The value of $\lambda$ such that the sum of the squares of the roots of the quadratic equation $x^2 + (3 - \lambda)x + 2 = \lambda$ has the least value is:

The equation $x^{(3/4)(\log_2 x)^2 + (\log_2 x) - 5/4} = \sqrt{2}$ has

$Quantity \, 1:$ Profit percent for a shopkeeper. The shopkeeper uses a faulty scale while buying cloth from a wholesaler. His scale measures $10 \%$ extra than the actual measurement. While selling,the shopkeeper gives $10 \%$ less and offers a discount of $10 \%$ to the customer.
$Quantity \, 2:$ Profit percent for a man who calculates his profit on the selling price. He marks up the cost of his item by $25 \%$ and then sells the item at a discount of $10 \%$.

Solve the given two equations and select the correct option based on the relationship between $x$ and $y$. The equations are: $8x + 7y = 135$ and $5x + 6y = 99$.

Find the roots of the equation $\sqrt{3y+1} = \sqrt{y-1}$.

Solve the given two equations and select the correct answer from the given options.
$I.$ $17 x^{2}+48 x=9$
$II.$ $13 y^{2}=32 y-12$