Question

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.

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

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).

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 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.

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 equation of the tangent line to the curve y =x² -2x + 7 which is (i) parallel to the line 2x-y +9 =0 (ii) perpendicular to the line 5y - 15x = 13.