Show that the function \(f:R[x\in R:1 defined by \(f(x)=\frac { x }{ 1+|x|^{ ' } } ,x\in R\) is one-one and onto function Hence find \(f^{ -1 }(x)\)

If \(A=\left( \begin{matrix} 1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3 \end{matrix} \right) \), prove that A³ - 6A² + 7A + 2I = 0

If  \(A=\left[ \begin{matrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{matrix} \right] \) Prove that , A =\(\left[ \begin{matrix} { 3 }^{ x-1 } & { 3 }^{ x-1 } & { 3 }^{ x-1 } \\ { 3 }^{ x-1 } & { 3 }^{ x-1 } & { 3 }^{ x-1 } \\ { 3 }^{ x-1 } & { 3 }^{ x-1 } & { 3 }^{ x-1 } \end{matrix} \right] \) for every positive integer n.

A school wants to award its students for the value of Honesty, Regularity and Hard work with a total cash award of Rs. 6,000. Three times the award money for hard work added to that given for honesty amounts to Rs. 11,000.The award money given for honesty and hard work together is double the one given for Regularity. Represent the above situation algebraically and find the award money for each value, using matrix method. Apart from these values namely, honesty, regularity and hard work, suggest one more value which the school must include for awards.

If A^-1 = \(\left[ \begin{matrix} 3 & -1 & 1 \\ -15 & 6 & -5 \\ 5 & -2 & 2 \end{matrix} \right] \) and B =\(\left[ \begin{matrix} 1 & 2 & -2 \\ -1 & 3 & 0 \\ 0 & -2 & 1 \end{matrix} \right] \) find (AB)^-1 

Evaluate : \(I=\int _{ 0 }^{ \pi }{ \frac { x\sin { x } }{ 1+\cos ^{ 2 }{ x } } } dx\)

Solve the following differential equation:
\(xcos\left( \frac { y }{ x } \right) (ydx+xdy)=ysin\left( \frac { y }{ x } \right) (xdy-ydx)\)

Find the position vector of foot of perpendicular and the perpendicular distance from the point P with position vector \(2\hat{i}+3\hat{j}+4\hat{k}\) to the plane \(\overrightarrow{r}.(2\hat{i}+\hat{j}+3\hat{k})-26=0.\) Also find the image of P in the plane.

An urn contains 4 balls. Two balls are drawn at random from the urn (without replacement) and are found to be white, What is the probability that all the four balls in the urn are white?