Embibe Experts Solutions for Chapter: Algebra, Exercise 3: EXERCISE - 1.3

If $x^2+x+1=0$, find the absolute value of $x^{1999}+x^{2000}$.

If $a+b+c=0$, find $\frac{b^2+c^2+a^2}{b^2-ca}.$

For how many real values of $a$ will $x^2+2ax+2008=0$ has two integer roots?

If $x, y$ are positive real numbers satisfying the system of equations $x^2+y\sqrt{xy}=336, y^2+x\sqrt{xy}=112,$ then $x+y$ equals

What is the negative of the sum of the reciprocals of the roots of the equation $\frac{2003}{2004}x+1+\frac{1}{x}=0?$

Both roots of the quadratic equation $x^2-63x+k=0$ are prime numbers. The number of possible values of $k$ is

Let $a, b,$ and $c$ be real numbers such that $a-7b+8c=4$ and $8a+4b-c=7.$ Then $a^2-b^2+c^2$ is

The zeros of the function $f\left(x\right)=x^2-ax+2a$ are integers. What is the sum of all possible values of $a$?