A precision-engineered micro-shuttle follows a path CCC in a magnetic field. The position of the shuttle at time ttt, where ttt is a parameter in radians, is given by the parametric equations
x=4cos2t,y=8sin3t,−π2<t<π2 x = 4 \cos 2t, \quad y = 8 \sin^3 t, \quad -\frac{\pi}{2} < t < \frac{\pi}{2} x=4cos2t,y=8sin3t,−2π<t<2πThe shuttle passes through point PPP when t=π6t = \frac{\pi}{6}t=6π.
The line lll represents the tangent to the shuttle's path at point PPP.
Use parametric differentiation to show that (i) dydx=ksint\frac{\mathrm{d}y}{\mathrm{d}x} = k \sin tdxdy=ksint where kkk is a constant to be found. (ii) an equation for the tangent line lll is 3x+4y−10=03x + 4y - 10 = 03x+4y−10=0.
The path CCC is intersected again by the line lll at the point QQQ.
Using algebra and showing detailed reasoning, find the exact coordinates of QQQ.
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.