Quadratic Equations — Class 12 Mathematics PYQs

If $-\sqrt{2}$ and $\sqrt{3}$ are roots of the equation $a_0+a_{1}x+a_2{x}^2+a_3{x}^3+x^4=0$ where $a_0,a_1,a_2,a_3$ are integers, then which one of the following is correct?

If $a,b$ and $c$ ($a>0,c>0$) are in GP, then consider the following in respect of the equation $a{x}^2+bx+c=0$:

1. The equation has imaginary roots.
2. The ratio of the roots of the equation is $1:\omega$ where $\omega$ is a cube root of unity.
3. The product of roots of the equation is $\left(\frac{b^2}{a^2}\right)$.

Which of the statements given above are correct?

If $\alpha$ and $\beta$ are the distinct roots of equation $x^2-x+1=0$, then what is the value of $\left|\frac{{\alpha }^{100}+{\beta }^{100}}{{\alpha }^{100}-{\beta }^{100}}\right|$?

If $2-i\sqrt{3}$ where $i=\sqrt{-1}$ is a root of the equation $x^2+ax+b=0$, then what is the value of $(a+b)$?

For how many integral values of $k$, the equation $x^2-4x+k=0$, where $k$ is an integer has real roots and both of them lie in the interval $(0,5)$?

$\alpha$ and $\beta$ are distinct real roots of the quadratic equation $x^2+ax+b=0$. Which of the following statements is/are sufficient to find $\alpha$?

1. $\alpha +\beta =0,{\alpha }^2+{\beta }^2=2$
2. $\alpha {\beta }^2=-1,a=0$

Select the correct answer using the code given below :

Consider the following for the next two (02) items that follow :

Consider the equation $(1-x{)}^4+(5-x{)}^4=82$.

What is the number of real roots of the equation ?

Consider the following for the next two (02) items that follow :

Consider the equation $(1-x{)}^4+(5-x{)}^4=82$.

What is the sum of all the roots of the equation ?

Consider the following for the next two (02) items that follow :

A quadratic equation is given by $(a+b)x^2-(a+b+c)x+k=0$, where a, b, c are real.

If $k=\frac{c}{2}$, $(c\ne 0)$, then the roots of the equation are :

Consider the following for the next two (02) items that follow :

A quadratic equation is given by $(a+b)x^2-(a+b+c)x+k=0$, where a, b, c are real.

If $k=c$, then the roots of the equation are :