Paragraph: Let $S$ be the circle in the $xy$ – plane defined by the equation $x^2 + y^2 = 4.$

Question : Let $P$ be a point on the circle $S$ with both coordinates being positive. Let the tangent to $S$ at $P$ intersect the coordinate axes at the points $M$ and $MN$ Then, the mid-point of the line segment must lie on the curve

Let \(\mathbb{N}\) denote the set of all natural numbers, and \(\mathbb{Z}\) denote the set of all integers. Consider the functions \(f: \mathbb{N} \rightarrow \mathbb{Z}\) and \(g: \mathbb{Z} \rightarrow \mathbb{N}\) defined by
\(f(n)= \begin{cases}(n+1) / 2 & \text { if } n \text { is odd } \\ (4-n) / 2 & \text { if } n \text { is even }\end{cases}\) And
\(g(n)=\left\{\begin{array}{cc}3+2 n & \text { if } n \geq 0 \\ -2 n & \text { if } n <0\end{array}\right.\)
Define \((g \circ f)(n)=g(f(n))\) for all \(n \in \mathbb{N}\), and \((f \circ g)(n)=f(g(n))\) for all \(n \in \mathbb{Z}\).
Then which of the following statements is (are) TRUE?

For $\mathrm{x}\mathrm{ }\mathrm{ }\mathrm{ϵ}\mathrm{ }\mathrm{ }\left(0,\mathrm{ }\mathrm{\pi }\right),$ the equation $\sin \mathrm{x}+2\sin 2\mathrm{x}-\sin 3\mathrm{x}=3$ has

Let \(L_1\) be the line of intersection of the planes given by the equation- \(2 x+3 y+z=4 \text { and } x+2 y+z=5 .\)
Let \(L_2\) be the line passing through the point \(P(2,-1,3)\) and parallel to \(L_1\). Let \(M\) denote the plane given by the equation- \(2 x+y-2 z=6\)
Suppose that the line \(L_2\) meets the plane \(M\) at the point \(Q\). Let \(R\) be the foot of the perpendicular drawn from \(P\) to the plane \(M\).
Then which of the following statements is (are) TRUE?

If the sum of first \(n\) terms of an \(\mathrm{AP}\) is \(\mathrm{cn}^2\), then the sum of squares of these \(n\) terms is

The function \(f:[0,3] \rightarrow[1,29]\), defined by \(f(x)=2 x^{3}-15 x^{2}+36 x+1\), is

Let $ABC$ be the triangle with $AB=1,AC=3$ and $\angle BAC=\frac{\pi }{2}$. If a circle of radius $r>0$ touches the sides $AB,AC$ and also touches internally the circumcircle of the triangle $ABC$, then the value of $r$ is ______. (round-off the value to two decimal place)

Paragraph :
The capacitor of capacitance \(C\) can be charged (with the help of a resistance \(R\) ) by a voltage source \(V\), by closing switch \(S_1\) while keeping switch \(S_2\) open. The capacitor can be connected in series with an inductor \(L\) by closing switch \(S_2\) and opening \(S_1\).

Question :
After the capacitor gets fully charged, \(S_1\) is opened and \(S_2\) is closed so that the inductor is connected in series with the capacitor. Then,

A bag contains \(N\) balls out of which 3 balls are white, 6 balls are green, and the remaining balls are blue. Assume that the balls are identical otherwise. Three balls are drawn randomly one after the other without replacement. For \(i=1,2,3\), let \(W_i, G_i\), and \(B_i\) denote the events that the ball drawn in the \(i^{\text {th }}\) draw is a white ball, green ball, and blue ball, respectively. If the probability \(P\left(W_1 \cap G_2 \cap B_3\right)=\frac{2}{5 N}\) and the conditional probability \(P\left(B_3 \mid W_1 \cap G_2\right)=\frac{2}{9}\), then \(N\) equals ________.

The value of \(n\) in the molecular formula \(\mathrm{Be}_n \mathrm{Al}_2 \mathrm{Si}_6 \mathrm{O}_{18}\) is

Let \(P=\{\theta: \sin \theta-\cos \theta=\sqrt{2} \cos \theta\}\) and \(Q=\{\theta: \sin \theta+\cos \theta=\sqrt{2} \sin \theta\}\) be two sets. Then,

For an ideal gas, consider only P-V work in going from an initial state \(\mathrm{X}\) to the final state \(\mathrm{Z}\). The final state \(\mathrm{Z}\) can be reached by either of the two paths shown in the figure. Which of the following choice(s) is (are) correct?
[Take \(\Delta \mathrm{S}\) as change in entropy and w as work done].

Paragraph:
When a particle is restricted to move along \(x\)-axis between \(x=0\) and \(x=a\), where \(a\) is of nanometer dimension, its energy can take only certain specific values. The allowed energies of the particle moving in such a restricted region, correspond to the formation of standing waves with nodes at its ends \(x=0\) and \(x=a\). The wavelength of this standing wave is related to the liner momentum \(p\) of the particle according to the de Broglie relation. The energy of the particle of mass \(m\) is related to its linear momentum as \(E=\frac{p^2}{2 m}\). Thus, the energy of the particle can be denoted by a quantum number \(n\) taking values \(1,2,3, \ldots(n=1\), called the ground state) corresponding to the number of loops in the standing wave.
Use the model described above to answer the following three questions for a particle moving in the line \(x=0\) to \(x=a\) [Take \(h=6.6 \times 10^{-34} \mathrm{Js}\) and \(e=1.6 \times 10^{-19} \mathrm{C}\) ]
Question:
The speed of the particle that can take discrete values is proportional to