Two points `A(x_1, y_1)` and `B(x_2,y_2)` are chosen on the graph of `f(x) = ln x` with `0 lt x_1 lt x_2.` The points C and D trisect line segment AB with `AC lt CB.` Through C a horizontal line is drawn to cut the curve at `E(x_3, y_3).` If `x_1 = 1` and `x_2 = 1000` then the value of `x_3` equals

If the lines y=3x+1 and 2y=x+3 are equally inclined to the line y= mx +4 (1/2 lt m lt 3) then the values of m are

A(x_1, y_1) and B(x_2, y_2) are any two points on the parabola y=cx^2 + bx + a . If P(x_3, y_3) be the point on the arc AB where the tangent is parallel to the chord AB , then (A) x_2 is the A.M. between x_1 and x_3 (B) x_2 is the G.M. between x_1 and x_3 (C) x_2 is the H.M. between x_1 and x_3 (D) none of these

If the tangent to the curve , y =f (x)= x log_(e)x, ( x gt 0) at a point (c, f(c)) is parallel to the line - segment joining the point (1,0) and (e,e) then c is equal to :

The area of the region described by A = {(x,y) : x^2 + y^2 lt= 1and y^2 lt= 1- x} is

The area of the region described by A = {(x,y) : x^2 + y^2 lt= 1and y^2 lt= 1- x} is

Two point B(x_(1), y_(1)) and C(x_(2), y_(2)) are such that x_(1), x_(2) are the roots of the equation x^(2)+4x+3=0 and y_(1),y_(2) are the roots of the equation y^(2)-y-6=0 . Also x_(1)lt x_(2) and y_(1)gt y_(2) . Coordinates of a third point A are (3, -5). If t is the length of the bisector of the angle BAC, then l^(2) is equal to.

If y = f(x) is a curve and if there exists two points A(x_1, f(x_1)) and B (x_2,f(x_2)) on it such that f'(x_1) = -1/( f(x_2)) , then the tangent at x_1 is normal at x_2 for that curve. Number of such lines on the curve y=sinx is

If A(x_1, y_1), B (x_2, y_2) and C (x_3, y_3) are the vertices of a DeltaABC and (x, y) be a point on the internal bisector of angle A , then prove that : b|(x,y,1),(x_1, y_1, 1), (x_2, y_2, 1)|+c|(x, y,1), (x_1, y_1, 1), (x_3, y_3, 1)|=0 where AC = b and AB=c .