Question

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 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.

Show that the function f : R {x ∈ R – 1 < x < 1} defined by f(x) =${\frac{x}{1+\left|\,x\right|}},$ x ∈ R is one-one and onto function. Hence find f⁻¹(x).

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

Consider $f\colon R^{+}\to[-9,\infty)$ given by f(x) = 5x² + 6x – 9.Prove that f is invertible with f⁻¹(y) = $\left({\frac{{\sqrt{54+5y}}-3}{5}}\right)$ [where, R⁺ is the set of all nonnegative real numbers.]

Solve the differential equation $x\;\frac{d y}{d x}\;+y=$ x cos x + sin x, given that y = 1 when $x={\frac{\pi}{2}}$

Show that the relation R in the Set A = {1, 2, 3, 4,5} given by R = {(a, b) : |a – b| is divisible by 2} is an equivalence relation. Write all the equivalence classes of R.