CDS Mathematics Polynomials Medium Questions

Which of the following expressions can divide both the polynomials $x^3+2x^2-5x+2$ and $x^3+4x^2+x-6$ exactly?
I. $x-1$
II. $x+1$
III. $x+2$
Select the correct answer using the code given below:

If $\frac{a^2}{b^2+c^2}=\frac{b^2}{c^2+a^2}=\frac{c^2}{a^2+b^2}$, then what is the value of $a^4+b^4+c^4$ equal to ?

The LCM and HCF of two polynomials $p(x)$ and $q(x)$ are $(x+a)(x^3-a^3)$ and $(x^2-ax+a^2)$, respectively. If $p(x)=x^4+a^2{x}^2+a^4$, then what is $q(x)$ equal to ?

If $16\left(x^4+\frac{1}{x^4}\right)-257=0$. What is $\left(x^3+\frac{1}{x^3}\right)$ equal to ?

Let $p(x)$ be a polynomial. When $p(x)$ is divided by $(x-1)$, it leaves 2 as the remainder. When $p(x)$ is divided by $(x-2)$, it leaves 1 as the remainder. What is the remainder when $p(x)$ is divided by $(x-1)(x-2)$?

If $(x+1)(x+p)(x^2+p^2)=x^4-1$, then what is the value of $p$?

If $\frac{1}{a}+\frac{1}{b}=\frac{5}{6}$ and $\frac{1}{a^2}+\frac{1}{b^2}=\frac{13}{36}$, then what is $\frac{1}{a^3}+\frac{1}{b^3}$ equal to?

What is the remainder when $x^6$ is divided by $x^2+1$?

(x+2) is a factor of which one of the following?

What is the HCF of $x^3+y^3+3xy-1$ and $(x+y{)}^4-1$?

What is $\frac{(a+b{)}^2}{(c-a)(c+a+b)}+\frac{(a+b)c}{c^2+bc-a^2-ab}-\frac{(a+2b+c)}{2(c-a)}$, $a\ne b,b\ne c,c\ne a$ equal to?

If $x-\frac{1}{x}=2$, $x>0$; then what is $x^2-\frac{1}{x^2}$ equal to?

If $(a-b{)}^2+(b-c{)}^2+(c-a{)}^2=6$ and $a^2+b^2+c^2=29$, then what is $(a+b+c)$ equal to?

Which of the following is/are the factor(s) of $(3x+y{)}^2+(3x+y)(x+5y)-20(x+5y{)}^2$?

(1) $(4x+13y)$

(2) $(x+19y)$

Select the correct answer using the code given below.

What is $\frac{\frac{x}{x-y}+\frac{y}{y-z}+\frac{z}{z-x}}{\frac{x+y}{x-y}+\frac{y+z}{y-z}+\frac{z+x}{z-x}+3}$ equal to?

A Question is given followed by two Statements I and II. Consider the Question and the Statements.

Question :

What is the remainder when $x^{2n}-y^{2n}+1$ is divided by $x^n+y^n$, where $n$ is a natural number?

(1) $n$ is odd.

(2) $n$ is even.

Which one of the following is correct in respect of the above Question and the Statements?