Examine whether the two quadrilaterals obtained by joining the vertices 'F' and C of a regular hexagon ABCDEF are. cyclic or not? Why?

Prove that the quadrilateral obtained by joining any two alternate vertices of a regular pentagon is cyclic.

In the picture below, the top vertex of a triangle is joined to the midpoint of the bottom side of the triangle and then the midpoint of this line is joined to the other two vertices. Prove that the areas of all four triangles obtained thus are equal to a fourth of the area of the whole triangle.

In the picture below, the top vertex of a triangle is joined to the midpoint of the bottom side of the triangle and then the midpoint of this line is joined to the other two vertices. Prove that the areas of all four triangles obtained thus are equal to a fourth of the area of the whole triangle.

State,with reasons,whether the following algebraic operations with scalar and vector physical quantities are meaningful: a)adding any two scalars,b)adding a scalar to a vector of the same dimensions,c)multiplying any vector by any scalar,d)multiplying any two scalars,e)adding any two vectors,f)adding a component of a vector to the same vector.

A horizontal metal wire is fixed on a state of tension between two vertical supports when plucked it gives a fundamental frequency f_0 . Obtain a mathematical expression for f_0 .

An organic compound A on treatment with CHCl_3 and KOH gives two compounds B and C . Both B an C give the same product (D) when distilled with zinc dust. Oxidation of D gives E . having molecular formula C_7 H_6 O_2 . The sodium salt of E on heating with soda lime gives F which may also be obtained by distilling A with zinc dust. Identify A to F ,

A horizontal metal wire is fixed on a state of tension between two vertical supports when plucked it gives a fundamental frequency f_0 . A 5.5 m wire has a mass of 0.035 kg ,if the tension of the string is 77 N ,the speed of wave on the string is a) 110 ms^(-1) b) 11sqrt10 ms^(-1) c) 77 ms^(-1) d) 11 ms^(-1) e) 102 ms^(-1) .