Let
$$ \alpha = \left( 1 - 2\cos\left(\frac{\pi}{11}\right) \right) \left( 1 - 2\cos\left(\frac{3\pi}{11}\right) \right) \left( 1 - 2\cos\left(\frac{9\pi}{11}\right) \right) \left( 1 - 2\cos\left(\frac{27\pi}{11}\right) \right) \left( 1 - 2\cos\left(\frac{81\pi}{11}\right) \right). $$
Then the value of $5 - \alpha^2$ is ______________.
Match each entry in List-I to the correct entry in List-II and choose the correct option.
| List-I | List-II |
|---|---|
| (P) 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 | (1) $x^2 + x + 1 = 0$ |
| (Q) 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 | (2) $x^2 - x + 1 = 0$ |
| (R) 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 | (3) $x^2 + x - 1 = 0$ |
| (S) 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 | (4) $-1$ |
| (5) $-4$ |
| List-I | List-II |
|---|---|
|
(P) The number of elements in the set $$\left\{x \in [-\pi,\pi] : \sin^6 x + \cos^4 x = 1 \right\}$$ |
(1) is 1 |
|
(Q) The number of elements in the set $$\left\{x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] : \sin^2 x + \cos^6 x = 1 \right\}$$ |
(2) is 2 |
|
(R) The number of elements in the set $$\left\{x \in [-\pi,\pi] : \cos^2\left(\frac{x}{2}\right) - \sin^2 x = \frac{1}{2} \right\}$$ |
(3) is 3 |
|
(S) The number of elements in the set $$\left\{x \in [-2\pi,2\pi] : 6\sin^2\left(\frac{x}{2}\right) - \cos 3x = 3 \right\}$$ |
(4) is 4 (5) is 5 |
For real numbers $\alpha$, $\beta$, $\gamma$, $\delta$ and $\mu$, consider the matrix
$$ M = \begin{bmatrix} \alpha & \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \beta & \frac{1}{\sqrt{3}} \\ \gamma & \delta & \mu \end{bmatrix}. $$
Suppose that $MM^{T} = I$, where $M^{T}$ is the transpose of the matrix $M$, and $I$ is the $3 \times 3$ identity matrix. Let
$$ \vec{u} = \alpha\,\hat{i} + \frac{1}{\sqrt{3}}\hat{j} + \gamma\,\hat{k}, \qquad \vec{v} = \frac{1}{\sqrt{2}}\hat{i} + \beta\hat{j} + \delta\hat{k} \qquad \text{and} \qquad \vec{w} = -\frac{1}{\sqrt{2}}\hat{i} + \frac{1}{\sqrt{3}}\hat{j} + \mu\hat{k}. $$
Match each entry in List-I to the correct entry in List-II and choose the correct option.
| List-I | List-II |
|---|---|
| (P) The value of $$\gamma^2 + \delta^2$$ is | (1) 0 |
| (Q) If $$x\vec{u} + y\vec{v} + z\vec{w} = \hat{j}$$ for some real numbers $x$, $y$ and $z$, then the value of $x$ is | (2) 1 |
| (R) The value of $$\left|\vec{u} \cdot (\vec{v} \times \vec{w})\right|$$ is | (3) $$\frac{1}{\sqrt{2}}$$ |
| (S) The value of $$\left|\vec{u} \times (\vec{v} \times \vec{w})\right|$$ is | (4) $$\frac{1}{\sqrt{3}}$$ |
| (5) $$\frac{5}{6}$$ |
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