A shipment service provider uses three types of containers for shipping materials. The height and width of the three containers are the same. The containers' width is $0.5 \mathrm{m}$ more than their height, and the volume of the smallest container is $500 {\mathrm{m}}^3$.

Which of the following statements is true?

Let $f\left(x\right)$ be a cubic monic polynomial. A monic polynomial is a polynomial in one variable in which the coefficient of the term of highest degree is equal to $1$.

$p\left(x\right), q\left(x\right), r\left(x\right)$ are quadratic polynomials which satisfy the following system of equations below:

$f\left(x\right)=p\left(x\right)\left(x-a\right)+1$

$f\left(x\right)=q\left(x\right)\left(x-b\right)+1$

$f\left(x\right)=r\left(x\right)\left(x-c\right)+1$

Find the value of $f\left(x\right)$ for $x=m$.

Two students in class of IX named Manasa and Uma were assigned a polynomial by their maths teacher. The polynomial was $p\left(x\right)=x^2-7x+12$.

They were asked to express this polynomial as product of factors. Both applied factorisation by splitting the middle term and got different answer.

Manasa's answer: $\left(x-3\right)\left(x-4\right)$

Uma's answer: $\left(x+3\right)\left(x-4\right)$.

Find out whose answer is correct and show factorisation.

Miss Mary gave a polynomial $f\left(x\right)=x^3+2x^2+3x+2$ to the class and said:

"When $f\left(x\right)$ is divided by another polynomial $g\left(x\right)$, it leaves quotient $q\left(x\right)$ and remainder $r\left(x\right)$, where $q\left(x\right)=r\left(x\right)\ne 1$. All the polynomials $g\left(x\right), q\left(x\right), r\left(x\right)$ have integral coefficients."

Using the above data, find the value of $g\left(5\right)$.

Usha learned about 'Remainder Theorem' in her Maths class today. Back home, she was practicing finding the remainders using the remainder theorem while dividing polynomials. She came across a polynomial $Q\left(x\right)=x^3+m{x}^2-3x+7$ which when divided by $\left(x+3\right)$, leaves the remainder $-2$.

Now, she is curious to find the remainder when $Q\left(x\right)$ is divided by $\left(x-4\right)$. What will be the remainder?

Rishav donated a certain amount of money to an NGO. His friend Pankaj wanted to know the amount donated by him. But, Rishav did not disclose the amount he donated and instead gave him a hint that if $\left(x+\frac{1}{x}\right)=₹9$, then the amount donated by him is $₹\left(x^3+\frac{1}{x^3}\right)$. Find the amount donated by Rishav to the NGO in $₹$