If ${x^2} + px + 1$ is a factor of the expression $a{x^3} + bx + c$, then
${a^2} + {c^2} = - ab$
${a^2} - {c^2} = - ab$
${a^2} - {c^2} = ab$
None of these
Let $x, y, z$ be positive integers such that $HCF$ $(x, y, z)=1$ and $x^2+y^2=2 z^2$. Which of the following statements are true?
$I$. $4$ divides $x$ or $4$ divides $y$.
$II$. $3$ divides $x+y$ or $3$ divides $x-y$.
$III$. $5$ divides $z\left(x^2-y^2\right)$.
Consider the cubic equation $x^3+c x^2+b x+c=0$ where $a, b, c$ are real numbers. Which of the following statements is correct?
The number of real solutions of the equation $|{x^2} + 4x + 3| + 2x + 5 = 0 $are
If $a, b, c$ are real numbers such that $a+b+c=0$ and $a^2+b^2+c^2=1$, then $(3 a+5 b-8 c)^2+(-8 a+3 b+5 c)^2$ $+(5 a-8 b+3 c)^2$ is equal to
The number of solutions of $\frac{{\log 5 + \log ({x^2} + 1)}}{{\log (x - 2)}} = 2$ is