Question

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.]

Consider the experiment of tossing a coin. If the coin shows head, toss is done again, but if it shows tail, then throw a die. Find the conditional probability of the events that ‘the die shows a number greater than 4’, given that ‘there is atleast one tail’.

Find the particular solution of the differential equation (x – sin y) dy + (tan y) dx = 0, given that y = 0 when x = 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.

Solve the following differential equation. ${\sqrt{1+x^{2}+y^{2}+x^{2}y^{2}}}+x y{\frac{d y}{d x}}\,=0$

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.

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