GATE 2023 Engg. Mathematics Quiz 3

A_3x3 has an eigenvalue –1. The matrix A + I reduced to corresponding to eigenvalue –1, all the eigenvectors of A are non-zero, then the eigenvectors of the form are __________.

The function satisfies the Lagrange’s Mean Value theorem in the interval (1,4) at a value of x=_________(rounded to two decimal places)

Consider x,y,z are positive real numbers such that x + 2y + 3z = 1.
Let is the maximum value of , then the value of

The following differential equation is an example of

An ordinary differential equation is given below

The general solution of the differential equation is

Find the value of

f(z) = u (x,y) + i v(x,y) is an analytic function of complex variable z = x + i y, where i = . If u(x,y) = 3xy, then v(x,y) may be expressed as