Statement 1: if `A={2,4,6,},B{1,5,3}w h e r eAa n dB` are the events of numbers occurring on a dice, then `P(A)+P(B)=1.` Statement 2: `A_1, A_2, A_3, A_n` are all mutually exclusive events, then `P(A_1)+P(A_2)++P(A_n)=1.`

If A_1, A_2, …, A_n are n independent events, such that P(A_i)=(1)/(i+1), i=1, 2,…, n , then the probability that none of A_1, A_2, …, A_n occur, is

Statement 1 : veca = 3 veci + p vecj +3veck and vecb = 2veci + 3vecj + qveck are parallel vectors if p = 9//2 and q =2 . Statement 2 : If veca= a_1 veci + a_2 vecj + a_3 veck and vecb = b_1 veci + b_2 vecj + b_3veck are parallel, then (a_1)/(b_1) = (a_2)/(b_2)= (a_3)/(b_3) .

If 1, a_1,a_2,a_3 ,…, a_(n-1) are the nth roots of unity then prove that : 1+a_1+a_2+…+a_(n-1) =0.

If a_1,a_2 …. a_n are positive real numbers whose product is a fixed real number c, then the minimum value of 1+a_1 +a_2 +….. + a_(n-1) + a_n is :

If 1, a_1,a_2,a_3 ,…, a_(n-1) are the nth roots of unity then prove that : (1-a_1)(1-a_2)(1-a_3)...(1-a_(n-1)) =n.

A_1 A_2 A_3............A_n is a polygon whose all sides are not equal, and angles too are not all equal. theta_1 is theangle between bar(A_i A_j +1) and bar(A_(i+1) A_(i +2)),1 leq i leq n-2.theta_(n-1) is the angle between bar(A_(n-1) A_n) and bar(A_n A_1), while theta_n, is the angle between bar(A_n A_1) and bar(A_1 A_2) then cos(sum_(i=1)^n theta_i) is equal to-