Find the equation of the tangent to the curve $y = \sqrt{3x - 2}$ which is parallel to the line $4x - 2y + 5 = 0$.

$y=x^2$ is the given curve. Imagine that this curve is dragged along the positive $X$-axis to a distance of '$a$' units. If the acute angle between the curves at two positions is $\theta$, then

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

The slope of the normal to the curve $y=2x^{2}+3\sin x$ at $x=0$ is

Find the equation of the normal to the curve $y = \frac{x-7}{(x-2)(x-3)}$ at the point where it cuts the $X$-axis.

Equation of the normal to the curve $y=x^2+x$ at the point $(1,2)$ is