Algebra — CDS Elementary Mathematics PYQs

A real number $M$ is squared to give the value $N$. What is the minimum value of $(M+N)$?

If $\alpha$ and $\beta$ are the roots of the equation $x+a+b=\frac{abx}{ab+ax+bx}$, then what is $(\alpha \beta +\alpha +\beta )$ equal to?

What is ${\log }_{10}2000+{\log }_{10}400+4{\log }_{10}25+5{\log }_{10}20$ equal to?

If $\frac{{\log }_{10}(100001-4^x)}{5-x}=1$, then what is $x$ equal to?

What is $\frac{(a-b{)}^2}{(b-c)(c-a)}+\frac{(b-c{)}^2}{(c-a)(a-b)}+\frac{(c-a{)}^2}{(a-b)(b-c)}-3$ equal to, where $a\ne b\ne c$?

If $x^2+y^2+z^2=3$, where $x$, $y$ and $z$ are integers, then how many values can $(xy+yz+zx)$ have?

If $x$, $y$, $z$ are real numbers such that $x+y+z=10$ and $xy+yz+zx=18$, then what is the value of $x^3+y^3+z^3-3xyz$?

What is $\sqrt{17-4\sqrt{15}}+\sqrt{8-2\sqrt{15}}$ equal to?

What is the maximum value of the sum of the numbers 36, 33, 30, 27, 24, ...?

If $2s=a+b+c$, then what is $s(s-a)(s-b)(s-c)\left[\frac{1}{s-a}+\frac{1}{s-b}+\frac{1}{s-c}-\frac{1}{s}\right]$ equal to?