Question

Let N denote the set of all natural numbers and R be the relation on N × N defined by (a, b) R (c, d) if ad(b + c) = bc(a + d). Show that R is an equivalence relation.

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.]

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.

Solve the differential equation : ${\frac{d y}{d x}}-3y\cot x=\sin2x\operatorname{given}y=2,{\mathrm{~when~}}x={\frac{\pi}{2}}.$

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.

Show that the function f:R → R defined by f(x) = ${\frac{x}{{\boldsymbol{x}}^{2}+1}},\ \ \forall\ x\in{\boldsymbol{R}}$ is neither one-one nor onto. Also, if g:R → R is defined as g(x) = 2x – 1, find fog(x).

In a hockey match, both teams A and B scored same number of goals up to the end of the game, so to decide the winner, the referee asked both the captains to throw a die alternately and decided that the team, whose captain gets a six first, will be declared the winner. If the captain of team A was asked to start, find their respective probabilities of winning the match and state whether the decision of the referee was fair or not.