If T = (2n+1)C1 + (2n+1)C2 + ... + (2n+1)Cn = 255. Find the value of n.

The value of $i^{1+3+5+\ldots . .+(2 n+1)}$ is:

A. $i$ if $n$ is even, $-i$ if $n$ is odd B. 1 if $n$ is even, -1 if $n$ is odd C. 1 if $n$ is odd, -1 if $n$ is even D. $i$ if $n$ is even, -1 if $n$ is odd

If $\sum_{r=1}^{n} T_{r} = \frac{n}{8}(n+1)(n+2)(n+3)$, and $\sum_{r=1}^{n} \frac{1}{T_{r}} = \frac{n^{2}+3n}{4 \sum_{k=1}^{p} k}$, then $p$ is equal to

(A) $n+1$

(B) $n$

(C) $n-1$

(D) $2n$

If water drops are falling at regular time intervals from a ceiling of height H, then the position of the drop from the ceiling is given by:

A $\quad \frac{(n-1)^{2}}{(n-r)^{2}} H$
B $\frac{(n-r)^{2}}{(n-1)^{2}} H$
C $\quad \frac{(n-r)^{2}}{(n+1)^{2}} H$
D $\frac{(n+1)^{2}}{(n+r)^{2}} H$

$$ \begin{array}{l} V = 3 + 9t \text{ if } t < 9 \ V = 8 - 2t \text{ if } t = 10 \end{array} $$

Find the value of $V(\text{m/s})$ when $t = 10$.

A. $V = 10$ m/s

B. $V = -10$ m/s

C. $V = -5$ m/s

D. $V = -12$ m/s

A body is projected at an angle $\theta$ so that its range is maximum. If $T$ is the time of flight, then the value of maximum range is (acceleration due to gravity = $g$).

A. $\frac{g^{2}T}{2}$
B. $\frac{gT}{2}$
C. $\frac{gT^{2}}{2}$
D. $\frac{g^{2}T^{2}}{2}$