Show that the following lines intersect. Also, find their point of intersection:
Line 1: \[ \frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4} \]
Line 2: \[ \frac{x - 4}{5} = \frac{y - 1}{2} = z \]
Two concentric circles are of radii $8\ \text{cm}$ and $5\ \text{cm}$. Find the length of the chord of the larger circle which touches (is tangent to) the smaller circle.
Find mean of the following frequency table:
Prove that $\dfrac{1+\sec\theta}{\sec\theta}=\dfrac{\sin^2\theta}{1-\cos\theta}$.
Prove that $7\sqrt{5}$ is an irrational number.
The modal class of the following table will be: \[ \begin{array}{|c|c|} \hline \text{Class Interval} & \text{Frequency} \\ \hline 0-5 & 5 \\ \hline 5-10 & 8 \\ \hline 10-15 & 12 \\ \hline 15-20 & 10 \\ \hline 20-25 & 7 \\ \hline \end{array} \]
Median class of the following frequency distribution will be: \[ \begin{array}{|c|c|} \hline \text{Class Interval} & \text{Frequency} \\ \hline 0-10 & 7 \\ \hline 10-20 & 12 \\ \hline 20-30 & 18 \\ \hline 30-40 & 15 \\ \hline 40-50 & 10 \\ \hline 50-60 & 3 \\ \hline \end{array} \]
