3.5.4 Differentiation (A-level only)

A specialized surveillance drone follows a path $C$ in a 2D plane defined by the parametric equations

3t^2 + 1, \quad y = 2t^3 - 15t + k

where $k$ is a constant and $\ge 0$ represents time.

Find an expression for $dydx\frac{dy}{dx}$ in terms of $t$ .

[2]

The line $l$ is the normal to the path at point $A$ where $t = 1$ .

Given that $l$ is also a tangent to the path at point $B$ where $t = T$ ,

show that $T$ is a solution of the equation

6T^2 - 4T - 15 = 0

[4]

Hence find the $x$ -coordinate of point $B$ , justifying your answer.

[3]

Given that the $y$ -intercept of the line $l$ is $13\frac{1}{3}$ ,

find the value of $k$ .

[3]

3.5.4 Differentiation (A-level only) Questions