Question

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.

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.

LetA = {x ∈ Z:0≤ x ≤12}. Show that R={(a,b):a,b ∈ A, |a-b| is divisible by 4} is an equivalence relation. Find the set of all elements related to 1. Also write the equivalence class [2].

Find the equation of tangent and normal to the curve x = 1 – cos θ, y = θ – sin θ at θ = ${\frac{\pi}{4}}\,.$

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.

Find the equations of tangents to the curve 3x² – y² = 8, which passes through the point $\left({\frac{4}{3}},\,0\right).$