The number of points in the rectangle which lie on the curve and at which the tangent to the curve is parallel to the ‐axis is

Tangent drawn at any point to the curve always makes

If the normal to the curve y=f(x) at the point (3, 4) makes an angle with the positive ‐axis then

The point(s) on the curve where the tangent is vertical is (are)

The radius and height of a cylinder are equal to the radius of sphere The ratio of the rates of change of the volume of the sphere and cylinder is

Here we used the concept of differentiation and the formulas of cylinder and sphere to find the ratio. Given that the ratio of the rates of growth of the sphere's and cylinder's volumes is 3:4, the ratio of the rates of growth of the cylinder's volume to that of the sphere's volume is 4:3.

The equation of normal to y=x log x is parallel to 2x-2y+3=0 then the equation of the normal is

Three normals are drawn to the parabola from the point (c, 0) These normals are real and distinct when

The points of contact of the vertical tangents to are

The curve touches the ‐axis at P(-2,0) and cuts the y ‐axis at a point where its gradient is 3 Then a+2b+c=-

If is to be square root of a determinant of the two rowed unit matrix, then should satisfy the relation

Matrix Ais given by then the determinant of is

If then

If then the number of positive divisors of the number p is

The number of positive integral solutions of the equation is

If be real matrices (not necessasrily square) such that forthematrix consider the statements

If , then
Statement because
Statement :A.M for