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\(\text{Yb}^{2+}\): Yb (Z=70) = [Xe] 4f¹⁴ 6s². Yb²⁺ = [Xe] 4f¹⁴ → completely filled f subshell → all electrons paired → diamagnetic ✓
Ce²⁺ = [Xe]4f², Sm²⁺ = [Xe]4f⁶, Eu²⁺ = [Xe]4f⁷ — all have unpaired f-electrons and are paramagnetic.
Let $n$ be a positive integer divisible by 6. Compare:
Quantity A: The remainder when $n$ is divided by 12
Quantity B: The remainder when $n$ is divided by 18
Since $n$ is divisible by 6, $n$ can be $6, 12, 18, 24, 30, 36, \ldots$
Example 1: $n = 12$. Remainder ÷12 $= 0$. Remainder ÷18 $= 12$. B > A.
Example 2: $n = 18$. Remainder ÷12 $= 6$. Remainder ÷18 $= 0$. A > B.
Example 3: $n = 36$. Remainder ÷12 $= 0$. Remainder ÷18 $= 0$. Equal.
The relationship changes, so it cannot be determined.
The maximum range is $R_{max} = \frac{u^2}{g}$.
- Area covered is a circle: $A = \pi R_{max}^2 = \pi (\frac{u^2}{g})^2 \propto u^4$.
- Ratio: $\frac{A_1}{A_2} = (\frac{u_1}{u_2})^4 = (\frac{1}{2})^4 = 1:16$.
To travel straight north, the swimmer must aim upstream (westward) to cancel river drift.
If $\phi$ is the angle west of north:
$v_{\text{swim}}\sin\phi = v_{\text{river}} \Rightarrow 20\sin\phi = 10 \Rightarrow \sin\phi = \dfrac{1}{2} \Rightarrow \phi = \mathbf{30°}$ west of north
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