The following table shows the ages of the patients admitted in a hospital during a year. Find the mode and the median of these data.\[\begin{array}{|c|c|c|c|c|c|c|} \hline Age (in years) & 5-15 & 15-25 & 25-35 & 35-45 & 45-55 & 55-65 \\ \hline \text{Number of patients} & \text{6} & \text{11} & \text{21} & \text{23} & \text{14} & \text{5} \\ \hline \end{array}\]
Find the unknown frequency if 24 is the median of the following frequency distribution:\[\begin{array}{|c|c|c|c|c|c|} \hline \text{Class-interval} & 0-10 & 10-20 & 20-30 & 30-40 & 40-50 \\ \hline \text{Frequency} & 5 & 25 & 25 & \text{$p$} & 7 \\ \hline \end{array}\]
The value of $\dfrac{1+\cot^2 \theta}{1+\tan^2 \theta}$ will be:
$PQ$ is a chord of length $4\ \text{cm}$ of a circle of radius $2.5\ \text{cm}$. The tangents at $P$ and $Q$ intersect at a point $T$. Find the length of $TP$.