In the figure, if `B_(1), B_(2), B_(3)`, and `A_(1), A_(2), A_(3), …..` have been marked at equal distances. In what ratio C divides AB?

In the figure, if B1, B2, B3,…... and A1,A2, A3,….. have been marked at equal distances. In what ratio C divides AB?

To divide a line segment AB in the ratio 5:6, draw a ray AX such that angleBAX is an acute angle, the draw a ray BY parallel to AX and the points A_(1),A_(2),A_(3),….." and " B_(1),B_(2),B_(3),….. are located to equal distances on ray AX and BY, respectively. Then, the points joined are

A_(1), A_(2), A_(3), ……, A_(n) " and " B_(1), B_(2), B_(3), …., B_(n) are non-singular square matrices of order n such that A_(1)B_(1) = I_(n), A_(2)B_(2) = I_(n), A_(3)B_(3) = I_(n),……A_(n)B_(n) = I_(n) " then"(A_(1) A_(2)A_(3)….. A_(n))^(-1) = ______.

If the arithmetic mean of a_(1),a_(2),a_(3),"........"a_(n) is a and b_(1),b_(2),b_(3),"........"b_(n) have the arithmetic mean b and a_(i)+b_(i)=1 for i=1,2,3,"……."n, prove that sum_(i=1)^(n)(a_(i)-a)^(2)+sum_(i=1)^(n)a_(i)b_(i)=nab .

Let a_(1), a_(2), a_(3).... and b_(1), b_(2), b_(3)... be arithmetic progression such that a_(1) = 25, b_(1) = 75 and a_(100) + b_(100) = 100 . Then

If vectors baraequiv(a_(1),a_(2),a_(3)) and barbequiv(b_(1),b_(2),b_(3)) are collinear/coincident/parallel, then, in general,

Let A and B be two sets such that n(A) = 5 and n(B) = 2 . If (a_(1), 2), (a_(2), 3), (a_(3), 2), (a_(4), 3), (a_(5), 2) are in A xx B and a_(1), a_(2), a_(3), a_(4) and a_(5) are distinct, find A and B.

If a and b are distinct positive real numbers such that a, a_(1), a_(2), a_(3), a_(4), a_(5), b are in A.P. , a, b_(1), b_(2), b_(3), b_(4), b_(5), b are in G.P. and a, c_(1), c_(2), c_(3), c_(4), c_(5), b are in H.P., then the roots of a_(3)x^(2)+b_(3)x+c_(3)=0 are

if Delta=det[[a_(1),b_(1),c_(1)a_(2),b_(2),c_(2)a_(3),b_(3),c_(3)]]

if a,A_(1),A_(2),A_(3)......,A_(2n),b are in A.P.then sum_(i=1)^(2n)A_(i)=(a+b)