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 Edexcel A Level Maths Differentiation with exam-style questions for A Level Maths. 311 questions covering 9.1 Differentiating sin x and cos x, 9.2 Differentiating exponentials and logarithms, 9.3 The Chain Rule, 9.4 The Product Rule, 9.5 The Quotient Rule, 9.6 Differentiating Trigonometric Functions, 9.7 Parametric Differentiation, 9.8 Implicit Differentiation, 9.9 Using Second Derivatives, and 9.10 Rates of Change, matched to the Edexcel A Level Maths (9MA0) 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.
