If the line $y=4x-5$ touches the curve $y^2=ax^3+b$ at the point $(2,3)$,then $7a+2b=$

The angle between the curves $y=\sin 2x$ and $y=\cos 2x$ is

Consider a curve $y=y(x)$ in the first quadrant as shown in the figure. Let the area $A_{1}$ be twice the area $A_{2}$. Then the normal to the curve perpendicular to the line $2x - 12y = 15$ does $NOT$ pass through the point.

The normal to the curve $x = a(1 + \cos \theta), y = a \sin \theta$ at the point $\theta$ always passes through the point:

Show that the tangents to the curve $y=7x^3+11$ at the points where $x=2$ and $x=-2$ are parallel.

If $f: R \rightarrow R$ is a function defined for all $x \in R$ by $f(x)=x^3+f^{\prime}(1) x^2+f^{\prime \prime}(2) x-f^{\prime \prime \prime}(3)$,then the area (in sq. units) of the triangle formed by the $X$-axis,the tangent,and the normal drawn to the curve $y=f(x)$ at $x=0$ is