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.]

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

Show that the normal at any point θ to the curve x = a cos θ + aθ sin θ, y = a sin θ – aθ cos θ is at a constant distance from the origin.

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.

If A = $\left(\begin{array}{l l l }1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3\end{array}\right)$, prove that A³ – 6A² + 7A + 2I = 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 = ${\frac{\pi}{4}}\,.$