Matrix A𝐴 is such that A2=2A−I,𝐴2=2𝐴-𝐼, where I𝐼 is the indentity matrix, then for n≥2,An is equal to

f nth 𝑛th  root of unity be 1,a1,a2,…an−1,1,𝑎1,𝑎2,…𝑎𝑛-1, then ∑n−1r=112+ar∑𝑟=1𝑛-112+𝑎𝑟 is equal toAn⋅2n−12n−1−1

If for a square matrix A, A2 3A + I = O and A 1 = xA + yI, then thevalue of x + y is :

Two waves of equal amplitude A𝐴, and equal frequency travel in the same direction in a medium. The amplitude of the resultant wave is

If B = [bij] is a scalar matrix of n  n order such that bii = a for all i = 1, 2, …, n, then the value of determinant Bis equal to

Let f(x)=∣∣2x2+5∣∣x|−3|,x∈R. If m and n denote the number of points where f is not continuous and not differentiable respectively, then m+n is equal to :