Question

Let A = {1, 2, 3, . ...., 9} and R be the relation in A × A defined by (a, b) R (c, d) if a + d =b + c for (a, b), (c, d) in A × A. Prove that R is an equivalence relation. Also obtain the equivalence class [(2, 5)].

Find the equations of the normal to the curve x² = 4y which passes through the point (1, 2).

If y(x) is a solution of the differential equation $\left(\frac{\dot{2}+\sin x}{1+y}\right)\frac{d y}{d x}\;=-\cos x\;\mathrm{and}\;y\;(0)=1,$ then find the value of $y\left({\frac{\pi}{2}}\right).$

Find the particular solution of the following differential equation given that at x = 2, y = 1 $x\,{\frac{d y}{d x}}\,+\,2y=x^{2},\,(x\,\neq\,0).$

Let R be a relation defined on the set of natural numbers N as follow : R = {(x, y) : $x\in N,y\in N$ and 2x + y = 24} Find the domain and range of the relation R. Also,find if R is an equivalence relation or not.

If f(x) =$\frac{4x+3}{6x-4}\,,\;\;x\;\;\neq\;\;\frac{2}{3}\,,$ then Show that fof(x) = x for all $x\neq{\frac{2}{3}}\,.$ What is the inverse of f?

Find a vector of magnitude 5 units and parallel to the resultant of $\stackrel{\rightarrow}{a}\$ = $2\stackrel{\wedge}{i}+3\stackrel{\wedge}{j} -\stackrel{\wedge}{k}$ and $\stackrel{\rightarrow}{b}\$ = $\stackrel{\wedge}{i}-2\stackrel{\wedge}{j} +\stackrel{\wedge}{k}$.