Identify the sum and product of the roots of the given quadratic equation.

a)

Step 1: Identify the sum and product of the roots of the given quadratic equation.
The given quadratic equation is $2x^2 - 5x + k = 0$.
The roots are $x_1 = 1+\alpha$ and $x_2 = 1+\beta$.
For a quadratic equation $ax^2+bx+c=0$, the sum of roots is $-\frac{b}{a}$ and the product of roots is $\frac{c}{a}$.
Sum of roots:

$$(1+\alpha) + (1+\beta) = -\frac{-5}{2}$$ $$2 + \alpha + \beta = \frac{5}{2}$$ $$\alpha + \beta = \frac{5}{2} - 2$$ $$\alpha + \beta = \frac{1}{2}$$

Product of roots:

$$(1+\alpha)(1+\beta) = \frac{k}{2}$$ $$1 + \alpha + \beta + \alpha\beta = \frac{k}{2}$$

Substitute $\alpha + \beta = \frac{1}{2}$:

$$1 + \frac{1}{2} + \alpha\beta = \frac{k}{2}$$ $$\frac{3}{2} + \alpha\beta = \frac{k}{2}$$

Multiply by 2:

$$3 + 2\alpha\beta = k$$ $$2\alpha\beta = k-3$$ $$\alpha\beta = \frac{k-3}{2}$$

Step 2: Find the sum and product of the roots of the new quadratic equation.
The roots of the new quadratic equation are $y_1 = 1+\alpha^2$ and $y_2 = 1+\beta^2$.
Let $S'$ be the sum of the new roots and $P'$ be the product of the new roots.
Sum of new roots $S'$:

$$S' = (1+\alpha^2) + (1+\beta^2)$$ $$S' = 2 + \alpha^2 + \beta^2$$

We know that $\alpha^2 + \beta^2 = (\alpha+\beta)^2 - 2\alpha\beta$.
Substitute the values of $\alpha+\beta$ and $\alpha\beta$:

$$\alpha^2 + \beta^2 = \left(\frac{1}{2}\right)^2 - 2\left(\frac{k-3}{2}\right)$$ $$\alpha^2 + \beta^2 = \frac{1}{4} - (k-3)$$ $$\alpha^2 + \beta^2 = \frac{1}{4} - k + 3$$ $$\alpha^2 + \beta^2 = \frac{1+12}{4} - k$$ $$\alpha^2 + \beta^2 = \frac{13}{4} - k$$

Now substitute this back into the expression for $S'$:

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