A uniform rope of length L and mass m₁ hangs vertically from a rigid support. A block of mass m₂ is attached to the free end of the ropes. A transverse pulse of wavelength λ₁ is produced at the lower end of the rope. The wavelength of the pulse when it reaches the top of the rope is λ₂. The ratio λ₂/λ₁ is-

1. $\sqrt{\frac{m_1+m_2}{m_2}}$

2. $\sqrt{\frac{m_2}{m_1}}$

3. $\sqrt{\frac{m_1+m_2}{m_1}}$

4. $\sqrt{\frac{m_1}{m_2}}$

If n₁, n₂ and n₃ are, are the fundamental frequencies of three segments into which a string is divided, then the original fundamental frequency n of the string is given by

(1) 1/n=1/n₁+1/n₂+1/n₃

(2) 1/√n=1/√n₁+1/√n₂+1/√n₃

(3) √n=√n₁+√n₂+√n₃

(4) n=n₁+n₂+n₃

When a string is divided into three segments of lengths $l_1, l_2 and l_3,$ the fundamental frequencies of these three segments are $v_1, v_2 and v_3$ respectively. The original fundamental frequency (v) of the string is 

(1) $\sqrt{v}=\sqrt{v_1}+\sqrt{v_2}+\sqrt{v_3}$

(2) $v=v_1+v_2+v_3$

(3) $\frac{1}{v}=\frac{1}{v_1}+\frac{{\displaystyle 1}}{{\displaystyle v_2}}+\frac{{\displaystyle 1}}{{\displaystyle v_3}}$

(4) $\frac{1}{\sqrt{v}}=\frac{1}{\sqrt{v_1}}+\frac{{\displaystyle 1}}{{\displaystyle \sqrt{v_2}}}+\frac{{\displaystyle 1}}{{\displaystyle \sqrt{v_3}}}$

A wave in a string has an amplitude of 2 cm. The wave travels in the +ve direction of x-axis with a speed of $128 {\mathrm{ms}}^{-1}$ and it is noted that 5 complete waves fit in 4 m length of the string. The equation describing the wave is :

1. $y=\left(0.02\right)\mathrm{m} \sin \left(7.85\mathrm{x} +1005\mathit{ }t \right)$

2. $y=\left(0.02\right)\mathrm{m} \sin \left(15.7\mathrm{x} -2010\mathit{ }t \right)$

3. $y=\left(0.02\right)\mathrm{m} \sin \left(15.7\mathrm{x} +2010\mathit{ }t \right)$

4. $y=\left(0.02\right)\mathrm{m} \sin \left(7.85\mathrm{x} -1005\mathit{ }t \right)$

The equation of a wave traveling in a string can be written as $y=3\cos \pi (100t-x)$. Its wavelength is :

(1) 100 cm

(2) 2 cm

(3) 5 cm

(4) None of the above