Exercise

Find the ratio in which the segment joining the point $\left(1,-3\right)$ and $\left(4,5\right)$ is divided by $x‐$axis? Also, find the coordinates of this point of the $x‐$axis.

Find the area of the shaded region in Fig. $2$, where arcs drawn with centres $\mathrm{A},\mathrm{B},\mathrm{C}$ and $\mathrm{D}$ intersect in pairs at mid-points $\mathrm{P},\mathrm{Q},\mathrm{R} \text{and} \mathrm{S}$ of the sides $\mathrm{AB},\mathrm{BC},\mathrm{CD} \text{and} \mathrm{DA}$ respectively of a square $\mathrm{ABCD}$ of side $12 \mathrm{cm}$. $[\text{Use} \mathrm{\pi }=3.14]$

Show that every positive odd integer is of the form $\left(4q+1\right)$ or $\left(4q+3\right)$, where $q$ is some integer.

As observed from the top of a $100 \mathrm{m}$ high light house from the sea level, the angles of depression of two ships are $30°$ and $45°$. If one ship is exactly behind the other on the same side of the light house, find the distance between the two ships. $[\text{Use} \sqrt{3}=1.732]$

A vertical tower stands on a horizontal plane and is surmounted by a vertical flag staff of height $6 \mathrm{m}$. At a point on the plane, the angle of elevation of the bottom and top of the flag staff are $30°$ and $45°$ respectively. Find the height of the tower. Take $\sqrt{3}=1.73$

In the figure, two tangents $\mathrm{TP} \text{and} \mathrm{TQ}$ are drawn to a circle with centre $\mathrm{O}$ from an external point $\mathrm{T}$. Prove that $\angle \mathrm{PTQ}=2\angle \mathrm{OPQ}$.

Find the midpoint of pair of points: $(4,2)\text{ and }(6,12)$.

Solve the equation $3x^2-48=0$ leaving your answers exact.