The trajectory of a specialized particle in a particle accelerator is modeled by the curve C C\,C with equation
x=ye−4y,y∈R x = y e^{-4y}, \quad y \in \mathbb{R} x=ye−4y,y∈RShow that, for points on C C\,C with y≠0,14\displaystyle y\ne 0,\frac14y=0,41,
dydx=yx(1−4y) \frac{\mathrm{d} y}{\mathrm{d} x} = \frac{y}{x(1 - 4y)} dxdy=x(1−4y)yand state separately the value of dydx\displaystyle \frac{\mathrm{d}y}{\mathrm{d}x}dxdy at the origin (0,0)(0,0)(0,0).
Given that a vertical detector strip at x=kx = kx=k, where k k\,k is a constant, detects the particle at exactly two distinct locations on the curve CCC,
find the range of possible values for kkk.
425 exam-style questions on OCR A Level Maths 1.7 Differentiation, covering 1.7.1 Derivative as gradient of the tangent, 1.7.2 Gradient of the tangent at a point, 1.7.3 Sketching the gradient function, 1.7.4 Second derivatives, 1.7.5 Second derivative as rate of change of gradient, 1.7.6 Convex, concave and points of inflection (A-level only), 1.7.7 Differentiation from first principles for powers of x, 1.7.8 Differentiation from first principles for sin x and cos x (A-level only), 1.7.9 Differentiating x^n, 1.7.10 Differentiating e^(kx) and a^(kx) (A-level only), 1.7.11 Differentiating trigonometric functions (A-level only), 1.7.12 Derivative of ln x (A-level only), 1.7.13 Tangents and normals, 1.7.14 Stationary points, 1.7.15 Increasing and decreasing functions, 1.7.16 Points of inflection (A-level only), 1.7.17 Product and quotient rules (A-level only), 1.7.18 Chain rule (A-level only), 1.7.19 Parametric and implicit differentiation (A-level only), 1.7.20 Constructing differential equations (A-level only), and 1.7 Differentiation. Each one has a worked solution and a mark scheme showing where the marks go.