Question

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

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].

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).

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.

Let N denote the set of all natural numbers and R be the relation on N × N defined by (a, b) R (c, d) if ad(b + c) = bc(a + d). Show that R is an equivalence relation.

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

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.