Class 12

Math

Calculus

Application of Derivatives

Show that the altitude of a right circular cone of maximum volume that can be inscribed in a sphere of radius r is 4r/3..

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Prove that there exist exactly two non-similar isosceles triangles $ABC$ such that $tanA+tanB+tanC=100.$

In the curve $y=ce_{ax}$ , the sub-tangent is constant sub-normal varies as the square of the ordinate tangent at $(x_{1},y_{1})$ on the curve intersects the x-axis at a distance of $(x_{1}−a)$ from the origin equation of the normal at the point where the curve cuts $y−aξs$ is $cy+ax=c_{2}$

If $a>b>0,$ with the aid of Lagranges mean value theorem, prove that $nb_{n−1}(a−b)1.$ $nb_{n−1}(a−b)>a_{n}−b_{n}>na_{n−1}(a−b),if0<n<1.$

Let $y=f(x)$ be a polynomial of odd degree $(≥3)$ with real coefficients and (a, b) be any point. Statement 1: There always exists a line passing through $(a,b)$ and touching the curve $y=f(x)$ at some point. Statement 2: A polynomial of odd degree with real coefficients has at least one real root.

Find the angle between the curves $x_{2}−3y_{2} =a_{2}andC_{2}:xy_{3}=c$

If in a triangle $ABC,$ the side $c$ and the angle $C$ remain constant, while the remaining elements are changed slightly, show that $cosAda +cosBdb =0.$

Separate the intervals of monotonocity for the function $f(x)=−2x_{3}−9x_{2}−12x+1$

Find the angle of intersection of the curves $xy=a_{2}andx_{2}+y_{2}=2a_{2}$