CUET PG 2023 — Mathematics PYQ

The value $a=\cos \frac{2\pi }{7}+i\sin \frac{2\pi }{7}$ is the $7^{th}$ root of unity.

Key properties:

• $a^7=1$

• $1+a+a^2+a^3+a^4+a^5+a^6=0$ (Sum of roots)

2. Find the Sum of Roots ($\alpha +\beta$)

Given:

Adding them together:

Using the sum of roots property:

3. Find the Product of Roots ($\alpha \beta$)

Multiplying the terms:

Since $a^7=1$:

• $a^8=a^7\cdot a=a$

• $a^9=a^7\cdot a^2=a^2$

• $a^{10}=a^7\cdot a^3=a^3$

Substituting these back:

Rearranging the terms:

Since $a+a^2+a^3+a^4+a^5+a^6=-1$:

4. Form the Quadratic Equation

The general form of a quadratic equation with roots $\alpha$ and $\beta$ is:

Substitute the values:

Final Answer:

The equation whose roots are $\alpha$ and $\beta$ is: