Let $\alpha, \beta$ be the roots of the quadratic equation $x^2 + px + p^3 = 0$ $(p \neq 0)$. If $(\alpha, \beta)$ is a point on the parabola $y^2 = x$,then the roots of the quadratic equation are:

Find two consecutive positive odd integers whose squares have the sum $290.$

The equation $\log_e x + \log_e(1 + x) = 0$ can be written as

If the graph of $y = ax^3 + bx^2 + cx + d$ is symmetric about the line $x = k$,then:

$Quantity \, 1:$ 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 answer from the given options.
$I.$ $35x^{2} - 53x + 20 = 0$
$II.$ $56y^{2} - 97y + 42 = 0$