Question

Solve the following differential equation : $\mathrm{cosec}\,x\,\mathrm{log}\,y\,\frac{d y}{d x}+x^{2}y^{2}=0\,.$

Find the equation of tangents to the curve y = x³ + 2x – 4 which are perpendicular to the line x + 14y – 3 = 0.

Find the point on the curve y = x³ – 11x + 5 at which the equation of tangent is y = x – 11.

Let A = R – {3}, B = R – {1}. Let f : A → B be defined by f(x) =${\frac{x-2}{x-3}},~~\forall~x\in~A.$ Show that f is bijective. Also, find (i) x, if $f^{-1}(x)$ = 4 (ii) $f^{-1}(7)$

Let f : N → N be defined as $f(n)={\left\{\begin{array}{l l}{\displaystyle{\frac{n+1}{2}},{\mathrm{when~}}n{\mathrm{~is~odd}}}\\ {\displaystyle{\frac{n}{2}},{\mathrm{when~}}n{\mathrm{iseven}}}\end{array}\right.}$ for all n ∈ N. State whether the function f is bijective. Justify your answer.

Find the position vector of a point R, which divides the line joining two points P and Q whose position vectors are $2\stackrel{\rightarrow}{a}+\stackrel{\rightarrow}{b}$ and $\stackrel{\rightarrow}{a}-3\stackrel{\rightarrow}{b}$ respectively, externally in the ratio 1 : 2 Also, show that P is the mid-point, of line segment RQ.

Form the differential equation representing family of curves given by (x – a)² + 2y²= a², where a is an arbitrary constant.