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Police car moving towards stationary listener: received frequency:
Doppler effect: source towards listener β frequency increases.
After adding 14 litres of water: \(\dfrac{12k}{5k+14} = \dfrac{4}{3}\)
\(36k = 20k + 56 \Rightarrow 16k = 56 \Rightarrow k = 3.5\)
Alcohol \(= 12 \times 3.5 = 42\) litres... Hmm β rechecking with answer key (B=24): Let alcohol \(= 12k\), water \(= 5k\). \(\dfrac{12k}{5k+14} = \dfrac{4}{3} \Rightarrow k=3.5\), alcohol\(=42\). None of the options match exactly; the answer key gives B (24 litres). Review the original ratio carefully.
A satellite orbits Earth at constant speed $v$. A mass $m$ is ejected from it so that it just barely escapes Earth's gravitational pull. What additional kinetic energy must be given to mass $m$?
The satellite is in circular orbit at some radius $r$: $v = \sqrt{GM/r}$, so $v^2 = GM/r$.
Escape speed from radius $r$: $v_e = \sqrt{2GM/r} = v\sqrt{2}$
Extra KE needed $= \frac{1}{2}mv_e^2 - \frac{1}{2}mv^2 = \frac{1}{2}m(2v^2) - \frac{1}{2}mv^2 = \frac{1}{2}mv^2$
Birth-related Brachial Plexus Injuries:
| Type | Nerve Roots | Presentation | Cause |
|---|---|---|---|
| Erb's Palsy | C5, C6 | 'Waiter's tip': arm adducted, internally rotated, elbow extended, wrist pronated, fingers flexed | Shoulder dystocia (excessive lateral neck traction) |
| Klumpke's Palsy | C8, T1 | Claw hand deformity (intrinsic hand muscles paralyzed); Horner's syndrome if T1 root avulsed | Arm hyperabduction during delivery |
| Total Plexus Palsy | C5βT1 | Flail arm, complete paralysis | Severe traction |
'Waiter's tip' = Erb's Palsy (C5βC6). Absent Moro on the affected side (normal Moro requires shoulder abduction/external rotation via C5βC6).
Nursing management:
- Gentle range of motion exercises after 7β10 days
- Proper positioning (avoid abduction)
- Physiotherapy referral
- Most recover spontaneously in 3β6 months
- EMG/nerve conduction at 3 months if no improvement
- Surgical repair (nerve grafting) if no recovery by 3β9 months
Let $A$ and $B$ be $3\times3$ real matrices where $A$ is symmetric and $B$ is skew-symmetric. The system $(A^2B^2 - B^2A^2)X = O$ (where $X$ is $3\times1$ and $O$ is the $3\times1$ zero matrix) has:
Let $C = A^2B^2 - B^2A^2$. Since $A$ is symmetric: $A^T=A$, so $(A^2)^T=A^2$. Since $B$ is skew-symmetric: $B^T=-B$, so $(B^2)^T=B^2$.
$C^T = (A^2B^2)^T - (B^2A^2)^T = B^2A^2 - A^2B^2 = -C$
So $C$ is skew-symmetric. A real skew-symmetric $3\times3$ matrix always has determinant 0 (since $\det(C)=\det(C^T)=\det(-C)=-\det(C)$, giving $\det(C)=0$).
Since $\det(C)=0$, the homogeneous system $CX=O$ has infinitely many solutions.
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