If $a, b, c$ are real numbers such that $a+b+c=0$ and $a^2+b^2+c^2=1$,then $(3a+5b-8c)^2+(-8a+3b+5c)^2+(5a-8b+3c)^2$ is equal to

Let $S = \{x \in [-6, 3] \setminus \{-2, 2\} : \frac{|x+3|-1}{|x|-2} \geq 0\}$ and $T = \{x \in \mathbb{Z} : x^2 - 7|x| + 9 \leq 0\}$. Then the number of elements in $S \cap T$ is $....$

If $\alpha, \beta$ are the roots of $x^2-a(x-1)+b=0$,then the value of $\frac{1}{\alpha^2-a \alpha}+\frac{1}{\beta^2-a \beta}+\frac{2}{a+b}$ is:

If $x^2+p x+1$ is a factor of $a x^3+b x+c$,then

If $a, b, c \in \mathbb{R}$ are such that $4a + 2b + c > 0$ and $ax^2 + bx + c = 0$ has no real roots,then the value of $(c + a)(c + b)$ is

Let the transformed equation of $2x^4-8x^3+3x^2-1=0$ such that the term containing the cubic power of $x$ is absent be $2x^4+bx^2+cx+d=0$. Then $b=$