If zero is inserted between the two digits of a two-digit number whose unit's place is $y$ and ten's place is $x$,then the new number is $\ldots \ldots \ldots \ldots$

In a prize-winning competition,$x+2$ students are qualified for the prize. If a total of $3x^3 + 10x^2 + 7x - 2$ prizes are distributed among them equally,find the number of prizes left undistributed,where $x \in N$.

Find the division of the following by synthetic division method: $p(x) = x^{4} + 1$ by $x + 1$.

On dividing $x^{3}-3 x^{2}+x+2$ by a polynomial $g(x)$,the quotient and remainder were $x-2$ and $-2 x+4$,respectively. Find $g(x)$.

The total cost of $(x+1)$ cameras is ₹ $(x^{3}+3x^{2}+5x+3)$. Find the cost of one camera.

Answer the following and justify: Can $x^{2}-1$ be the quotient on division of $x^{6}+2 x^{3}+x-1$ by a polynomial in $x$ of degree $5$?