MAH-CET 2026 — Mathematics PYQ

Step 1: Identify the roots of the given equation

The given quadratic equation is:

$$x^2+x+1=0$$

The roots of this specific equation are the imaginary cube roots of unity, commonly denoted as $\omega$ and ${\omega }^2$.

Therefore, we can let:

$$\alpha =\omega \text{and}\beta ={\omega }^2$$

Properties of cube roots of unity:

• ${\omega }^3=1$

• $1+\omega +{\omega }^2=0$

Step 2: Simplify the new roots (${\alpha }^{19}$ and ${\beta }^7$)

Now, we find the values of the new roots by reducing their powers using the property ${\omega }^3=1$.

• For the first new root:

  $${\alpha }^{19}={\omega }^{19}={\omega }^{18+1}=({\omega }^3{)}^6\cdot \omega =(1{)}^6\cdot \omega =\omega$$

• For the second new root:

  $${\beta }^7=({\omega }^2{)}^7={\omega }^{14}={\omega }^{12+2}=({\omega }^3{)}^4\cdot {\omega }^2=(1{)}^4\cdot {\omega }^2={\omega }^2$$

So, the new roots are simply $\omega$ and ${\omega }^2$, which are exactly the same as the original roots ($\alpha$ and $\beta$).

Step 3: Form the new quadratic equation

Since the new roots (${\alpha }^{19}=\omega$ and ${\beta }^7={\omega }^2$) are identical to the original roots, the required quadratic equation will remain unchanged.

The equation is:

$$x^2+x+1=0$$

Correct Answer:

(d) $x^2+x+1=0$