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Differentiation

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Question 563

The trajectory of a particle in a high-energy magnetic field is described by the implicit curve C C\,C defined by the equation

x2sin⁡y+y2cos⁡x=K x^2 \sin y + y^2 \cos x = K x2siny+y2cosx=K

where 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​).

a.

Show that K=3π24\displaystyle K = \frac{3\pi^2}{4}K=43π2​.

[2]
bi.

Show that dydx=y2sin⁡x−2xsin⁡yx2cos⁡y+2ycos⁡x\displaystyle \frac{dy}{dx} = \frac{y^2 \sin x - 2x \sin y}{x^2 \cos y + 2y \cos x}dxdy​=x2cosy+2ycosxy2sinx−2xsiny​.

[4]
bii.

Hence, determine the numerical gradient of the trajectory at point PPP.

[2]
biii.

The tangent to the trajectory at P P\,P intersects the xxx-axis at the point QQQ. Find the exact xxx-coordinate of QQQ.

[3]
Markscheme

Differentiation Questions

  1. A Level
  2. /Maths
  3. /Differentiation

717 exam-style questions on Edexcel A Level Maths Differentiation, 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. Each one has a worked solution and a mark scheme showing where the marks go.

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