Question

Let A = R – {2}, B = R – {1}. If f : A → B is a function defined by $f(x)\,=\,\left({\frac{x-1}{x-2}}\right)\!,$ show that f is one-one and onto. Hence find $f^{-1}.$

Using differentials, find the approximate value of ${\sqrt{49\cdot5}}\ .$

For the following matrices A and B, verify that [AB]’ = B’A’ where, A = $\left(\begin{array}{l }1 \\ -4 \\ 3\end{array}\right)$ , B = $\left(\begin{array}{l l l }-1 & 2 & 1\end{array}\right)$

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

Show that f : N → N, given by f(x) = |``x+1, if x is odd x- 1, if x is even is both one-one and onto.``|

A and B throw a pair of dice alternately. A wins the game if he gets a total of 9 and B wins if he gets a total of 7. If A starts the game, find the probability of winning the game by B.

Evaluate : ($\large\textstyle\int(3x-2){\sqrt{x^{2}+x+1}}\,d x~.$