Question

Find the value of p for which the curves x² = 9p(9 – y) and x² = p(y + 1) cut each other at right angles.

Find the particular solution of the differential equation $\left(\tan^{-1}y-x\right)d y=(1+y^{2})\,d x,$ given that when x = 0, y = 0.

Find the equation of the tangent to the curve $\begin{array}{r c l}{y}&{=}&{{\sqrt{3x-2}}}\end{array}$ which is parallel to the line 4x – 2y + 5 = 0.

Find the equation of tangent and normal to the curve x = 1 – cos θ, y = θ – sin θ at θ = ${\frac{\pi}{4}}\,.$

Find the equation of the normal at a point on the curve x² = 4y which passes through the point (1, 2). Also find the equation of the corresponding tangent.

Consider $f:R-\left\{-{\frac{4}{3}}\right\}\to R-\left\{{\frac{4}{3}}\right\}$ given by f(x) = ${\frac{4x+3}{3x+4}}\,.$ Show that f is bijective. Find the inverse of f and hence find f⁻¹(0) and x such that f⁻¹(x) = 2.

Find the particular solution of the differential equation $(1+y^{2})+(x-e^{\mathrm{tan}^{-1}y})\ {\frac{d y}{d x}}\,=0,$ given that y = 0 when x = 1.