Explanation:

Velocity of wave \(v=n \lambda\)
Where \(n = \) frequency of wave \(\Rightarrow n=\dfrac{\nu}{\lambda}\)
\({n_1} = \frac{{{v_1}}}{{{\lambda _1}}} = \frac{{396}}{{99 \times {{10}^{ - 2}}}} = 400\;Hz\)
\({n_2} = \frac{{{v_2}}}{{{\lambda _2}}} = \frac{{396}}{{100 \times {{10}^{ - 2}}}} = 396\;Hz\)
No. of beats \(=n_{1}-n_{2}=4\)

Explanation:

Explanation:

Given, \({y_1} = 5\sin (300\,\pi t)\) and
\({y_2} = 4\sin (302\,\pi t)\)
Comparing with standard eqns.
\({y_{1}=A_{1} \sin \omega_{1} t}\) and \({y_{2}=A_{2} \sin \omega_{2} t}\)
\({A_1} = 5,{A_2} = 4,{\omega _1} = 300\,\pi ,{\omega _2} = 302\,\pi \)
\({\because \quad A_{\max }=A_{1}+A_{2}, A_{\text {min }}=A_{1}-A_{2}}\)
and \({A_{\text {max }}^{2}={A}_{1}^{2}+{A}_{2}^{2}+2 A_{1} A_{2}}\)
\({A_{\text {min }}^{2}=A_{1}^{2}+{A}_{2}^{2}-2 A_{1} A_{2}}\)
\({\because \quad I=k A^{2}}\) or \({A^{2}=\dfrac{I}{k}}\)
\(\therefore \;\;\;\frac{{{I_{\max }}}}{k} = A_1^2 + A_2^2 + 2{A_1}{A_2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(1)\)
\(\therefore \;\;\;\frac{{{I_{{\text{min }}}}}}{k} = A_1^2 + A_2^2 - 2{A_1}{A_2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(2)\)
From eqn. (1) and (2)
\({\dfrac{I_{\max }}{I_{\min }}=\dfrac{A_{1}^{2}+A_{2}^{2}+2 A_{1} A_{2}}{A_{1}^{2}+A_{2}^{2}-2 A_{1} A_{2}}=\dfrac{5^{2}+4^{2}+2 \times 5 \times 4}{5^{2}+4^{2}-2 \times 5 \times 4}}\)
\({\Rightarrow \dfrac{I_{\max }}{I_{\min }}=\dfrac{81}{1}}\).
So correct option is (3)

Explanation:

Since \(\phi=\dfrac{\pi}{2} \Rightarrow A=\sqrt{a_{1}^{2}+a_{2}^{2}}=\sqrt{(4)^{2}+(3)^{2}}=5\).

Explanation:

Velocity of wave \(v=n \lambda\)
Where \(n = \) frequency of wave \(\Rightarrow n=\dfrac{\nu}{\lambda}\)
\({n_1} = \frac{{{v_1}}}{{{\lambda _1}}} = \frac{{396}}{{99 \times {{10}^{ - 2}}}} = 400\;Hz\)
\({n_2} = \frac{{{v_2}}}{{{\lambda _2}}} = \frac{{396}}{{100 \times {{10}^{ - 2}}}} = 396\;Hz\)
No. of beats \(=n_{1}-n_{2}=4\)

Explanation:

Explanation:

Given, \({y_1} = 5\sin (300\,\pi t)\) and
\({y_2} = 4\sin (302\,\pi t)\)
Comparing with standard eqns.
\({y_{1}=A_{1} \sin \omega_{1} t}\) and \({y_{2}=A_{2} \sin \omega_{2} t}\)
\({A_1} = 5,{A_2} = 4,{\omega _1} = 300\,\pi ,{\omega _2} = 302\,\pi \)
\({\because \quad A_{\max }=A_{1}+A_{2}, A_{\text {min }}=A_{1}-A_{2}}\)
and \({A_{\text {max }}^{2}={A}_{1}^{2}+{A}_{2}^{2}+2 A_{1} A_{2}}\)
\({A_{\text {min }}^{2}=A_{1}^{2}+{A}_{2}^{2}-2 A_{1} A_{2}}\)
\({\because \quad I=k A^{2}}\) or \({A^{2}=\dfrac{I}{k}}\)
\(\therefore \;\;\;\frac{{{I_{\max }}}}{k} = A_1^2 + A_2^2 + 2{A_1}{A_2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(1)\)
\(\therefore \;\;\;\frac{{{I_{{\text{min }}}}}}{k} = A_1^2 + A_2^2 - 2{A_1}{A_2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(2)\)
From eqn. (1) and (2)
\({\dfrac{I_{\max }}{I_{\min }}=\dfrac{A_{1}^{2}+A_{2}^{2}+2 A_{1} A_{2}}{A_{1}^{2}+A_{2}^{2}-2 A_{1} A_{2}}=\dfrac{5^{2}+4^{2}+2 \times 5 \times 4}{5^{2}+4^{2}-2 \times 5 \times 4}}\)
\({\Rightarrow \dfrac{I_{\max }}{I_{\min }}=\dfrac{81}{1}}\).
So correct option is (3)

Explanation:

Since \(\phi=\dfrac{\pi}{2} \Rightarrow A=\sqrt{a_{1}^{2}+a_{2}^{2}}=\sqrt{(4)^{2}+(3)^{2}}=5\).

