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### The ranges and heights for two projectiles projected ....

The ranges and heights for two projectiles projected with the same initial velocity at angles $42^\circ$ and $48^\circ$ with the horizontal are $R_1\,,R_2$ and $H_1\,,H_2$ respectively. Choose the correct option:

(A) $R_1 < R_2$ and $H_1 < H_2$
(B) $R_1 > R_2$ and $H_1 = H_2$
(C) $R_1 = R_2$ and $H_1 = H_2$
(D) $R_1 = R_2$ and $H_1 < H_2$

Solution

$R=\frac {u^2 sin 2\theta }{g}$, $H=\frac {u^2 sin^2 \theta }{2g}$

$\frac {R_1}{R_2} = \frac {sin 2 \times 42^\circ}{sin 2\times 48^\circ} = \frac {cos 6^\circ}{cos 6^\circ} = 1$

$\frac {H_1}{H_2}= \frac {sin^2 42^\circ }{sin^2 48^\circ } = tan^2 42^\circ < 1$

### Sum of the coefficients in the expansion of $(x+y)^n$ ....

If the sum of the coefficients in the expansion of $(x+y)^n$ is 4096, then the greatest coefficient in the expansion is _ _ _ _ . Solution $C_0 + C_1 + C_2 + C_3 + ......................... + C_n =4096$ $\therefore 2^n = 4096 =2^{12}$ $\Rightarrow n = 12$ Greatest coefficient = ${}^{12}{C_6} = 924$