The line $bx + ay = ab$ will touch the curve $y = b \cdot e^{-x/a}$ at the point:

If $y=4x-5$ is a tangent to the curve $y^2=px^3+q$ at $(2,3)$,then $p-q$ is

If the tangent to the curve $2y^3 = ax^2 + x^3$ at the point $(a, a)$ cuts off intercepts $\alpha$ and $\beta$ on the coordinate axes,where $\alpha^2 + \beta^2 = 61$,then the value of $|a|$ is

The line which is parallel to the $x$-axis and intersects the curve $y = \sqrt{x}$ at an angle of $45^\circ$ is given by:

If the slope of the tangent drawn at any point $(x, y)$ to the curve $y=f(x)$ is $3x^2-5$ and $f(1)=2$,then the tangent at $(1, 2)$ to the curve $y=f(x)$ intersects the curve at the point

The curve $y=ax^3+bx^2+cx+5$ touches the $X$-axis at $P(-2,0)$ and cuts the $Y$-axis at a point $Q$,where the gradient is $3$. Then the values of $a, b, c$ are