Given:The diagonals of a cyclic quadrilateral are diameters of the circle through the vertices of the quadrilateral.To do:We have to prove that it is a rectangle.Solution:Let $PR$ and $QS$ be the diagonals of a cyclic quadrilateral $PQRS$.This implies, $PR$ and $QS$ are diameters of the circle.$PR=QS$$OP=OQ=OR=OS$    (Radii of the ... Read More

Given:\( \mathrm{ABCD} \) is a cyclic quadrilateral whose diagonals intersect at a point \( \mathrm{E} \).  \( \angle \mathrm{DBC}=70^{\circ} \), \( \angle \mathrm{BAC} \) is \( 30^{\circ} \)\( \mathrm{AB}=\mathrm{BC} \)To do:We have to find \( \angle \mathrm{BCD} \) and \( \angle \mathrm{ECD} \).Solution:We know that, The angles in the same ... Read More

Given:The diameter of a sphere is decreased by \( 25 \% \).To do:We have to find the per cent by which the curved surface area decreases.Solution:Let $d$ be the diameter of the sphere initially.This implies, Radius of the sphere initially $r=\frac{d}{2}$Surface area of the sphere initially $=4 \pi(\frac{d}{2})^{2}$$=4\times \pi \times ... Read More

Given:A, B, C and D are four points on a circle. \( \mathrm{AC} \) and \( \mathrm{BD} \) intersect at a point \( \mathrm{E} \) such that \( \angle \mathrm{BEC}=130^{\circ} \) and \( \angle \mathrm{ECD}=20^{\circ} \). To do:We have to find \( \angle \mathrm{BAC} \).Solution:We know that, The angles in the ... Read More

 Given:A heap of wheat is in the form of a cone whose diameter is \( 10.5 \mathrm{~m} \) and height is \( 3 \mathrm{~m} \).To do:We have to find its volume and the canvas cloth required to cover the heap.Solution:Diameter of the conical heap of wheat $= 10.5\ m$This implies, ... Read More

Given:Radius of a sphere is(i) \( 7 \mathrm{~cm} \)(ii) \( 063 \mathrm{~m} \).To do:We have to find the volumes of the sphere in each case.Solution:We know that, Volume of a sphere of radius $r$ is $\frac{4}{3} \pi r^3$Therefore, (i) Volume of the sphere of radius $7\ cm= \frac{4}{3} \pi (7)^3$$=\frac{4}{3} ... Read More

Given:Diameter of a solid spherical ball is(i) \( 28 \mathrm{~cm} \)(ii) \( 0.21 \mathrm{~m} \).To do:We have to find the amount of water displaced by the solid spherical ball in each case.Solution:(i) Diameter of the  solid spherical ball $= 28\ cm$Radius of the solid spherical ball $r = \frac{28}{2}\ cm$$= 14\ cm$Water ... Read More

Given:The diameter of a metallic ball is \( 4.2 \mathrm{~cm} \).The density of the metal is \( 8.9 \mathrm{~g} \) per \( \mathrm{cm}^{3} \).To do:We have to find the mass of the ball.Solution:Diameter of the metallic ball $= 4.2\ cm$This implies, Radius of the metallic ball $r = \frac{4.2}{2}\ cm$$= ... Read More

Given:The diameter of a hemispherical bowl is \( 10.5 \mathrm{~cm} \).To do:We have to find the volume of water it can hold.Solution:Diameter of the hemispherical bowl $= 10.5\ cm$This implies, Radius of the hemispherical bowl $r = \frac{10.5}{2}\ cm$$= 5.25\ cm$Therefore,  Volume of the hemispherical bowl $=\frac{2}{3} \pi r^{3}$$=\frac{2}{3} \times \frac{22}{7} ... Read More

 Given:A hemispherical tank is made up of an iron sheet $1\ cm$ thick.The inner radius is $1\ m$.To do:We have to find the volume of the iron used to make the tank.Solution:Thickness of the hemispherical tank $= 1\ cm$Inner radius of the tank $(r) = 1\ m$$= 100\ cm$This implies, ... Read More