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Options

A. `((1)/(sqrt(2)),(7)/(sqrt(2)))`
B. `(-sqrt(2),7sqrt(2))`
C. `(-(1)/(sqrt(2)),(7)/(sqrt(2)))`
D. `(sqrt(2),7sqrt(2))`

Explanation

Let `B,C,D` be the position of the point `A(4,1)` after the three operations `I,II` and `III` respectively. Then, `B` is `(1,4), C(1+2,4)` i.e. `(3,4)` . The point `D` is obtained from `C` by rotationg the coordinate axes through an angle `pi//4` in anti-clockwise direction.
Therefore, the coordinates of `D` are given by
`X=3cos.(pi)/(4)-4sin.(pi)/(4)=-(1)/(sqrt(2))`
and `Y=3sin.(pi)/(4)+4cos.(pi)/(4)=(7)/(sqrt(2))`
`:.` Coordinates of `D` are `(-(1)/(sqrt(2)),(7)/(sqrt(2)))`.