Q 1 :

A value of $b$ for which the equations

$x^2+bx-1=0$

$x^2+x+b=0$

have one root in common is                          [2011]

• $-\sqrt{2}$

   

• $-i\sqrt{3}$

   

• $i\sqrt{5}$

   

• $\sqrt{2}$

   

Q 2 :

For all $\text{'}x\text{'},x^2+2ax+10-3a>0,$ then the interval in which $\text{'}a\text{'}$ lies is                   [2004]

• $a<-5$

   

• $-5<a<2$

   

• $a>5$

   

• $2<a<5$

   

Q 3 :

Let $f\left(x\right)=x^4+a{x}^3+b{x}^2+c$ be a polynomial with real coefficients such that $f\left(1\right)=-9$. Suppose that $i\sqrt{3}$ is a root of the equation $4x^3+3a{x}^2+2bx=0$ where $i=\sqrt{-1}$. If ${\alpha }_1,{\alpha }_2,{\alpha }_3$ and ${\alpha }_4$ are all the roots of the equation $f\left(x\right)=0$, then $|{\alpha }_1{|}^2+|{\alpha }_2{|}^2+|{\alpha }_3{|}^2+|{\alpha }_4{|}^2$ is equal to _______.            [2024]

Q 4 :

Let $S$ be the set of all non-zero real numbers $\alpha$ such that the quadratic equation $\alpha x^2-x+\alpha =0$ has two distinct real roots $x_1$ and $x_2$ satisfying the inequality $|x_1-x_2|$$<1$. Which of the following intervals is(are) $a$ subset(s) of $S$?                [2015]

• $(-\frac{1}{2},-\frac{1}{\sqrt{5}})$

   

• $(-\frac{1}{\sqrt{5}},0)$

   

• $(0,\frac{1}{\sqrt{5}})$

   

• $(\frac{1}{\sqrt{5}},\frac{1}{2})$