Problem 2

1. What would be the approximate length of the day if the earth spun so fast that bodies floated on the equator? Take the radius of the earth $R=6\times 10^6 m$ and $g=9.8 m/s^2$
   (A) 12 hours (B) 6 hours (C) 3 hours (D) 1.5 hour
   Answer: Body remains on earth's surface because
   $gravitational~ pull ~i.e.~ weight ~of~ the~ body\geq centrifugal~ force ~due ~to~ rotation$
   Body will start floating when
   $centrifugal~force~due~ to ~rotation> gravitational~pull~i.e. ~weight~of~the~ body$
   $$m\omega^2r>mg$$ or $$\omega>\sqrt{\frac{g}{r}}$$ But $$\omega=\frac{2\pi}{T}$$ $$T(in~ seconds)>2\pi\sqrt{\frac{r}{g}}$$
   $$T(in~ hours)>2\pi\sqrt{\frac{r}{g}}\frac{1}{3600}=\frac{2\times3.14\times1000\times\sqrt{6}}{3600\times\sqrt{9.8}}=1.5~ hours$$
2. The real matrix $A=\begin{pmatrix}a&-f&-g\\ f&a&h\\g&-h&a\end{pmatrix}$ is skew symmetric when
   (a) $a=0$ (b) $f=0$ (c) $g=h$ (d) $f=g$ Answer: Matrix $A$ is skew symmetric when $A_{ij}=-A_{ji}$. Hence  $a=0$
3. If $A$ and $B$ are matrices such that $AB=B$ and $BA=A$ then $A^2+B^2$ equals
   (A)2AB (B) 2BA (C) A+B (D) AB
     Answer: $$A^2+B^2=(BA)^2+(AB)^2=BABA+ABAB$$ $$A^2+B^2=B\underline{AB}A+A\underline{BA}B=B\underline{BA}+A\underline{AB}=BA+AB=A+B$$
4.  The average value of function $f(x)=4x^3$ in the interval 1 to 3 is

        (a) 15 (b) 20 (c) 40 (d) 40
Answer: $<f(x)>=\frac{\int_1^3f(x)dx}{\int_1^3dx}=\frac{\int_1^34^3dx}{\int_1^3dx}=\frac{[x^4]_1^3}{[x]_1^3}=\frac{81-1}{3-1}=40$