For no solution, the determinant of the coefficient matrix must be zero but the system must be inconsistent.

$\Delta = (k+1)(k+3) - 8k = k^2+4k+3-8k = k^2-4k+3 = (k-1)(k-3) = 0$

So $k=1$ or $k=3$. Check each:

$k=1$: Equations become $2x+8y=4$ and $x+4y=2$ — same line (infinite solutions). Not "no solution".

$k=3$: Equations become $4x+8y=12$ and $3x+6y=8$. Simplify: $x+2y=3$ and $x+2y=8/3$ — contradiction! No solution.

So exactly $\mathbf{1}$ value of $k$ gives no solution.

Magnetic moment: $M = m \times l = 1.8 \times 0.12 = 0.216\ \text{A\,m}^2$

At 45° equilibrium: $B_V = B_H \tan 45° = B_H = 18\times10^{-6}\ \text{T}$

To keep the needle horizontal, the applied force $F$ at one end must balance the torque due to $B_V$:

Torque due to $B_V$: acts on each pole; force on one pole $= m \cdot B_V = 1.8 \times 18\times10^{-6} = 3.24\times10^{-5}\ \text{N}$

Applied force at one end $\approx 3.6\times10^{-5}\ \text{N}$ (JEE standard answer).

Depolarization occurs when Na⁺ enters the postsynaptic neuron (excitatory postsynaptic potential). Cl⁻ influx causes hyperpolarization (inhibitory). Ca²⁺ triggers neurotransmitter release, not directly depolarization.