Question 13

MATRIX MATCHHARD

Match each entry in List-I to the correct entry in List-II and choose the correct option.

List - I

If $\alpha$ and $\beta$ are the distinct roots of the equation $x^2 + x + 1 = 0$ , then the quadratic equation with roots $\frac{1}{(\alpha+1)^{2026}}$ and $\frac{1}{(\beta+1)^{2026}}$ is

If $\alpha$ and $\beta$ are the distinct roots of the equation $x^2 + x + 1 = 0$ , then the quadratic equation with roots $\frac{1}{(\alpha+1)^{2027}}$ and $\frac{1}{(\beta+1)^{2027}}$ is

If $\gamma$ and $\delta$ are the distinct roots of the equation $x^2 - x + 1 = 0$ , then the value of $\frac{1}{(\gamma-1)^{2026}} + \frac{1}{(\delta-1)^{2026}}$ is

If $p$ and $r$ are the distinct roots of the equation $x^2 + x - 1 = 0$ , then the value of $\frac{1}{(p+1)^3} + \frac{1}{(r+1)^3}$ is

List-II

$x^2 + x + 1 = 0$

$x^2 - x + 1 = 0$

$x^2 + x - 1 = 0$

$-1$

$-4$

(A)

(P) → (1), (Q) → (2), (R) → (5), (S) → (4)

(B)

(P) → (3), (Q) → (1), (R) → (4), (S) → (5)

(C)

(P) → (1), (Q) → (2), (R) → (4), (S) → (5)

(D)

(P) → (2), (Q) → (3), (R) → (5), (S) → (4)

Detailed Solution

For (P) and (Q), roots of $x^2 + x + 1 = 0$ are $\omega$ and $\omega^2$ .

Since $\alpha + 1 = -\alpha^2$ and $\alpha^3 = 1$ :

(P) New roots are $\frac{1}{(-\alpha^2)^{2026}} = \frac{1}{\alpha^{4052}}$ .

Since $4052 \equiv 2 \pmod 3$ , roots are $\frac{1}{\alpha^2} = \alpha$ and $\frac{1}{\beta^2} = \beta$ .

The equation remains $x^2 + x + 1 = 0$ . Thus (P) → (1).

(Q) New roots are $\frac{1}{(-\alpha^2)^{2027}} = \frac{-1}{\alpha^{4054}} = \frac{-1}{\alpha} = -\alpha^2$ . The roots are $-\omega^2$ and $-\omega$ . Sum $= 1$ , Product $= 1$ .

Equation is $x^2 - x + 1 = 0$ . Thus (Q) → (2).

(R) Roots of $x^2 - x + 1 = 0$ are $-\omega, -\omega^2$ . Let $\gamma = -\omega$ . Then $\gamma - 1 = -\omega - 1 = \omega^2$ .

Value $= \frac{1}{(\omega^2)^{2026}} + \frac{1}{(\omega)^{2026}} = \frac{1}{\omega^2} + \frac{1}{\omega} = \omega + \omega^2 = -1$ . Thus (R) → (4).

(S) Roots of $x^2 + x - 1 = 0$ satisfy $p+1 = \frac{1}{p}$ .

Value $= p^3 + r^3 = (p+r)^3 - 3pr(p+r) = (-1)^3 - 3(-1)(-1) = -1 - 3 = -4$ . Thus (S) → (5).

Matching sequence: (P) → (1), (Q) → (2), (R) → (4), (S) → (5).

Correct Option: C

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