If The Tangent At (1, 1) On y2 = x (2 – x)2 meets The Curve Again At P, Then P is

If at each point of the curve y = x³ – ax² + x + 1 the tangents is inclined at an acute angle with the positive direction of the x-axis, a lies in the interval.

Two variable curves C₁ : y² = 4a (x – b₁) and C₂ : x² = 4a (y – b₂) where ‘a’ is a given positive real no. and b₁ and b₂ are variable such that C₁ and C₂ are tangents to each other at point of contact then locus of point of contact is:

f : R âŸ¶ R be a differentiable function ∀ x Ïµ R. If tangent drawn to the curve at any point x Ïµ (a, b) always lie below the curve then

A lamp of negliligible height is placed on the ground ‘l₁’ m away from a wall. A man ‘l₂’ m tall is walking at a speed of  m/sec from the lamp to the nearest point on the well. When he is midway between the lamp and the wall, the rate of change in the length of his shadow on the wall is

Consider the parabola y² = 4x. A = (4, –4) and B = (9, 6) be two fixed points on the parabola. Let ‘C’ be moving point on the parabola between A and B such that the area of triangle ABC is maximum, then coordinate of ‘C’ is

If the rate of change of volume of a sphere is the same as rate of change of its radius, then radius, is equal to

A spherical balloon is pumped at the constant rate of 3 m³/min. The rate of increase of its surface area as certain instant is found to be 5 m²/min. At this instant it’s radius is equal to

The third derivative of a function f’’(x) vanishes for all x. If f (0) = 1, f’ (1) = 2 and f’’ = –1, then f (x) is equal to 

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 

The distance between the origin and the normal to the curve

y = e^2x + x² at x = 0 is