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Pakistan's Expanded Programme on Immunization (EPI) currently protects children against 12 vaccine-preventable diseases:
- Tuberculosis (BCG)
- Poliomyelitis (OPV/IPV)
- Diphtheria
- Pertussis (Whooping cough)
- Tetanus
- Hepatitis B
- Haemophilus influenzae type b (Hib)
- Pneumococcal disease (PCV)
- Rotavirus diarrhea
- Measles
- Rubella
- Typhoid (recently added)
As a Head Nurse, knowing the current EPI schedule is mandatory for supervising vaccination services at community level.
If $\alpha$ and $\beta$ are the two roots of $x^2+2x+2=0$, find $\alpha^{15}+\beta^{15}$.
Roots: $x=\frac{-2\pm\sqrt{4-8}}{2}=-1\pm i$. So $\alpha=-1+i, \beta=-1-i$.
In polar form: $|\alpha|=\sqrt{2}$, $\arg(\alpha)=\frac{3\pi}{4}$. So $\alpha=\sqrt{2}\,e^{i3\pi/4}$.
$\alpha^{15}=(\sqrt{2})^{15}e^{i\cdot45\pi/4}=2^{15/2}e^{i\pi/4}$ (since $45\pi/4=11\pi+\pi/4$, so $e^{i45\pi/4}=e^{i\pi/4}\cdot(-1)^{11}=-e^{i\pi/4}$... careful: $45/4=11.25$, $11\pi+\pi/4$, $e^{i(11\pi+\pi/4)}=e^{i\pi}\cdot e^{i\pi/4} \cdot (-1)^{10}... $)
More cleanly: $\alpha^{15}+\beta^{15}=2\,\text{Re}(\alpha^{15})=2(\sqrt{2})^{15}\cos\!\left(\frac{45\pi}{4}\right)$. Since $\cos(45\pi/4)=\cos(\pi/4+11\pi)=-\cos(\pi/4)=-\frac{1}{\sqrt{2}}$: $=2\cdot2^{15/2}\cdot(-\frac{1}{\sqrt{2}})=-2^{15/2+1-1/2}=-2^8=-\mathbf{256}$
If force ($F$), velocity ($V$), and time ($T$) are taken as fundamental units, then the dimensions of mass in this system are:
From Newton's second law: $F = ma = m\dfrac{v}{t}$
So $m = \dfrac{Ft}{v}$
In terms of fundamental units F, V, T:
$[m] = \dfrac{[F][T]}{[V]} = FV^{-1}T$
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