State the following statement is True or False If both the roots of the equation $x^2-6ax+2-2a+9a^2=0$ exceed $3$, then $a<\dfrac {11}9$

Let the roots of the equation be alpha and beta, which can be written as alpha equals 3 plus m and beta equals 3 plus n, where m and n are both positive. The sum of roots gives 6a equals 6 plus m plus n, which yields a equals 1 plus the quantity (m plus n) divided by 6. The product of roots gives 2 minus 2a plus 9a squared equals 9 plus 3m plus 3n plus mn, and substituting (m plus n) with 6a minus 6 simplifies this to m times n equals 2a squared plus 4a minus 5. For m and n to be strictly positive, both their sum (6a minus 6) and product (2a squared plus 4a minus 5) must be greater than zero, leading to the condition a is greater than 5 divided by 4. Therefore, the statement that a is less than 11 divided by 9 is false.

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