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.
825 exam-style questions on OCR (MEI) A Level Maths 1.9 Calculus, covering 1.9.1 Gradient of a curve at a point, 1.9.2 Gradient as limit of chord gradient, 1.9.3 Derivative as gradient of tangent, 1.9.4 Sketch the gradient function, 1.9.5 Differentiate y = kx^n, 1.9.6 Second derivative as rate of change of gradient, 1.9.7 Stationary points: maxima and minima, 1.9.8 Increasing and decreasing functions, 1.9.9 Tangent and normal at a point, 1.9.10 Differentiate e^kx, a^kx and ln x (A-level only), 1.9.11 Differentiate trigonometric functions (A-level only), 1.9.12 Product rule (A-level only), 1.9.13 Quotient rule (A-level only), 1.9.14 Chain rule (A-level only), 1.9.15 Rates of change with the chain rule (A-level only), 1.9.16 Implicit differentiation (A-level only), 1.9.17 Concavity and the second derivative (A-level only), 1.9.18 Points of inflection (A-level only), 1.9.19 Integration as reverse of differentiation, 1.9.20 Integrate kx^n, 1.9.21 Constant of integration, 1.9.22 Indefinite and definite integrals, 1.9.23 Area between a graph and the x-axis, 1.9.24 Integrate e^kx, 1/x, sin kx, cos kx (A-level only), 1.9.25 Integration as the limit of a sum (A-level only), 1.9.26 Area between two curves (A-level only), 1.9.27 Integration by substitution (reverse chain rule) (A-level only), 1.9.28 Integration by substitution (other cases) (A-level only), 1.9.29 Integration by parts (A-level only), 1.9.30 Integration using partial fractions (A-level only), 1.9.31 Formulate first order differential equations (A-level only), 1.9.32 Solve first order differential equations (A-level only), and 1.9.33 Interpret solutions of differential equations (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.