Question

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

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.

Show that the relation R in the Set A = {1, 2, 3, 4,5} given by R = {(a, b) : |a – b| is divisible by 2} is an equivalence relation. Write all the equivalence classes of R.

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

Find the particular solution of the differential equation $\left(\tan^{-1}y-x\right)d y=(1+y^{2})\,d x,$ given that when x = 0, y = 0.

Find the angle of intersection of the curves y² = 4ax and x²= 4by.

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.