Prove that `e^x-2=0` has a solution between 0 and 1.

So that x^5+x+1=0 for some value of x between -1 and 0.

If a system of equations lamda x +3y=0 x+(lamda-2)y=0 has infinitely many solutions, then find the values of lamda .

Given the general solution as y = (x^2 + c) e^(-x) of a differential equation, what is the particular solution if y = 0 when x = 1.

Prove that the line 4x-2y-1=0 touches the parabola whose focus it at (0,0) and directrix is the line y=2x-1.

Prove that In(1+x) lt x for x gt 0.

If the pair of lines represented by x^2 - 2pxy - y^2= 0 and x^2 - 2qxy - y^2 = 0 be such that each pair bisects the angle between the other pair, then prove that pq = -1 .