`A_1,A_2,..., A_n` are the vertices of a regular plane polygon with n sides and O as its centre. Show that `sum_(i=1)^n vec (OA)_i xx vec(OA)_(i+1)=(1-n)(vec (OA)_2 xx vec(OA)_1)`

A_(1),A_(2), …. A_(n) are the vertices of a regular plane polygon with n sides and O ars its centre. Show that sum_(i=1)^(n-1) (vec(OA_(i))xxvec(OA)_(i+1))=(n-1) (vec(OA)_1 xx vec(OA)_(2))

Match the following: List - I, List - II Let y(x)=cos(3cos^(-1)x) x in [-1,1],x!=+-(sqrt(3))/2 Then 1/(y(x)){(x^2-1)(d^2y(x))/(dx^2)+(dy(x))/(dx)} equals, 1 Let A_1,A_2, A_n(n >2) be the vertices of a regular polygon of n sides with its centre at the origin. Let vec a_k be the position vector of the point A_k ,k=1,2, ndot If |sum_(k=1)^(n-1)( vec a_k x vec a_(k+1))|=|sum_(k=1)^(n-1)( vec a_kdot vec a_(k+1))|, then the minimum value of n is, 2 If the normal from the point P(h ,1) on the ellipse (x^2)/6+(y^2)/3=1 is perpendicular to the line x+y=8 , then the value of h is, 8 Number of positive solutions satisfying the equation tan^(-1)(1/(2x)+1)+tan^(-1)(1/(4x+1))=tan^(-1)(2/(x^2)) is, 9

Let A_1,A_2,....A_n be the vertices of an n-sided regular polygon such that , 1/(A_1A_2)=1/(A_1A_3)+1/(A_1A_4) . Find the value of n.

Let O be the centre of the regular hexagon ABCDEF then find vec(OA)+vec(OB)+vec(OD)+vec(OC)+vec(OE)+vec(OF)

ABCD is a quadrilateral and E is the point of intersection of the lines joining the middle points of opposite side. Show that the resultant of vec (OA) , vec(OB) , vec(OC) and vec(OD) = 4 vec(OE) , where O is any point.

Let O be the centre of a regular pentagon ABCDE and vec(OA) = veca , then vec(AB) +vec(2BC) + vec(3CD) + vec(4DE) + vec(5EA) is equals:

Let O be the centre of a regular pentagon ABCDE and vec(OA) = veca , then vec(AB) +vec(2BC) + vec(3CD) + vec(4DE) + vec(5EA) is equals:

Let O, O' and G be the circumcentre, orthocentre and centroid of a Delta ABC and S be any point in the plane of the triangle. Statement -1: vec(O'A) + vec(O'B) + vec(O'C)=2vec(O'O) Statement -2: vec(SA) + vec(SB) + vec(SC) = 3 vec(SG)

Show that ("lim")_(n vec oo)(1/(n+1)+1/(n+2)++1/(6n))=log6

If | vec(a) | = | vec( b) | =1 and | vec(a ) xx vec( b)| =1 , then