If a root of the given equation $a(b - c)x^2 + b(c - a)x + c(a - b) = 0$ is $1$,then the other root will be

Quantity $I$: Overall profit percentage if the cost prices of two shirts are equal. One shirt is sold for $20 \%$ profit and the other is sold for $10 \%$ loss.
Quantity $II$: Profit percentage made in selling each meter if the profit made in selling $20 \, m$ of a cloth equals the cost price of $5 \, m$ of that cloth.

Form a quadratic equation whose one root is $3-\sqrt{5}$ and the sum of roots is $6$.

Sachin and Rahul attempted to solve a quadratic equation. Sachin made a mistake in writing down the constant term and ended up with roots $(4, 3).$ Rahul made a mistake in writing down the coefficient of $x$ and got roots $(3, 2).$ The correct roots of the equation are:

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3 + 2x - 5 = 0$ and the equation $x^3 + bx^2 + cx + d = 0$ has roots $2\alpha + 1, 2\beta + 1, 2\gamma + 1$,then the value of $|b + c + d|$ is (where $b, c, d$ are constants):

If $x=\frac{\sqrt{3}+1}{\sqrt{3}-1}$ and $y=\frac{\sqrt{3}-1}{\sqrt{3}+1}$,find the value of $x^{2}+y^{2}$.