Embibe Experts Solutions for Exercise 5: Assignment

If $a$ and $b$ are the non-zero roots of equation $x^2+ax+b=0,$ then $(a+b)$ is equal to

$\left({\alpha }_1, {\alpha }_2\right), \left({\alpha }_2, {\alpha }_3\right)$ and $\left({\alpha }_3, {\alpha }_1\right)$ are respectively the roots of $x^2-2ax+2=0,x^2-2bx+3=0$ and $x^2-2cx+6=0.$ If ${\alpha }_1, {\alpha }_2, {\alpha }_3\in R^{+},$ then the value of $a+b+c$ is equal to

The number of irrational roots of the equation $\frac{4x}{x^2+x+3}+\frac{5x}{x^2-5x+3}=-\frac{3}{2}$ is $\underline{ }$

For all $x\in (-2, 3)$ number of solutions of the equation ${\left(x^2+x+1\right)}^2+1=\left(x^2+x+1\right)\left(x^2-x-5\right)$ is $\underline{ }$

The value of $m$ for which one of the roots of $x^2-3x+2m=0$ is double of one of the roots of $x^2-x+m=0$ is $\underline{ }$

If $\alpha , \beta , \gamma$ are such that $\Sigma \alpha =2;\Sigma {\alpha }^2=6;\Sigma {\alpha }^3=8,$ then ${\alpha }^4+{\beta }^4+{\gamma }^4$is equal to $\underline{ }$

The number of integral values of $a$ for which the quadratic equation $(x+a)(x+1991)+1=0$ has integral roots $\underline{ }$

If $x=2+\sqrt{3}$ then value of expression $x^4-8x^3+18x^2-8x+2$ is $\underline{ }$