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(D) For system $(a)$: The effective spring constant is $K_{eff} = K$. The time period is $T_a = 2 \pi \sqrt{\frac{m}{K}}$.For system $(b)$: The two sp

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$T_b=2 T_c$

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(D) For system $(a)$: The effective spring constant is $K_{eff} = K$. The time period is $T_a = 2 \pi \sqrt{\frac{m}{K}}$.For system $(b)$: The two springs are in series. The effective spring constant is $\frac{1}{K_{eff}} = \frac{1}{K} + \frac{1}{K} = \frac{2}{K}$,so $K_{eff} = \frac{K}{2}$. The time period is $T_b = 2 \pi \sqrt{\frac{m}{K/2}} = 2 \pi \sqrt{\frac{2m}{K}} = \sqrt{2} T_a$.For system $(c)$: The two springs are in parallel. The effective spring constant is $K_{eff} = K + K = 2K$. The time period is $T_c = 2 \pi \sqrt{\frac{m}{2K}} = \frac{1}{\sqrt{2}} (2 \pi \sqrt{\frac{m}{K}}) = \frac{T_a}{\sqrt{2}}$.From $T_b = \sqrt{2} T_a$ and $T_c = \frac{T_a}{\sqrt{2}}$,we can write $T_a = \sqrt{2} T_c$.Substituting this into the expression for $T_b$: $T_b = \sqrt{2} (\sqrt{2} T_c) = 2 T_c$.

List-$I$	List-$II$
$(A)$ $[\mathbf{a} \mathbf{b} \mathbf{c}]$	$1. \|\mathbf{a}\|\|\mathbf{b}\|\cos(\mathbf{a}, \mathbf{b})$
$(B)$ $(\mathbf{c} \times \mathbf{a}) \times \mathbf{b}$	$2. (\mathbf{a} \cdot \mathbf{c})\mathbf{b} - (\mathbf{a} \cdot \mathbf{b})\mathbf{c}$
$(C)$ $\mathbf{a} \times (\mathbf{b} \times \mathbf{c})$	$3. \mathbf{a} \cdot \mathbf{b} \times \mathbf{c}$
$(D)$ $\mathbf{a} \cdot \mathbf{b}$	$4. \|\mathbf{a}\|\|\mathbf{b}\|$
	$5. (\mathbf{b} \cdot \mathbf{c})\mathbf{a} - (\mathbf{a} \cdot \mathbf{b})\mathbf{c}$

All the springs in figures $(a)$,$(b)$,and $(c)$ are identical,each having a force constant $K$. The mass attached to each system is $m$. If $T_a, T_b$,and $T_c$ are the time periods of oscillations of the three systems respectively,then:

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