Explanation:

Velocity of wave \(v=n \lambda\)
Where \(n = \) frequency of wave \(\Rightarrow n=\dfrac{\nu}{\lambda}\)
\({n_1} = \frac{{{v_1}}}{{{\lambda _1}}} = \frac{{396}}{{99 \times {{10}^{ - 2}}}} = 400\;Hz\)
\({n_2} = \frac{{{v_2}}}{{{\lambda _2}}} = \frac{{396}}{{100 \times {{10}^{ - 2}}}} = 396\;Hz\)
No. of beats \(=n_{1}-n_{2}=4\)

Explanation:

Explanation:

Given, \({y_1} = 5\sin (300\,\pi t)\) and
\({y_2} = 4\sin (302\,\pi t)\)
Comparing with standard eqns.
\({y_{1}=A_{1} \sin \omega_{1} t}\) and \({y_{2}=A_{2} \sin \omega_{2} t}\)
\({A_1} = 5,{A_2} = 4,{\omega _1} = 300\,\pi ,{\omega _2} = 302\,\pi \)
\({\because \quad A_{\max }=A_{1}+A_{2}, A_{\text {min }}=A_{1}-A_{2}}\)
and \({A_{\text {max }}^{2}={A}_{1}^{2}+{A}_{2}^{2}+2 A_{1} A_{2}}\)
\({A_{\text {min }}^{2}=A_{1}^{2}+{A}_{2}^{2}-2 A_{1} A_{2}}\)
\({\because \quad I=k A^{2}}\) or \({A^{2}=\dfrac{I}{k}}\)
\(\therefore \;\;\;\frac{{{I_{\max }}}}{k} = A_1^2 + A_2^2 + 2{A_1}{A_2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(1)\)
\(\therefore \;\;\;\frac{{{I_{{\text{min }}}}}}{k} = A_1^2 + A_2^2 - 2{A_1}{A_2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(2)\)
From eqn. (1) and (2)
\({\dfrac{I_{\max }}{I_{\min }}=\dfrac{A_{1}^{2}+A_{2}^{2}+2 A_{1} A_{2}}{A_{1}^{2}+A_{2}^{2}-2 A_{1} A_{2}}=\dfrac{5^{2}+4^{2}+2 \times 5 \times 4}{5^{2}+4^{2}-2 \times 5 \times 4}}\)
\({\Rightarrow \dfrac{I_{\max }}{I_{\min }}=\dfrac{81}{1}}\).
So correct option is (3)

Explanation:

Since \(\phi=\dfrac{\pi}{2} \Rightarrow A=\sqrt{a_{1}^{2}+a_{2}^{2}}=\sqrt{(4)^{2}+(3)^{2}}=5\).

Explanation:

Velocity of wave \(v=n \lambda\)
Where \(n = \) frequency of wave \(\Rightarrow n=\dfrac{\nu}{\lambda}\)
\({n_1} = \frac{{{v_1}}}{{{\lambda _1}}} = \frac{{396}}{{99 \times {{10}^{ - 2}}}} = 400\;Hz\)
\({n_2} = \frac{{{v_2}}}{{{\lambda _2}}} = \frac{{396}}{{100 \times {{10}^{ - 2}}}} = 396\;Hz\)
No. of beats \(=n_{1}-n_{2}=4\)

Explanation:

Explanation:

Given, \({y_1} = 5\sin (300\,\pi t)\) and
\({y_2} = 4\sin (302\,\pi t)\)
Comparing with standard eqns.
\({y_{1}=A_{1} \sin \omega_{1} t}\) and \({y_{2}=A_{2} \sin \omega_{2} t}\)
\({A_1} = 5,{A_2} = 4,{\omega _1} = 300\,\pi ,{\omega _2} = 302\,\pi \)
\({\because \quad A_{\max }=A_{1}+A_{2}, A_{\text {min }}=A_{1}-A_{2}}\)
and \({A_{\text {max }}^{2}={A}_{1}^{2}+{A}_{2}^{2}+2 A_{1} A_{2}}\)
\({A_{\text {min }}^{2}=A_{1}^{2}+{A}_{2}^{2}-2 A_{1} A_{2}}\)
\({\because \quad I=k A^{2}}\) or \({A^{2}=\dfrac{I}{k}}\)
\(\therefore \;\;\;\frac{{{I_{\max }}}}{k} = A_1^2 + A_2^2 + 2{A_1}{A_2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(1)\)
\(\therefore \;\;\;\frac{{{I_{{\text{min }}}}}}{k} = A_1^2 + A_2^2 - 2{A_1}{A_2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(2)\)
From eqn. (1) and (2)
\({\dfrac{I_{\max }}{I_{\min }}=\dfrac{A_{1}^{2}+A_{2}^{2}+2 A_{1} A_{2}}{A_{1}^{2}+A_{2}^{2}-2 A_{1} A_{2}}=\dfrac{5^{2}+4^{2}+2 \times 5 \times 4}{5^{2}+4^{2}-2 \times 5 \times 4}}\)
\({\Rightarrow \dfrac{I_{\max }}{I_{\min }}=\dfrac{81}{1}}\).
So correct option is (3)

Explanation:

Since \(\phi=\dfrac{\pi}{2} \Rightarrow A=\sqrt{a_{1}^{2}+a_{2}^{2}}=\sqrt{(4)^{2}+(3)^{2}}=5\).