The trajectory of a particle in a high-energy magnetic field is described by the implicit curve C C\,C defined by the equation
x2siny+y2cosx=K x^2 \sin y + y^2 \cos x = K x2siny+y2cosx=Kwhere K K\,K is a constant. The particle is observed to pass through the point P(π,π2)\displaystyle P\left(\pi, \frac{\pi}{2}\right)P(π,2π).
Show that K=3π24\displaystyle K = \frac{3\pi^2}{4}K=43π2.
Show that dydx=y2sinx−2xsinyx2cosy+2ycosx\displaystyle \frac{dy}{dx} = \frac{y^2 \sin x - 2x \sin y}{x^2 \cos y + 2y \cos x}dxdy=x2cosy+2ycosxy2sinx−2xsiny.
Hence, determine the numerical gradient of the trajectory at point PPP.
The tangent to the trajectory at P P\,P intersects the xxx-axis at the point QQQ. Find the exact xxx-coordinate of QQQ.
Practise AQA A Level Maths 1.10 G: Differentiation with exam-style questions for A Level Maths. 321 questions covering 1.10.1 The derivative and second derivative, 1.10.2 Differentiating standard functions, 1.10.3 Applications of differentiation, 1.10.4 Product, quotient and chain rules (A-level only), 1.10.5 Implicit and parametric differentiation (A-level only), and 1.10.6 Constructing differential equations (A-level only), matched to the AQA A Level Maths (7357) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.
