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

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) $$\frac{5}{6}$$

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

List-I	List-II
(P) The circle with centre $(1,2)$ and touching the straight line $$3x + 4y = 1$$ passes through	(1) the point $(1,1)$
(Q) The common tangent to the circle $$x^2 + y^2 = 2$$ and the parabola $$y^2 = 8x$$ with positive slope, passes through	(2) the point $(7,9)$
(R) Let $M$ be the end point of the latus rectum of the ellipse $$3x^2 + 4y^2 = 48$$ such that $M$ lies in the first quadrant. Then the normal to the ellipse drawn at $M$ passes through	(3) the point $(3,2)$
(S) Let $H$ be the hyperbola whose centre is at the origin, one of the foci is at $(5,0)$, and one directrix is $$5x + 16 = 0$$ Then $H$ passes through	(4) the point $(2,5)$
	(5) the point $(8, 3\sqrt{3})$

Consider a large disk of radius R and two smaller disks, each of radius r = R / 50, lying on its circumference, as shown in the figure. The smaller disks are initially in contact with each other, with an angular separation Δθ between their centers. They are made to roll without slipping in opposite directions, with constant angular velocities ω and 2ω while the large disk is held stationary. The time τ at which the smaller disks are again in contact is:
[Use sin(Δθ)=Δθ and ignore gravity.]