A Cylindrical Gas Container Is Closed At The Top And Open At The Bottom; If The Iron Plate Forming The Cylindrical Sides. The Ratio Of The Height To Diameter Of The Diameter Of The Cylindrical Using Minimum Material For The Same Capacity Is

The chord joining the points where x = p and x = q on the curve y = ax² + bx + c is parallel to the tangent at the point on the curve whose abscissa is 

If the tangent at (1, 1) on y² = x (2 – x)² meets the curve again at P, then P is

The distance between the origin and the normal to the curve

y = e^2x + x² at x = 0 is

If the line ax + by + c = 0 is a normal to the curve xy = 1 then

The point of intersection of the tangents drawn to the curve x²y = 1 – y at the points where it is meet by the curve xy = 1 – y is given by

The slope of the normal at the point with abscissa x = –2 of the graph of the function f (x) = | x² – x | is

The tangent to the graph of the function y = f (x) at the point with abscissax = 1 form an angle of π/6 and at the point x = 2 an angle of π/3 and at the point x = 3 an angle of π/4. The value of 

The equations of the tangents to the curve y = x⁴ from the point (2, 0) not on the curve, are given by

The value of parameter a so that line (3 – a) x + ay + (a²ⁿ – 1) = 0 is normal to the curve xy = 1, may lie in the interval

The critical points of the function f (x) where