G Tewani Solutions for Chapter: Theory of Equations, Exercise 6: DPP

If $f\left(x\right)$ is a polynomial of degree four with the leading coefficient one satisfying $f\left(1\right)=1$, $f\left(2\right)=2$ and $f\left(3\right)=3$, then $\left[\frac{f\left(-1\right)+f\left(5\right)}{f\left(0\right)+f\left(4\right)}\right]$ (where $\left[·\right]$ represents the greatest integer function) is equal to

Let $P\left(x\right)=x^6-x^5-x^3-x^2-x$ $\&$ $Q\left(x\right)=x^4-x^3-x^2-1$. If $\alpha$, $\beta$, $\gamma$ and $\delta$ are the roots of $Q\left(x\right)=0$ then the value of $P\left(\alpha \right)+P\left(\beta \right)+P\left(\gamma \right)+P\left(\delta \right)$ is equal to

The line $y=mx+1$ touches the curve $y=-x^4+2x^2$ $+x$ at two points $P\left(x_1, y_1\right)$ and $Q\left(x_2, y_2\right)$. The value of $x_1^2+x_2^2+y_1^2+y_2^2$  is

If $a+b+c=24$, $a^2+b^2+c^2=210$ and $abc=440$, then the least value of $a-b-c$ is

If the roots of $x^4+q{x}^2+kx+225=0$ are in arithmetic progression, then the value of $q$ is

If $\alpha$, $\beta$ and $\gamma$ are the roots of the equation $9x^3-7x+6=0$, then the equation $x^3+A{x}^2+Bx+C=0$ has the roots $3\alpha +2$, $3\beta +$ $2$ and $3\gamma +2$, where

Let $m$ be a real number and assume that two of the three solutions of the cubic equation $x^3+3x^2-34x=m$ differ by $1$. Then, the possible value(s) of $m$ is/are

Let $f\left(x\right)=x^3+x+1$. Let $p\left(x\right)$ be a cubic polynomial such that the roots of $p\left(x\right)=0$ are the squares of the roots of $f\left(x\right)=0$. Then