If $a, b, c, d$ and $p$ are distinct real numbers such that $(a^2 + b^2 + c^2)p^2 - 2p(ab + bc + cd) + (b^2 + c^2 + d^2) \le 0$,then

If $\alpha$ and $\beta$ are the roots of $6x^2 - 6x + 1 = 0$,then the value of $\frac{1}{2}[a + b\alpha + c\alpha^2 + d\alpha^3] + \frac{1}{2}[a + b\beta + c\beta^2 + d\beta^3]$ is

Solve the given two equations and select the correct answer from the given options.
$I.$ $4x + 7y = 209$
$II.$ $12x - 14y = -38$

Find the value of $a^{2}+b^{2}+c^{2}-2ab+2ac-2bc$,if $a=x+y$,$b=x-y$,and $c=2x-1$.

Solve the given two equations and select the correct answer from the given options:
$I. 6x^{2} + 29x + 35 = 0$
$II. 3y^{2} + 19y + 30 = 0$

$Quantity \, 1:$ 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 $10 \%$.