}{a b}

3

\frac{\sqrt{a} + \sqrt{b}}{a b}

4

\frac{\sqrt{a} \pm \sqrt{b}}{a b}

Explanation:

$y = \frac{1}{\sqrt{a}} \sin ω t \pm \frac{1}{\sqrt{b}} \sin (ω t + \frac{π}{2})$
Here phase difference $= \frac{π}{2}$
Resultant amplitude
$= \sqrt{{(\frac{1}{\sqrt{b}})}^{2} + {(\frac{1}{\sqrt{b}})}^{2}} = \sqrt{\frac{1}{a} + \frac{1}{b}} = \sqrt{\frac{a + b}{a b}}$

PHXI15:WAVES

355082 The ratio of intensities of two waves is 16:9. If they produce interference, then the ratio of maximum and minimum will be

1

4 : 3

2

49 : 1

3

64 : 27

4

81 : 49

Explanation:

Since, $\frac{I_{1}}{I_{2}} = \frac{16}{9} = \frac{a_{1}^{2}}{a_{2}^{2}}$
$\Rightarrow \frac{a_{1}}{a_{2}} = \sqrt{\frac{16}{9}} = \frac{4}{3}$
$\Rightarrow a_{1} = \frac{4}{3} a_{2}$
$∴ \frac{I_{max}}{I_{min}} = \frac{{(a_{1} + a_{2})}^{2}}{{(a_{1} - a_{2})}^{2}} = \frac{{(\frac{4}{3} a_{2} + a_{2})}^{2}}{(\frac{4}{3} a_{2} - a_{2})}$
$= 49 : 1$

PHXI15:WAVES

355079 Assertion :
If two waves of same amplitude, produce a resultant wave of same amplitude, then the phase difference between them will be $120^{\circ}$ .
Reason :
The resultant amplitude of two waves is equal to sum of amplitude of two waves.

1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.

2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.

3 Assertion is correct but Reason is incorrect.

4 Assertion is incorrect but reason is correct.

Explanation:

$\sqrt{A_{1}^{2} + A_{2}^{2} + 2 A_{1} A_{2} \cos ϕ}$
$A_{1} = A_{2} \Rightarrow A_{n e t} = 2 A \cos \frac{ϕ}{2}$
$ϕ = 120^{\circ} \Rightarrow A_{n e t} = 2 A \cos 60^{\circ} = A$
So correct option is (3).

PHXI15:WAVES

355080 Three coherent waves of equal frequencies having amplitude $10 μ m, 4 μ m$ and $7 μ m$ respectively, arrive at a given point with succesive phase difference of $π / 2$ . The amplitude of the resulting wave in $μ m$ is given by:

1 5

2 6

3 3

4 4

PHXI15:WAVES

355081 The amplitude of a wave represented by displacement equation $y = \frac{1}{\sqrt{a}} \sin ω t \pm \frac{1}{\sqrt{b}} \cos ω t$ will be

1

\sqrt{\frac{a + b}{a b}}

2

\frac{a + b}{a b}

3

\frac{\sqrt{a} + \sqrt{b}}{a b}

4

\frac{\sqrt{a} \pm \sqrt{b}}{a b}

Explanation:

$y = \frac{1}{\sqrt{a}} \sin ω t \pm \frac{1}{\sqrt{b}} \sin (ω t + \frac{π}{2})$
Here phase difference $= \frac{π}{2}$
Resultant amplitude
$= \sqrt{{(\frac{1}{\sqrt{b}})}^{2} + {(\frac{1}{\sqrt{b}})}^{2}} = \sqrt{\frac{1}{a} + \frac{1}{b}} = \sqrt{\frac{a + b}{a b}}$

PHXI15:WAVES

355082 The ratio of intensities of two waves is 16:9. If they produce interference, then the ratio of maximum and minimum will be

1

4 : 3

2

49 : 1

3

64 : 27

4

81 : 49

Explanation:

Since, $\frac{I_{1}}{I_{2}} = \frac{16}{9} = \frac{a_{1}^{2}}{a_{2}^{2}}$
$\Rightarrow \frac{a_{1}}{a_{2}} = \sqrt{\frac{16}{9}} = \frac{4}{3}$
$\Rightarrow a_{1} = \frac{4}{3} a_{2}$
$∴ \frac{I_{max}}{I_{min}} = \frac{{(a_{1} + a_{2})}^{2}}{{(a_{1} - a_{2})}^{2}} = \frac{{(\frac{4}{3} a_{2} + a_{2})}^{2}}{(\frac{4}{3} a_{2} - a_{2})}$
$= 49 : 1$

PHXI15:WAVES

355079 Assertion :
If two waves of same amplitude, produce a resultant wave of same amplitude, then the phase difference between them will be $120^{\circ}$ .
Reason :
The resultant amplitude of two waves is equal to sum of amplitude of two waves.

1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.

2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.

3 Assertion is correct but Reason is incorrect.

4 Assertion is incorrect but reason is correct.

Explanation:

$\sqrt{A_{1}^{2} + A_{2}^{2} + 2 A_{1} A_{2} \cos ϕ}$
$A_{1} = A_{2} \Rightarrow A_{n e t} = 2 A \cos \frac{ϕ}{2}$
$ϕ = 120^{\circ} \Rightarrow A_{n e t} = 2 A \cos 60^{\circ} = A$
So correct option is (3).

PHXI15:WAVES

355080 Three coherent waves of equal frequencies having amplitude $10 μ m, 4 μ m$ and $7 μ m$ respectively, arrive at a given point with succesive phase difference of $π / 2$ . The amplitude of the resulting wave in $μ m$ is given by:

1 5

2 6

3 3

4 4

PHXI15:WAVES

355081 The amplitude of a wave represented by displacement equation $y = \frac{1}{\sqrt{a}} \sin ω t \pm \frac{1}{\sqrt{b}} \cos ω t$ will be

1

\sqrt{\frac{a + b}{a b}}

2

\frac{a + b}{a b}

3

\frac{\sqrt{a} + \sqrt{b}}{a b}

4

\frac{\sqrt{a} \pm \sqrt{b}}{a b}

Explanation:

$y = \frac{1}{\sqrt{a}} \sin ω t \pm \frac{1}{\sqrt{b}} \sin (ω t + \frac{π}{2})$
Here phase difference $= \frac{π}{2}$
Resultant amplitude
$= \sqrt{{(\frac{1}{\sqrt{b}})}^{2} + {(\frac{1}{\sqrt{b}})}^{2}} = \sqrt{\frac{1}{a} + \frac{1}{b}} = \sqrt{\frac{a + b}{a b}}$

PHXI15:WAVES

355082 The ratio of intensities of two waves is 16:9. If they produce interference, then the ratio of maximum and minimum will be

1

4 : 3

2

49 : 1

3

64 : 27

4

81 : 49

Explanation:

Since, $\frac{I_{1}}{I_{2}} = \frac{16}{9} = \frac{a_{1}^{2}}{a_{2}^{2}}$
$\Rightarrow \frac{a_{1}}{a_{2}} = \sqrt{\frac{16}{9}} = \frac{4}{3}$
$\Rightarrow a_{1} = \frac{4}{3} a_{2}$
$∴ \frac{I_{max}}{I_{min}} = \frac{{(a_{1} + a_{2})}^{2}}{{(a_{1} - a_{2})}^{2}} = \frac{{(\frac{4}{3} a_{2} + a_{2})}^{2}}{(\frac{4}{3} a_{2} - a_{2})}$
$= 49 : 1$